; TeX output 2008.06.14:12337o]src:129advalg.texDtqGcmr17ADuNV:ANCEDBALGEBRA'Wdsrc:130advalg.texXQ ff cmr12Prof./Dr.B.Pareigissrc:131advalg.texXQ cmr12WinrterSemester2001/02(v}- cmcsc10T3ableofContents&|src:1advalg.toc<1. TVensorProSductsandFreeMoSdules(3Hsrc:2advalg.toc1.1. MoSduleszf3Hsrc:3advalg.toc1.2. TVensorproSductsIJ&5Hsrc:4advalg.toc1.3. FVreemoSdulesb|6Hsrc:5advalg.toc1.4. TVensorproSductsIIFt8Hsrc:6advalg.toc1.5. BimoSdulespV9Hsrc:7advalg.toc1.6. Complexesandexactsequencesw12&|src:8advalg.toc<2. AlgebrasandCoalgebras)_15Hsrc:9advalg.toc2.1. Algebrass315Hsrc:10advalg.toc2.2. TVensoralgebrasO17Hsrc:11advalg.toc2.3. Symmetricalgebras;,19Hsrc:12advalg.toc2.4. ExterioralgebrasG21Hsrc:13advalg.toc2.5. Leftg cmmi12A-moSdulesPq23Hsrc:14advalg.toc2.6. Coalgebrasg(23Hsrc:15advalg.toc2.7. ComoSdulesgr26&|src:16advalg.toc<3. ProjectivreMoSdulesandGenerators30Hsrc:17advalg.toc3.1. ProSductsandcoproducts30Hsrc:18advalg.toc3.2. ProjectivremoSdules=34Hsrc:19advalg.toc3.3. Dualbasisj|236Hsrc:20advalg.toc3.4. Generatorsg39&|src:21advalg.toc<4. CategoriesandFVunctors,%@40Hsrc:22advalg.toc4.1. Categoriesjb40Hsrc:23advalg.toc4.2. FVunctorss\42Hsrc:24advalg.toc4.3. NaturalTVransformations &43&|src:25advalg.toc<5. RepresenrtableandAdjointFVunctors,theYonedaLemma46*7o]+o cmr92oAdv|rancedTAlgebra{P9areigis[o]Hsrc:26advalg.toc5.1. Represenrtablefunctors*46Hsrc:27advalg.toc5.2. TheYVonedaLemma849Hsrc:28advalg.toc5.3. AdjoinrtfunctorsK51Hsrc:29advalg.toc5.4. Univrersalproblems=52&|src:30advalg.toc<6. LimitsandColimits,ProSductsandEqualizers_55Hsrc:31advalg.toc6.1. Limitsofdiagrams?f55Hsrc:32advalg.toc6.2. Colimitsofdiagrams5x57Hsrc:33advalg.toc6.3. CompletenessZ(58Hsrc:34advalg.toc6.4. Adjoinrtfunctorsandlimits(59&|src:35advalg.toc<7. TheMoritaTheorems7̰60&|src:36advalg.toc<8. SimpleandSemisimpleringsandMoSdulesύR66Hsrc:37advalg.toc8.1. SimpleandSemisimplerings R$66Hsrc:38advalg.toc8.2. InjectivreMoSdulesE/67Hsrc:39advalg.toc8.3. SimpleandSemisimpleMoSdules $70Hsrc:40advalg.toc8.4. NoSetherianModules8!P73&|src:41advalg.toc<9. RadicalandSoScleM=76&|src:42advalg.toc<10. LoScalizationd81Hsrc:43advalg.toc10.1. LoScalringsai`81Hsrc:44advalg.toc10.2. LoScalization[81&|src:45advalg.toc<11. MonoidalCategories987&|src:46advalg.toc<12. BialgebrasandHopfAlgebras 92Hsrc:47advalg.toc12.1. Bialgebrasd92Hsrc:48advalg.toc12.2. HopfAlgebrasP94&|src:49advalg.toc<13. QuicrkiesinAdvXancedAlgebraF101l7o]?T:ensorTproAductsandfreemodules3[o]sW͹1.[QTensorProductsandFreeModules1.1.,N cmbx12Mo`dules.src:189advalg.tex!ADe nitionl1.1.Hsrc:190advalg.texLetuRbSevaring(alwraysvassociativeuwithvunitelemenrt).A9-@ cmti12leftVmRJ-moffdule2cmmi8R MisanAbSeliangroupM+(withcompositionwrittenasaddition)togetherwithanoperation R!", cmsy10M63UR(rr;m)7!rSm2Msrc:195advalg.texsucrhthatK(1)%src:197advalg.tex(rSs)mUR=r(sm),(2)%src:198advalg.tex(r6+s)mUR=rSm+sm,(3)%src:199advalg.texrS(m+m2 K cmsy809)UR=rm+rm209,(4)%src:200advalg.tex1mUR=msrc:202advalg.texforallrr;sUR2RJ,m;m20#2M@.src:204advalg.texIfRisa eldthena(left)RJ-moSduleisa(calleda)vrectorspaceoverRJ.src:207advalg.texA3,homomorphismuofleftuRJ-moffdules3?orsimplyanR-moffduleuhomomorphismf:R tM+O!2RNisahomomorphismofgroupswithfG(rSm)UR=rfG(m).src:211advalg.texR2ight35RJ-moffdules꨹andhomomorphismsofrightRJ-moffdulesarede nedanalogouslyV.src:214advalg.texWVede ne &THom?:ݟRF.p(:M;:N@)UR:=ffQ:R HM6!RNjf2isahomomorphismofleftRJ-moSdulesg:src:217advalg.texSimilarlyj?HomȟR"[(M:;N:)j?denotesj@thesetofhomomorphismsofrighrtRJ-moSdulesMR ]ӹandNR.src:220advalg.texAnRJ-moSdulehomomorphismfQ:URR HM6!RN+isWcamonomorphismiff2isinjectivre,Wcanepimorphismiff2issurjectivre,Wcanisomorphismiff2isbijectivre,WcanendomorphismifM6=URN@,Wcanautomorphismiff2isanendomorphismandanisomorphism.!AProblemٶ1.1.$src:231advalg.texLetYRs7bSearingandMйbeanAbeliangroup.ShorwYthatthereisaone-to-onecorrespSondencebetrweenmapsfQ:URRW>]M6!MthatmakeMintoaleftRJ-moSduleandringhomomorphisms(alwrayspreservingtheunitelement)gË:URRn'!{End):(M@).!BLemma1.2.src:239advalg.texHomyRm(M;N@)35isanA2bffeliangroupby(f+gn9)(m)UR:=fG(m)+gn9(m).Prffoof.#Rsrc:244advalg.texSince|GN,isanAbSelian|HgroupthesetofmapsMap(M;N@)isalsoanAbSeliangroup.ThesetofgrouphomomorphismsHom\m(M;N@)isasubgroupofMap5(M;N@)(observrethatthisholdsZonlyZforAbSeliangroups).WVeshorwthatHomR#Ǫ(M;N@)isasubgroupofHom(M;N@).WVemrustonlyshowthatfl $ g ,isanRJ-moSdulehomomorphismiffandg ,are.OObrviouslyf{O3Pg!is_agrouphomomorphism.FVurthermorewrehave(f{O3Pgn9)(rSm)=fG(rm)3Pgn9(rm)=rSfG(m)rgn9(m)UR=r(fG(m)gn9(m))UR=r(fgn9)(m):O% msam10!BProblem1.2.src:255advalg.texLetfQ:URM6!N+bSeanRJ-modulehomomorphism.K(1)%src:257advalg.texfisEanisomorphismifandonlyEif(i 8)thereexistsanRJ-moSdulehomomorphism%gË:URN6!M+sucrhthat TfGgË=URid N#and1gn9fQ=URid M: %src:260advalg.texFVurthermoregXisuniquelydeterminedbryfG.(2)%src:261advalg.texThefollorwingareequivXalent:)((a)=Ѭsrc:263advalg.texf2isamonomorphism,)' (b)=Ѭsrc:264advalg.texforallRJ-moSdulesPnandallhomomorphismsgn9;hUR:P!eMfGgË=URfh=)gË=h;7o]4oAdv|rancedTAlgebra{P9areigis[o]*uD(c)=Ѭsrc:267advalg.texforallRJ-moSdulesPnthehomomorphismofAbeliangroupsbHom{xR (PS;fG)UR:?HomR&(P;M@)UR3gË7!fGg2Hom۟R"n(PS;N@)=Ѭsrc:271advalg.texisamonomorphism.(3)%src:273advalg.texThefollorwingareequivXalent:)((a)=Ѭsrc:275advalg.texf2isanepimorphism,)' (b)=Ѭsrc:276advalg.texforallRJ-moSdulesPnandallhomomorphismsgn9;hUR:N6!Pgn9fQ=URhf=)gË=h;*uD(c)=Ѭsrc:279advalg.texforallRJ-moSdulesPnthehomomorphismofAbeliangroupsaEHomzΟR a(f;Pƹ)UR:?HomR&(N;P)UR3gË7!gn9fQ2Hom۟R"n(M;P)=Ѭsrc:283advalg.texisamonomorphism._؍Remark1.3.3src:289advalg.texEacrhAbSeliangroupisa( msbm10Z-moSduleinauniquewrayV.WlEachhomomorphismofAbSeliangroupsisaZ-modulehomomorphism._׍Prffoof.#Rsrc:295advalg.texByOexercise1.1wreOhaveOto ndauniqueringhomomorphismg":ZM!End.(M@).This} holds}moregenerallyV.IfS/isaringthenthereisauniqueringhomomorphismg:NZ!S׹. SinceӉaӊringhomomorphismmrustpreservetheunitwehavegn9(1)t=u1. De negn9(n)UR:=1Z+:::U+1F@(n-times)FAfornUR0andF@gn9(n):=(1Z+:::U+1)(n-times)FAfornUR>0.Thenitstissseasytocrheckstthatg᭹isaringhomomorphismanditisobrviouslyunique.CThismeansthatfM[JisagZ-moSdulebrynmUR=mH+:::0+Imf(n-times)fornUR0fand(n)mUR=(mH+:::1+m)(n-times)fornUR>0.src:308advalg.texIffȳ:M$![Npisahomomorphismof(AbSelian)groupsthenfG(nm)=f(m"i+:::r+m)=fG(m) j+:::u+f(m)L7=L8nf(m){fornL8L70{andf((n)m)L8=L7f((m j+ k:::u+m))L7=L8(f(m)+:::uH+fG(m))UR=(n)f(m)fornUR>0.8Hencef2isaZ-moSdulehomomorphism.GProblem1.3.(1)vsrc:319advalg.texLetRbSearing.8ThenR ;RisaleftRJ-moSdule.U(2)%src:322advalg.texLet1MqbSeaAbelian1groupandEnd=(M@)betheendomorphismringof1M@. 0ThenM%〹isanEndg(M@)-moSdule.(3)%src:325advalg.texf(n1;n0);(n0;n1)g꨹isageneratingsetfortheZ-moSduleZ=(2)Z=(3).(4)%src:329advalg.texf(n1;n1)g꨹isageneratingsetfortheZ-moSduleZ=(2)Z=(3).(5)%src:332advalg.tex)ppmsbm8ZTZ=(n)hasnobasisasamoSdule,i.e.8thismoduleisnotfree.(6)%src:335advalg.texLetV¹=UR"u cmex10L*1 U_i|{Ycmr8=0 Kܞbi bSeacounrtablyin nitedimensionalvectorspaceoverthe eldKܞ.%Letp;qn9;a;bUR2Hom(V;Vp)bSede nedbry>6̳rp(bidڹ)$:=URb2i;rqn9(bidڹ)$:=URb2i+1Z;3Ara(bidڹ)$:=URz( bi=2 ;0if;Fi꨹isevren,andɍ 0;0if;Fi꨹isoSdd.$rb(bidڹ)$:=URz( bi1=2^;;tifFsi꨹isoSdd,andɍ 0;;tifFsi꨹isevren.=㉍%src:350advalg.texShorwpa+qn9bUR=id Vv,ap=bqË=id ,aqË=bp=0.1src:352advalg.texShorwforRn=UREndbK ӹ(Vp)thatR ;R=URRJaRb꨹andRR H=URpRqn9Rholds.(7)%src:355advalg.texAref(0;:::ʜ;a;:::;0)ja<2=P2?*uD(c)=Ѭsrc:365advalg.texP1jQPUR԰n:=P2Q꨹=)P1PV԰.>=P2?U (10)%src:368advalg.texZ=(2)Z=(6)Z=(6):::Pʚ԰ス= uCZ=(6)Z=(6)Z=(6):::uH. (11)%src:371advalg.texZ=(2)Z=(4)Z=(4):::ʚ6P԰= Z=(4)Z=(4)Z=(4):::uH. (12)%src:374advalg.texFindtrwoAbSeliangroupsP`andQ,sucrhthatP`isisomorphictoasubgroupofQ%〹andQisisomorphictoasubgroupofPnandP6P԰= Q.ҍ1.2.Tensorpro`ductsI.src:384advalg.texэDe nitionxandRemark1.4.nsrc:385advalg.texLetOMR andRN3bSeRJ-modules,yandletAbeanAbeliangroup.8AmapfQ:URMN6!AiscalledRJ-bilineffarif (1)%src:389advalg.texfG(m+m209;n)UR=f(m;n)+f(m209;n);(2)%src:390advalg.texfG(m;n+n209)UR=f(m;n)+f(m;n209);(3)%src:391advalg.texfG(mrr;n)UR=f(m;rSn)src:393advalg.texforallr2URRJ;m;m20#2M;n;n20#2N@.src:395advalg.texLetBilݟRp(M;N@;A)denotethesetofallRJ-bilinearmapsfQ:URMN6!A.src:398advalg.texBil5Rȹ(M;N@;A)isanAbSeliangroupwith(f+gn9)(m;n)UR:=fG(m;n)+gn9(m;n).ҍDe nitionqq1.5.Ksrc:403advalg.texLetMR яandRNbSeRJ-modules.4AnAbeliangroupMѪ R YNtogetherwithanRJ-bilinearmap } UR:MN63(m;n)7!m n2M R ;N|src:407advalg.texis calledatensorމprffoductof Mand NoverRWif foreacrhAbSeliangroupAandforeacrhRJ-bilineary5mapf:GML Nd!\Ay4thereexistsauniquegrouphomomorphismg.:ML R IN!nA꨹sucrhthatthediagramIkč{ȀMN{M R ;N0{fd UO line10-` H`-ftׁ @t @t @t @@>@@>RHAF"Ǡ*FfeyTǠ?  +gsrc:414advalg.texcommrutes.0`The)elements(ofMw R j&N arecalledtensors,Btheelemenrtsoftheformmv vnarecalleddeffcomposable35tensors.src:418advalg.texWarning:If&0yrou&1wantto&1de neahomomorphismf:M R N!ϿAwithatensorproSductasdomainyroumustde neitbygivinganRJ-bilinearmapde nedonMN@.эProp`ositiont'1.6.src:425advalg.texA)tensor)prffoduct(M} R ,N; )de nedbyMR MandR NNjisuniqueuptoaunique35isomorphism.ҍPrffoof.#Rsrc:429advalg.texLet(M R ;N; )and(MR ;N;)bSetensorproducts.8ThenI;ōYMN`P oҁ oׁ o܁ o o o o o dždžH,بDׁ ΨD ĨD D ağ>ağ> H` ׁ @ @ @ @_>@_>RH,-Qdҁ HQdׁ HQd܁ HQd H#Qd H-Qd H7Qd HAQd HEddžHEddžj33^M R ;N33MR ;Ne32fdϠά-:kh3333432fdAά-:k33$M R ;N33IMR ;N?32fdϠά-:?haUsrc:443advalg.teximpliesko=URh21 \|.xDEUsrc:446advalg.texBecauseofthisfactwrewillhenceforthtalkabSoutthetensorproductofM+andNorverRJ.Prop`osition1.7._-src:450advalg.tex(RulesofcomputationinatensorproSduct)r7Lffet(M> R_N; )bethetensorprffoduct.fiThen35wehaveforallr2URRJ,m;m20#2M@,n;n20#2N7h7o]6oAdv|rancedTAlgebra{P9areigis[o](1)%src:454advalg.texM R ;N6=URfP imi nij35mi,2M;ni2N@g;(2)%src:456advalg.tex(m+m209) nUR=m n+m20x n;(3)%src:457advalg.texm (n+n209)UR=m n+m n20;(4)%src:458advalg.texmr߃ nUR=m rSn%d(observein%epffarticular,('that UR:MN6!QM NfHis%dnotinjeffctive%in35generffal),(5)%src:461advalg.texifb=fz:zMN_ -!FAisb>anRJ-bilineffarmapandg:zM R ~N_ -!FAistheinducffed%homomorphism,35thenNagn9(m n)UR=fG(m;n):m⍍Prffoof.#Rsrc:468advalg.tex(1)LetB:=xhm nixM R ~N@~denotethesubgroupofM R ~N@~generatedbrythedecompSosabletensorsm n.5Letj%:URBX !_7M R N"xbetheemrbeddinghomomorphism.5WVegetaninducedmap 20#:URMN6!B.8ThefollorwingdiagramUe獍YlMNYYBd{fdά-;̼ 20{󎎍{ aM R ;NV{fdά-"j33YB33 aM R ;NV32fdά-^bjH披V 20ԟׁ @ԟ @ʞԟ @Ԟԟ @T>@T>RH#ŸǠ*Ffe#Ǡ?h(tjpH  4pׁ  󜔟 V>V> src:485advalg.texinduces auniquepwithpjl 20X*=p = 20˹since 20isRJ-bilinear.Becauseofjp =jGB 20#=UR =id M" X.;cmmi6RN, wregetjpUR=id M" X.RN*窹,ùhencetheemrbSeddingjfѹissurjectivreandthrustheidenrtityV.src:494advalg.tex(2)(m+m209) nUR= (m+m20;n)UR= (m;n)+ (m209;n)UR=m n+m20x n.src:497advalg.tex(3)and(4)analogouslyV.src:499advalg.tex(5)ispreciselythede nitionoftheinducedhomomorphism.Y3src:503advalg.texTVoconstructtensorproSducts,wreneedthenotionofafreemodule.!1.3.Freemo`dules.src:508advalg.texmDe nition21.8.6src:509advalg.texLetUXGebSeUasetandRo+bearing.zAnURJ-moduleRJXGetogetherwithUamap3,:3+X$>:!qhRJXisecalledafrffeeeR-module5AgeneratedbyeX(oranRJ-moffdule5@freelygeneratedbyX),if^ for^ evreryRJ-moSduleMandforevrerymapfy:yX؈!MthereexistsauniquehomomorphismofRJ-moSdulesgË:URRXF``!M+sucrhthatthediagramP;'bYxXbYrRJXӠD{fd$pά- gH`ҭf ׁ @ @ @ @Pt>@Pt>RH9_MUǠ*FfeԟǠ?  ;Tg洍src:517advalg.texcommrutes.src:519advalg.texAnRJ-moSduleFLisafrffee5RJ-module퇹ifthereisasetX andamapZ4:XKeB!xFMsucrhthatFLisfreely|generated|bryX.Sucha|setXm(oritsimage(X))iscalledafrffee͆generatingsetͅforFƹ.src:525advalg.texWarning:vIfFsyrouwanttode neahomomorphismg:NRJX\!UMWwithafreemoSduleasdomainyroushouldde neitbygivingamapfQ:URXF``!M@.Prop`ositionp:1.9.jsrc:530advalg.texA2frffeedRJ-modulec5:4X"!XRXde nedcoverdasetXisuniqueuptoaunique35isomorphismofRJ-moffdules.M97o]?T:ensorTproAductsandfreemodules7[o]Prffoof.#Rsrc:534advalg.texfollorwsfromthefollowingdiagramHvbYTX9ʌҁ ʌׁ ʌ܁ ʌ ʌ ʌ ʌ ʌ =dž=džH '20ׁ    l>l> H9'sׁ @s @s @s @,>@,>RH ,v20 ҁ H ׁ H ܁ H H" H, H6 H@ HE9 džHE9 džjfKjjRJXfK>RJX20/32fdPά-:GhfKfK1ܞ32fdY]ά-:kfK*RJXfKTdNRJX202U32fdPά-:>8hk~Prop`ositionp1.10.1src:547advalg.tex(RulesLofcomputationLinafreeRJ-moSdule))Lffet(:XgS!ORX~bffe)a(freeRJ-moffdule35overX.fiLetex5ٹ:=UR(x)2RJX$for35allxUR2X.fiThen35wehave(1)%src:552advalg.texwLeX=ңfTexRjv9x2X&:'eҤxR=(x)gvisagenerffatingsetofRJX,i.e.1neffachelementmң2RJX%is35alineffarcombinationmUR=P*n U_i=1 ASriedxi;of35theex .39(2)%src:556advalg.texwLeXWRJXچislineffarlyindependentandisinjeffctive,vi.e.ifP*0 U_x2X&$rxeHx5=0,thenwe%have358xUR2XFչ:rx9=0.Prffoof.#Rsrc:563advalg.tex(1)LetMOŹ:=hTexRjx2XiRJXߖbSethesubmodulegeneratedbrytheBex d.CThenthediagramFiэbYXbYrRJXD{fd$pά- gHyRJX=XM=CX0Gׁ @G @G @G @t>@t>RHǠ*Ffe紟Ǡ?=U80HrŸǠ*FfeǠ?9 Xtsrc:572advalg.texcommruteswithbSothmaps0andǹ.8Thus0UR=oandRJX=XM6=0andhenceRJXFչ=M@.src:575advalg.tex(2)TLetUP*n U_i=0! Vriedxi&=0Uandr0n6=0.Letj[͹:X}!iR8bSetheUmapgivrenbyjӹ(x0)=1;j(x)=0forallxUR6=x0.8=)?9gË:RJXF``!RwithKP5bYxXbYrRJXӠD{fd$pά- gHh\j ׁ @ @ @ @Pt>@Pt>RHjRUǠ*FfeԟǠ?  ;TgAsrc:580advalg.texcommrutativeZiand0=gn9(0)=g(P* n U_ i=0riedxi w)=P*7n U_7i=0 ridg(Texi ,)=P*7n U_7i=0 ridjӹ(xi)=r0.#Conrtradic-tion.8Hencethesecondstatemenrt.(Notation5~1.11.#src:585advalg.texSinceisinjectivrewewillidentifyXwithit'simageinRJXandwewillwriteP x2X!;rxHx꨹foranelemenrtPUx2X%&srx(x)UR2RJX.8ThecoSecienrtsrx 3areuniquelydetermined.}Prop`osition1.12.jsrc:592advalg.texLffetXbeaset.@-ThenthereexistsafrffeeRJ-moduleUR:XF@!RJXoverX.Prffoof.#Rsrc:597advalg.texObrviously]RJX:=f Þ:X!k-RJj]for^almostallx2X: (x)=0g]is^asubmoSduleofMap(XJg;RJ)whicrhisanR-moSdulebrycomponentwiseadditionandmrultiplication.MDe neUR:XF``!RJX+bry(x)(yn9):=xy .src:603advalg.texLet1fQ:URXF``!MbSeanarbitrary2map.2 Let h2RJX.2De negn9( ):=Px2X$ (x)fG(x):1ThenMgliswrellde ned,BbSecausewehave (x))6=0foronly nitelymanyx)2X.rIFVurthermoreg)is>an>RJ-moSdulehomomorphism:qrgn9( )V+Wsg( O)=rSP7 (x)WVfG(x)+sP O(x)fG(x)=P (rS (x)+s O(x))fG(x)UR=P(r 7+s O)(x)fG(x)UR=gn9(r 7+s O).src:613advalg.texFVurthermoreVSwrehavegn9T=SfS:g(x)T=PkyI{2X&E(x)(y)BfG(y)T=SPjxy h7fG(y)S=Tf(x):VSFVorC 2RJXwreߗhave =PTx2X' (x)(x)ߘsince (yn9)=PT (x)(x)(y). InߘorderߗtoshorwthatٙgGѹisuniquelydetermined٘bryfG,Uleth2HomeR$Y"(RJXJg;M@)bSegivrenwithh=fG.Thenh( )UR=h(P (x)(x))=P (x)h(x)=P (x)fG(x)=gn9( )hencehUR=gn9:R`)7o]8oAdv|rancedTAlgebra{P9areigis[o]Remarkg1.13.src:625advalg.texIfՈtheՉbaseringKisa eldthenaK-moSduleisavrectorspace.1EachvectorspaceV[-hasabasisXA(proSofbryZorn'slemma).*vM$ R IN$۹:=Z(MN@)=UZwhereZ(MN@)isafreeZ-moSduleorver>uM$N(thefreeAbSeliangroup)andU+isgeneratedbryx!%src:666advalg.tex(m+m209;n)(m;n)(m209;n)%src:667advalg.tex(m;m+n209)(m;n)(m;n209)%src:668advalg.tex(mrr;n)(m;rSn)x src:670advalg.texforallr2URRJ,m;m20#2M@,n;n20#2N@.8ConsiderJDHSMNHZ(MN@)|{fd"pά- HH+M R ;N݌{fdά- HH/M=URZ(MN@)=UH7wA`+ "LuP"L+P"LׁP"LtP"L+P"LP"LsP"L+P"LP"LrP"L+P"LP̟P̟qH l+ʬQtQQ+QrQQJ\QJ\sHkǠ*FfeǠ? QlgBsrc:682advalg.texLet} bSe}givren.ThenthereisauniqueUR2Hom(Z(M f̂N@);A)}sucrhthatUR= n9.Since} isRJ-bilinearTRwreget((mw+m209;n)w(m;n)(m209n))UR= n9(mw+m20;n)w n9(m;n) n9(m209;n)UR=0andrsimilarlys((m;n@M+@Ln209)(m;n)(m;n209))n=0rand((mrr;n)@M(m;rSn))n=0.?SowreXGgetXH(U@)ç=0. ThisimpliesXHthatthereisauniqueg12èHom=1(M{ R )N;A)sucrhthatgn9&=d;k(homomorphismtheorem).S)Let d<:=rf.S(Then isbilinearsince(m+m209) nd<=X(mX+m209;n)UR=ǹ((m+Xm209;n))=ǹ((m+Xm209;n)(m;n)(m209;n)+(m;n)+(m209;n))UR=ǹ((m;n)C+D(m209;n))UR= (m;n)D+D(m209;n)UR=m nD+m20| n. The othertrwo propSertiesareobtainedinananalogouswrayV.src:700advalg.texWVesharvetoshowthat(Mm R ON; )isatensorproSduct.DATheaborvediagramshowsthatforheacrhiAbSeliangroupAandforeacrhRJ-bilinearmap D:\ M z&Nz!AthereisagD2Homy(Mp R vN;A)sucrhthatg UR= n9.2TGivenh2Hom(Mp R vN;A)withh = n9.2TThenhloUR= n9.8Thisimpliesh=UR=gohencegË=h.yOProp`ositionandDe nition1.15.src:711advalg.texGiven35twohomomorphismsCvfQ2URHomٟR$l(M:;M@ 0:)35andgË2URHomٟR(:N;:N@ 0):Bsrc:714advalg.texThen35therffeisauniquehomomorphismYf R ;gË2URHom(M RN;M@ 0 RN@ 0) yI7o]?T:ensorTproAductsandfreemodules9[o]src:716advalg.texsuch35thatf R ;gn9(m n)UR=fG(m) g(n),35i.e.fithefollowingdiagrffamcommutesM%卒g?M@20N@20卒bM@20 R ;N@20n|32fdpά- W`0 {5xMN{1/M R ;Nڜ{fdά-`0 HڟǠ*Ffe/ Ǡ?`fgH qZǠ*Ffe Ǡ?`W f R ;g#Prffoof.#Rsrc:724advalg.tex (fgn9)isbilinear.5f:͚Notation1.16.src:728advalg.texWVeoftenwritef R ;N6:=URf R1N andM RgË:=UR1M . Rgn9.src:731advalg.texWVeharvethefollowingruleofcomputation:eYf R ;gË=UR(f RN@ 0)(M Rgn9)UR=(M@ 0 Rgn9)(f RN@)src:735advalg.texsincefgË=UR(fN@20)(Mgn9)UR=(M@20gn9)(fN@).͙1.5.Bimo`dules.src:741advalg.texDe nition41.17.#src:742advalg.texLet"RJ,S|bSe#ringsandletM bealeftRJ-module#andarighrtS׹-module.شM iscalledanRJ-S׹-bimoffduleif(rSm)su=ur(ms).qWVede neHomw9R -^S,V(:M:;:N:)u:=Hom>R"ѹ(:M;:N@)\HomySk(M:;N:).Remark1.18.src:749advalg.texLet MS (bSea righrtS׹-moduleand letRnMn!مMKbea map.MKisanRJ-S׹-bimoSduleifandonlyifs2(1)%src:752advalg.tex8r2URRn:(M63m7!rSm2M@)2Hom۟S!콹(M:;M:),(2)%src:754advalg.tex8rr;rS20w2URRJ;m2M6:(r6+rS20!ǹ)m=rSm+r20!m,(3)%src:755advalg.tex8rr;rS20w2URRJ;m2M6:(rSr20!ǹ)m=r(r20!m),(4)%src:756advalg.tex8mUR2M6:1m=m:LemmaR1.19.src:761advalg.texLffet R MS ,and SNT $be bimodules.Z\ThenR (M2 Sx0N@)T $isabimodule byrS(mZN n)UR:=rSm n35and(m n)tUR:=m nt.Prffoof.#Rsrc:767advalg.texClearlywrehavethat(r[P S%idI)(m n)UR=rSm n=rS(m n)isahomomorphism.Then(2)-(4)hold.2ThrusM SNisaleftRJ-moSdule.2SimilarlyitisarighrtTƹ-moSdule.2FinallywreharverS((m n)t)UR=r(m nt)UR=rm ntUR=(rm n)tUR=(r(m n))t.f͙Corollary1.20.qsrc:776advalg.texGivengbimoffdulesRZMS,3SQNT,3R'M2@0bS,3SN2@0bTandghomomorphismsf n2HomyR -?S)>3(:M:;:M@20:)andg P2 PHom"[ٟSr}-T2(:N:;:N@20:).6Thenwehavefs S gVg P2 PHom"[ٟR -?T(:M SȊN:;:M@20 SN@20:):Prffoof.#Rsrc:784advalg.texf SȊgn9(rSm nt)UR=fG(rm) gn9(nt)UR=r(f SȊgn9)(m n)t:Remark1.21.src:789advalg.texUnlessotherwisede nedKwillalwraysbSeacommrutativering.src:792advalg.texEvreryKmoSduleKMoverthecommutativeKringKandinparticularevreryvectorspaceKovera eldKisaK-K-bimoSdulebrymUR=m.7ObservethatthereareK-K-bimoSdulesthatdonotsatisfy1mq=rm.ZTVakreforexampleanautomorphism :Kq!oKandaleftK-moSduleMandde nemUR:= ()m.8ThenM+issucrhaK-K-bimoSdule.src:801advalg.texThetensorproSductM K ~ NRxoftrwoK-K-bimoSdulesMRwandNisagainaK-K-bimoSdule.IfwreMhave,lhowever,K-K-bimoSdulesMMsandNtarisingfromK-modulesMasaborve,li.e.satisfyingmUR=m,Sthen?their>tensorproSductMe KN"alsosatis esthisequation,TsoMe KN"comesfrom#a#(left)K-moSdule.Indeedwrehave#ms tnN=m sn=m sn=Om n.Thrus#wecanalsode neatensorproSductoftrwoleftK-modules.src:813advalg.texWVe5often5writethetensorproSductoftrwovector5spacesortrwo5leftmoSdulesMuandNorveracommrutativeringKasM N:iinsteadofM K mN:iandthetensorproSductorverKoftwoK-moSdulehomomorphismsf2andgXasf ginsteadoff K cgn9.(Warning:8Donotconfusethiswithatensorf gn9.Seethefollorwingexercise.) 7o]10oAdv|rancedTAlgebra{P9areigis[o]Problem1.6.src:823advalg.tex(1) Let MR,R xN@,M20bR,and R N20)bSe RJ-modules. Shorwthatthe followingisahomomorphismofAbSeliangroups:򧍑#UR:Hom۟R"n(M;M@ 0) ZHom R'(N;N@ 0)UR3f gË7!f R ;gË2Hom(M RN;M@ 0 RN@ 0):src:830advalg.tex(2)Findexampleswhereisnotinjectivreandwhereisnotsurjective.src:833advalg.tex(3)Explainwhryf gXisadecompSosabletensorwhereasf R ;gXisnotatensor.ۍTheorem1.22.3src:837advalg.texLffetlR MS,S ŜNT,andlT 1PU belbimodules.~Thenmtherearecanonicalisomor-phisms35ofbimoffdulesS(1)%src:841advalg.texAssoSciativitryLaw:fi h:UR(M SȊN@) T KPP԰=M S(N T KPƹ).(2)%src:843advalg.texLarwoftheLeftUnit:fiUR:R R ;MP6԰=@M@.(3)%src:845advalg.texLarwoftheRightUnit:fiUR:M SȊSP)԰!=M@.(4)%src:847advalg.texSymmetryLarw:LIfM@, N@arffeK-modulesthenthereisanisomorphismofK-moffdules%o:URM NP6԰=@N M@.(5)%src:850advalg.texExistenceKPofInnerKQHom-FVunctors:-LffetR RMT, S NT, andS PRbffebimodules.5Thenthere%arffe35canonicalisomorphismsofbimodulesL{ HomdSr}-Tuf׹(:PLn R ;M:;:N:)PUR԰n:=Hom(ySr}-R9>.(:PS:;:HomyT!(M:;N:):)35and鍍XӹHomp\Sr}-T(:PLn R ;M:;:N:)PUR԰n:=Hom(yR -?T9c(:M:;:HomyS i(:PS;:N@):):܍Prffoof.#Rsrc:860advalg.texWVeonlydescribSethecorrespondinghomomorphisms.src:863advalg.tex(1)Use1.7(5)tode ne ((m n) p)UR:=m (n p).src:866advalg.tex(2)De neUR:R R ;M6!M+bry(r6 m):=rSm.src:869advalg.tex(3)De neUR:M SȊS)!!wM+bry(m s)UR:=ms.src:872advalg.tex(4)De neW(m n)UR:=n m.src:874advalg.tex(5)FVorf:vPY R SM~!FN>ȹde ne(fG)v:P1`!pHom/ T5(M;N@)bry(fG)(p)(m)v:=f(p m)and n9(fG)UR:M6!Hom.~S4(PS;N@)bry (fG)(m)(p)UR:=f(p m).=lsrc:880advalg.texUsually.oneidenrti esthreefoldtensorproSductsalongthemap BBsothatwecanuseM SN} T3P:=UR(M| S{N@) TP=URM} Sz(N T3Pƹ).dFVor6thenotion5ofamonoidalortensorcategoryV,horwever,S?this>Tcanonical>Sisomorphism(naturaltransformation)isofcenrtralimpSortanceandwillbSediscussedlater.Problem1.7.src:888advalg.tex(1)LGivreaKcompleteproSofofTheorem1.22.UIn(5)shorwhowHomm՟T$(M:;N:)KbSecomesLanS׹-RJ-bimoSdule.src:893advalg.tex(2)GivreanexplicitproSofofM R ;(X+Yp)PUR԰n:=M RX+M RYp.src:896advalg.tex(3)7Shorwthatforevery nite7dimensionalvectorspaceVbthereisauniqueelementP*n U_i=1 #vi{ v2n9RAi2URVG Vp2 Gsucrhthatthefollowingholds78vË2URV¹:JX ㇍Mi"jv n9ڍi.=(vn9)vi,=v:! bsrc:900advalg.tex(Hinrt:8UseanisomorphismEndg(Vp)PUR԰n:=VG V2 GanddualbasesfvidgofVandfv2n9RAi.=gofVp2\t.)src:903advalg.tex(4)Shorwthatthefollowingdiagrams(cffoherence5diagrams6orcffonstraints)ofK-moSdulescommrute:BA'}@((A B) Cܞ) D(A (BE Cܞ)) D{生:2fd9ά-̯M (A;Bd;C) 1]L@A ((BE Cܞ) DS) O:2fd9ά-̯$o (A;Bd C;DProp`osition]1.23.)src:969advalg.texLffet!(RJXJg;)!beafree!RJ-moduleandS >MR bea!bimodule.0Theneveryelement35uUR2M R ;RJX$hasauniquerffepresentationuUR=Px2X$mx x.ƐPrffoof.#Rsrc:976advalg.texByn1.10Pʟx2X%rxHxisthegeneralelemenrtofRJX.?Hencewehaveu5=5P߾mii ) i=RPmi POrx;i 5x4=PqiֹP$fx+Kmidrx;i/8 x=5Pqxɹ(P imirx;i 5) x. TVoZ]shorwtheZ^uniquenessletP yI{2X my< /yܹ=Z0.sLet.x2Xuandfx :RJXL& !RxbSede nedbry/fxH((yn9))=fx(yn9):=xy .CThen(1M . R ;fxH)(Pmy yn9)UR=Pmy fxH(yn9)=mx 1=0forallxUR2X.8NorwletOY{MR{XM R ;R]{fd `ά-` H Vmrulttׁ @t @t @t @@>@@>RH)MF"Ǡ*FfeyTǠ?  +?src:989advalg.texbSegivren.gThen(mx 01)=mx01=mxꍹ=0hencewrehaveuniqueness.hFVrom1.22(3)wreknorwthatisanisomorphism.*'6Corollary1.24.isrc:995advalg.texLffet_$S }MR,jR ]Nbe_$(bi-)modules.5LetMbeafreeS-moduleoverYp,jandNbffe35afreeRJ-moduleoverX.fiThenM R ;NtisafreeS-moduleoverYGX.?Prffoof.#Rsrc:1002advalg.texConsiderthediagramOVY}[|YGXY5MN{fdJpά-0ʍqY eX{󎎍{󍍒%dS+M R ;N {fdGά-`P H3379S=WhU`fiuPi+PiׁPitPi+PiPisPi+PiPirP i+PiP=<P=<qH 4Lg g|ׁ @g| @ g| @*g| @,>@,>RH>*Ǡ*Ffe>\Ǡ? Ch N7o]12oAdv|rancedTAlgebra{P9areigis[o]src:1013advalg.texLetgfbSeangarbitrarymap. FVorallxUR2XXwregde nehomomorphismsgn9(-;x)UR2Hom۟S!콹(:M;:U@)brythecommutativediagramGf,{WY{󍍒SMጟ{fd%ά-6YH`9fG(-;x)ጟׁ @ጟ @ጟ @ጟ @( >@( >RH33󳖟SxU-:Ǡ*Ffe`lǠ?`gn9(-;x)src:1017advalg.texLet(eg N2URHom۟R"n(:N;:HomyS i(:MR;:U@))bSede nedbryL={CX{󍍒셟Ry2Nk{fd%ά-6}xH`egn9(-;-)Tׁ @T @T @T @ԟ>@ԟ>RHPRDHom S(:MR;:U@)!Ǡ*FfeT4Ǡ? 4egMsrc:1021advalg.texwithKx17!gn9(-;x). [ThenKwrede negn9(m;n)2:=e1g (n)(m)1=:h(m n). [ObservrethatgisadditivreXinXmandinn(bSecause^:egisadditivreinmandinn),t>andgisRJ-bilinear,t>bSecausegn9(mrr;n)==eg M(n)(mrS)=(rYegb(n))(m)=>eg N(rn)(m)=gn9(m;rn).TObrviously$g(y;x)=fG(yn9;x),hence|hzy Y h7X =e^fG.\FVurthermore|wrehaveh(smy zn)e^=jeg s(n)(sm)=s(eg(n)(m))=sh(m n),hencehisanS׹-moSdulehomomorphism.src:1033advalg.texLet,^kzbSeanS׹-module,]homomorphismsatisfyingk% YvX r۹=URfG,Rmthenk (-;x)UR=gn9(-;x),sincek$3 isS׹-linearinthe rstargumenrt.Thusk (m;n)d=eg (n)(m)=h(m n), }andhencehUR=kg.o֍Problem01.8.t src:1042advalg.tex(TVensors4 in4 phrysics:)ݒLetV|bSea nitedimensionalvrectorspaceorverthe4 eldKandletVp2 M.bSeitsdualspace.LettbeatensorinVH *:::" +V *Vp2 ::: +Vp2 ƹ=URVp2 rwt (Vp2\t)2 s .(1)ShorwthatforeachbasisB=(b1;:::ʜ;bnP)anddualbasisB2 p=(b21;:::ʜ;b2nP)thereisauniquelydeterminedscrheme(afamilyoran(r+zgs)-dimensionalmatrix)ofcoSecientsB(a(B)OiqAacmr61*;:::\;irájq1*;:::\;jsZ)witha(B)Oiq1*;:::\;irájq1*;:::\;js2URKsucrhthat](1)@tUR= )nX ㇍iq1*=1AM:::-nn'X"X ㇍' ir,p=1An;X ㇍:jq1*=1Op:::ekn_X ㇍_;pjs=1sa(B)Oiq1*;:::\;irájq1*;:::\;jsZbiq1 :) ::: bir ; b jq1 Q ::: b js: `ƍsrc:1059advalg.tex(2)avShorwauthatforeacrhchangeauofbasesL:B!C>withcj=P52iRAjf biO(withinrverseaumatrix*b(2iRAjf ))thefollorwingtrffansformation35formulaholdsAǍ(2)DEa(B)Oiq1*;:::\;irájq1*;:::\;js= {BndX'؍URkq1*=1ɀ:::/n)oX'؍(kr,p=1D>=n>'X'؍> lq1*=1Q:::g nanX'؍aeQls=1tniq1*kq1 :::Pߍirykr }nlq1Qٍjq1 :::Eߍls5hjsa(Cܞ)Okq1*;:::\;kr lq1*;:::\;lssrc:1070advalg.tex(3);Shorwa(B)Oiq1*;:::\;irájq1*;:::\;js 2K۹satisfyingthe>transformationformrula(2)de nesauniqueѦtensor(indepSendenrtofthechoiceofthebasis)tUR2Vp2 ru (Vp2\t)2 s\suchthat(1)holds.R2ule'for(physicists:At>UH`Dr ׁ       ԟ>ԟ> tsrc:1237advalg.texLetOAandBVbSeK-algebras. AhomomorphismofalgebrffasOfT: A &p!3XBUisaK-modulehomomorphismsucrhthatthefollowingdiagramscommute:O WA *8BV32fd6^ά- fYTA AYBE B<{fd@ά-if fHEǠ*Ffex̟Ǡ?rX.AH Ǡ*Ffe LǠ?rX.B`ɍsrc:1243advalg.texandBj⍒6pK'ɺ1X.Aׁ    С>С> H'X.BLׁ AL AL AL A >A >UAB:c32fd(Ѝά- VfRemark2.2.src:1252advalg.texEvreryK-algebraAisaringwiththemultiplications⍒ʿAA2 pURn!1A A2krpURn!A:usrc:1255advalg.texTheunitelemenrtisn9(1),where1istheunitelementofK.src:1258advalg.texObrviouslythecompSositionoftrwohomomorphismsofalgebrasisagainahomomorphismofalgebras.8FVurthermoretheidenrtitymapisahomomorphismofalgebras.}7o]16oAdv|rancedTAlgebra{P9areigis[o]Problem2.1.src:1264advalg.tex(1)ShorwthatEndgKI(Vp)isaK-algebra.src:1266advalg.tex(2)Shorwthat(A;r:A: :Aq!YA;,:Kq!YA)isaK-algebraifandonlyifAwiththexmrultiplication AA2K p!2A A2rp!2Aandtheunitn9(1)isa ringandʹ:K!Cenrt-H(A)isaringhomomorphisminrtothecffenterofA,whereCenteB(A)UR:=fa2Aj8b2A:ab=bag.src:1274advalg.tex(3)LetVYbSeaK-module.}ShorwthatD(Vp)UR:=KCCVYwiththemrultiplication(r1;v1)(r2;v2)UR:=(r1r2;r1v2j+r2v1)isacommrutativeK-algebra.Lemma72.3.Qsrc:1280advalg.texLffetXAYandBa_beXalgebras.ThenA Ba_isYanalgebraXwiththemultiplication(a1j b1)(a2 b2)UR:=a1a2j b1b2.񍍍Prffoof.#Rsrc:1286advalg.texCertainlyFthealgebrapropSertiescaneasilybeEcrheckedFbyasimplecalculationwithelemenrts.8FVorlaterapplicationswepreferadiagrammaticproSof.src:1290advalg.texLetlrA 36:URA An!1AlandrB :B* BX !_7Bxrdenotethemrultiplicationsofthetwoalgebras.ThenwthenewmrultiplicationiswrA B:=UR(rA rBN>)(1A " 1BN>)UR:A B[ A BX !_7A Bwhereq:B} (wAc!9=A B>wisptheqsymmetrymapfromTheorem1.22.c:Norwthefollorwingdiagramcommrutes py7AA BE A B A Bpy]A A BE B A B|mifd(ά-jE,y1 r 1-:3pypyrpA BE A BF.@񜌟>RHmwAǠ*FfeǠ? lgesrc:1405advalg.texcommrutes.src:1407advalg.texNote:qIfEyrouwanttode neahomomorphismgË:URTƹ(Vp)n!1AE˹withatensoralgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVp.Lemma2.5.src:1414advalg.texAMtensorralgebrffas(Tƹ(Vp);)de nedbysV=isuniqueuptoauniqueisomorphism.Prffoof.#Rsrc:1418advalg.texLet(Tƹ(Vp);)and(T20o(Vp);209)bSetensoralgebrasorverV.8ThenGōbYqAV9ϻҁ Żׁ ܁      .dž.džH 20ׁ d>d> H9dׁ @d @d @d @$>@$>RH ,g20ҁ Hׁ H܁ H H" H, H6 H@ HE*džHE*džjg.Tƹ(Vp)~TƟ20o(Vp)Ԟ32fd@`ά-:?hԞ32fdQά-:kTZTƹ(Vp)P}TƟ20o(Vp)5ꔞ32fd@`ά-:>(hf6src:1428advalg.teximpliesko=URh21 \|.xDProp`ositionH2.6.| src:1432advalg.tex(Rulesݻofcomputationݼinatensoralgebra)Lffet(Tƹ(Vp);)bethetensoralgebrffa35overVp.fiThenwehaves2(1)%src:1435advalg.texUR:VX-!Tƹ(Vp)"is#injeffctive(sowemayidentifytheelements(vn9)andv[forallvË2URVp),DY(2)%src:1437advalg.texTƹ(Vp)UR=fP n;\)ڟ^ciIviq1 :):::vinjgz' NiWy=(i1;:::ʜ;inP)35multiindexoflengthnng;35wherffevi8:j 2URV,>(3)%src:1440advalg.texiff9͹:V> !yAisahomomorphismofK-moffdules,IAisaK-algebrffa,Iandg`:Tƹ(Vp)%(I!7?A35istheinducffed35homomorphismofK-algebrffas,thenW{Qgn9(XTun;\)ڟ^ciUVviq1 :):::vin)UR=XT/n;\)ڟ^cifG(viq1):::f(vin):!种Prffoof.#Rsrc:1450advalg.tex(1)3FUsethe3EemrbSeddinghomomorphismj}:Vm\!WD(Vp),EmwhereD(Vp)is3Ede nedasin2.1(3)toconstructgË:URTƹ(Vp)n!1DS(V)sucrhthatgUR=jӹ.8Sincej{isinjectivesois.src:1455advalg.tex(2)!hLetBM:=fP n;\)ڟ^ciIviq1 _o:::j~vinjgz' Ni=(i1;:::ʜ;inP)mrultiindexoflengthoBng.!ObrviouslyBnisthe>subalgebraBofBTƹ(Vp)generatedbrytheelemenrtsofVp.A$Letj:BBH!Tƹ(V)bSeBtheemrbeddinghomomorphism.'Then:V\ !Tƹ(Vp)factorsthroughaK-moSdulehomomorphism20%:V!nB.8ThefollorwingdiagramOebYYVbYjB{fd*Fά-{XR20HHVTƹ(Vp)f{fd!wЍά-"O:jjBVTƹ(Vp)f32fd!wЍά-^bO:jH 20ׁ @ @Ư @Я @<>@<>RHǠ*FfeܟǠ?h$\jpH  p |ׁ | | | f>f> +src:1473advalg.texinducesauniquepwithpS!j20Ⱥ=p=20Ssince20isahomomorphismofK-moSdules.Because"\of"[jp~~g=j+206=g=id2T.:(V)%F"\wreget"[jpg=id2T.:(V)#Fڹ,pIhencethe"\emrbSeddingj/issurjectivreandthusj{istheidentityV.src:1482advalg.tex(3)ispreciselythede nitionoftheinducedhomomorphism..ޠ7o]18oAdv|rancedTAlgebra{P9areigis[o]Prop`osition2.7.src:1487advalg.texGiven35aK-moffduleVp.fiThenthereexistsatensoralgebra(Tƹ(Vp);).Prffoof.#Rsrc:1492advalg.texDe neTƟ2nJ(Vp)ur:=VX :::p V=V2 nu&tobSethen-foldtensorproductofVp.4De neTƟ20aʹ(Vp)UR:=K꨹andTƟ21(Vp)UR:=V.8WVede neoK?Tƹ(Vp)UR:=M i0wvT i(Vp)=KVG(V Vp)(V V Vp):::uF: }Esrc:1498advalg.texThecompSonenrtsTƟ2nJ(Vp)ofTƹ(V)arecalledhomoffgeneous35components.src:1501advalg.texThecanonicalisomorphismsTƟ2m (Vp) TƟ2nJ(V)PUR԰n:=TƟ2m+nkR(V)takrenasmultiplicationэʍQrUR:TƟ2m (Vp) TƟ2nJ(V)URn!1TƟ2m+nkR(V)rUR:Tƹ(Vp) T(Vp)URn!1T(V)rsrc:1507advalg.texandftheemrbSedding:(K=(TƟ20aʹ(Vp)B !jTƹ(V)finducethestructureofaK-algebraonTƹ(V).FVurthermorewrehavetheembSeddingUR:V M!`TƟ21aʹ(Vp)Tƹ(V).src:1511advalg.texWVeharvetoshowthat(Tƹ(Vp);)isatensoralgebra.{Letf?:{@V1;!}AbSeahomomorphismofK-moSdules.EacrhXMelementinTƹ(Vp)isaXNsumofdecompSosabletensorsv1S N:::= OvnP.De neg:NTƹ(Vp)M!p'AIbrygn9(v1Y :::R vnP)M:=fG(v1):::ʜf(vn)I(and(g:MTƟ20aʹ(Vp)!p(A)=(:K!8A)).`Byinductiononeseesthatgeisahomomorphismofalgebras.`Since(g:QTƟ21aʹ(Vp)!, A)=(fZ:V{!A)YwregetYgdT=fG.NIfYh:Tƹ(Vp), !>AisaYhomomorphismofalgebraswithhUR=f2wregeth(v1j ::: vnP)=h(v1):::ʜh(vn)=fG(v1):::ʜf(vn)hencehUR=gn9.Problemi2.2..src:1526advalg.tex(1)!Let!X\bSeasetandVO:=BKX\bSethefreeK-moSduleorver!X.rShowthatX :_!iVL!Tƹ(Vp)jde nesafrffeealgebrajorverjX,i.e.foreveryjK-algebraAandevrerymapf:Xu!DA#thereis$auniquehomomorphismofK-algebrasg ,:Tƹ(Vp)~!SrAsucrhthatthediagramE!0HXHTƹ(Vp)7{fd ဍά-H`ifѡ4ׁ @ۡ4 @4 @4 @紟>@紟>RHAǠ*Ffe Ǡ? Ҕgsrc:1532advalg.texcommrutes.src:1534advalg.texWVetwritesKhXi$:=#Tƹ(KX)andscallitthepffolynomial)[ringoverKinthenon-commutingvariablesX.src:1538advalg.tex(2)ԟLetԞTƹ(Vp)and㎹:V !}Tƹ(Vp)bSeatensoralgebra.RegardVqasasubsetofTƹ(Vp)bry.Shorw,thatthere,isauniquehomomorphismofalgebrasu:Tƹ(Vp)!vT(Vp)׃ T(Vp),with(vn9)UR=v 1+1 vXforallvË2URVp.src:1544advalg.tex(3)Shorwthat( 1)UR=(1 )UR:Tƹ(Vp)n!1T(Vp) T(Vp) T(Vp).src:1547advalg.tex(4)уShorwthatтthereisauniquehomomorphismofalgebras"UR:Tƹ(Vp)n!1Kwithу"(vn9)=0forallvË2URVp.src:1551advalg.tex(5)Shorwthat(" 1)UR=(1 ")UR=id T.:(V)!4M.src:1554advalg.tex(6)GShorwFthatthereisauniquehomomorphismofalgebrasSF&:OTƹ(Vp)!@*T(Vp)2opdwithS׹(vn9)w=v.9(Tƹ(Vp)2op x|is`the_oppffositealgebraof_Tƹ(Vp)withmrultiplicationstw:=wts_foralls;tUR2Tƹ(Vp)=T(Vp)2op ŹandwherestdenotestheproSductinT(Vp).)src:1560advalg.tex(7)Shorwthatthediagrams@>XzTƹ(Vp)5K:2fdנά-Í" JzTƹ(Vp):2fdנά-(7sTƹ(Vp) T(Vp)eTƹ(Vp) T(Vp)Ԟ32fdά-/u1 SuSr} 1rǠ@fe礟Ǡ??;rǠ@fe٤?`6?;$r؍src:1571advalg.texcommrute.Fנ7o]AlgebrasTandCoalgebrasS19[o]2.3.Symmetricalgebras.src:1576advalg.tex`ɍDe nition2.8.<src:1577advalg.texLetaKabSeacommrutativearing.oLetVbSeaK-module.pAadK-algebraS׹(Vp)togetherwithahomomorphismofK-moSdulesUR:V M!`S׹(Vp),3sucrhthat(vn9)M(v20@ɤ>RHAҟǠ*FfeǠ? gOˍsrc:1587advalg.texcommrutes.src:1589advalg.texNote:[rIf{yrou{wanttode neahomomorphismg׹:LS׹(Vp)f(!Awith{asymmetricalgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesfQ:URV M!`AsatisfyingfG(vn9)f(vn920d> H9dׁ @d @d @d @$>@$>RH ,g20ҁ Hׁ H܁ H H" H, H6 H@ HE*džHE*džjgzS׹(Vp)ʽSן20(Vp)j32fdά-:?h32fdRά-:kYS׹(Vp)P}Sן20(Vp)5d32fdά-:>(hϜsrc:1611advalg.teximpliesko=URh21 \|.xDProp`ositionx2.10.qsrc:1615advalg.tex(Rules-ofcomputationinasymmetricalgebra)4Lffet3(S׹(Vp);)bethesymmetric35algebrffaoverVp.fiThenwehave'卍(1)%src:1619advalg.texUR:VX-!S׹(Vp)35isinjeffctive(wewillidentifytheelements(vn9)andvnforallvË2URVp),DY(2)%src:1621advalg.texS׹(Vp)UR=fP n;\)ڟ^ciIviq1 :):::vinjgz' NiWy=(i1;:::ʜ;inP)35multiindexoflengthnng;>(3)%src:1624advalg.texifP7f: V'y!AisahomomorphismofK-moffdulessatisfyingfG(vn9)$%f(v20{Qgn9(XTun;\)ڟ^ciUVviq1 :):::vin)UR=XT/n;\)ڟ^cifG(viq1):::f(vin):(҉Prffoof.#Rsrc:1635advalg.tex(1) Usethe!emrbSeddinghomomorphismj%:URV M!`D(Vp),whereD(Vp)!isthecommruta-tivre_algebra`de nedin2.1(3)toconstructg˒:]YS׹(Vp)v!?DS(V)_sucrh`thatg]Y=jӹ.GSincej2isinjectivresois.src:1641advalg.tex(2)!hLetBM:=fP n;\)ڟ^ciIviq1 _o:::j~vinjgz' Ni=(i1;:::ʜ;inP)mrultiindexoflengthoBng.!ObrviouslyBnisthesubalgebraPofS׹(Vp)generatedbrytheelementsofPVp.jLetjϹ:B!S׹(V)bSetheemrbeddinghomomorphism.=Then8 : V) !nS׹(Vp)factorsthrougha7K-moSdulehomomorphism20:V^/7o]20oAdv|rancedTAlgebra{P9areigis[o]!nB.8ThefollorwingdiagramQqjbYYVbYjB{fd*Fά-{XR20HHS׹(Vp)f{fd!ά-"ZjjBS׹(Vp)f32fd!ά-^bZjH 20ׁ @ @Ư @Я @<>@<>RHǠ*FfeܟǠ?h$\jpH  p |ׁ | | | f>f> src:1659advalg.texinducesauniquepwithpkkj20ٹ=8p8=20ӹsince20ҹisahomomorphismofK-moSdulessatisfying209(vn9)[[20(v20@ɤ>RHAҟǠ*FfeǠ? g5src:1684advalg.texcffommutes.Prffoof.#Rsrc:1688advalg.texCommrutativityfollowsfromthecommutativityofthegenerators:Hvn9v20="v20the0)de ning0*conditionfG(vn9)-f(vn920@̟>RHAǠ*FfeM,Ǡ? g6src:1718advalg.texcommrutes.src:1720advalg.texThe(algebra)K[X]:=S׹(KX))is(calledthepffolynomialIringoverIKinthe(cffommuting)vari-ables35X.src:1724advalg.tex(2)CGLetCFS׹(Vp)and:VifforeacrhK-algebraAandYforYeacrhhomomorphismofK-moSdulesfZ2:3V.!cA,u]sucrhthatfG(vn9)226=30forallvl23Vp,thereexistsauniquehomomorphismofK-algebrasgË:URE(Vp)n!1A꨹sucrhthatthediagramJ免H&VHE(Vp){fd ά- wH`fׁ @ @ @ @`l>@`l>RH1WAeǠ*Ffe̟Ǡ? KLgesrc:1771advalg.texcommrutes.src:1773advalg.texThemrultiplicationinE(Vp)isusuallydenotedbyu^vn9.src:1776advalg.texNote:gIfyrouwanttode neahomomorphismg3:nE(Vp)!hAwithanexterioralgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVfsatisfyingfG(vn9)22V=UR0forallv;v202URVp.gOProblem2.4.)src:1784advalg.tex(1)d> H9dׁ @d @d @d @$>@$>RH ,g20ҁ Hׁ H܁ H H" H, H6 H@ HE*džHE*džjfE(Vp) E20P(Vp)=32fdOά-:?h]32fdQ ά-:k_E(Vp)P3E20P(Vp)6bԞ32fdOά-:>(hf6src:1809advalg.teximpliesko=URh21 \|.xDProp`osition3,2.15.src:1813advalg.tex(Rulesofcomputationinanexterioralgebra)Lffet(E(Vp);)betheexte-rior35algebrffaoverVp.fiThenwehaves2(1)%src:1817advalg.texUR:VX-!E(Vp)35isinjeffctive(wewillidentifytheelements(vn9)andvnforallvË2URVp),̠7o]22oAdv|rancedTAlgebra{P9areigis[(2)%src:1819advalg.texE(Vp)UR=fP n;\)ڟ^ciIviq1 :)^:::^vinjgz' NiWy=(i1;:::ʜ;inP)35multiindexoflengthnng;>(3)%src:1822advalg.texif˥fQ:URVX-!AisahomomorphismofK-moffdulessatisfyingfG(vn9)Ćf(vn920:AV- G@<>RHǠ*FfeܟǠ?h$\jpH  p |ׁ | | | f>f> ۶src:1856advalg.texinducesauniquepwithpkkj20ٹ=8p8=20ӹsince20ҹisahomomorphismofK-moSdulessatisfyingN209(vn9)l20(vn920 cmmi10n oiG^"*.src:1891advalg.tex(2)ShorwthatthesymmetricgroupSnopSerates(fromtheleft)onTƟ2nJ(Vp)bryn9(v1 :::XV vnP)UR=vI{1 ;;(1) ::: vI{1 ;;(n)F/withË2URSn andvi,2Vp.`src:1896advalg.tex(3)1A1tensor1aJ2ITƟ2nJ(Vp)iscalledatsymmetricttensorifn9(a)J=afor1all<2ISnP.Letx2^Sן2nѹ(Vp)bSethesubspaceofsymmetrictensorsinTƟ2nJ(Vp).src:1901advalg.texa)ShorwthatS;:URTƟ2nJ(Vp)3a7!PI{2Sn'Hn9(a)2TƟ2n(Vp)isalinearmap(symmetrization).src:1905advalg.texb)ShorwthatS hasitsimageinx^Sן2n(Vp).]܍src:1907advalg.texc)ShorwthatImq(Sb)UR=xVb^Sן2nj(Vp)ifn!isinvertibleinK.rjsrc:1910advalg.texd)JShorwJthatxK^Sן2n(Vp),!TƟ2nJ(V)2 S p!YSן2n['(V)JisanJisomorphismifn!isinrvertibleJinKandU:XTƟ2nJ(Vp)Xr!ʩSן2n['(V)istherestrictionof:XTƹ(Vp)r!ʨS׹(V),whereS(V)isthesymmetricalgebra.src:1916advalg.tex(4)tA9tensora*2+TƟ2nJ(Vp)iscalledanYantisymmetrictensoruiftn9(a)="()atforalluLc2SnwhereS "(n9)isthesignSofthepSermrutation.rLetx^E2nb(Vp)SbSethesubspaceofanrtisymmetrictensorsinTƟ2nJ(Vp).src:1922advalg.texa)ShorwthatE<:)TƟ2nJ(Vp)3)a7!PԑI{2Sn)m"(n9)(a)2)TƟ2nJ(Vp)isaK-moSdulehomomorphism(anrtisymmetrization).7o]AlgebrasTandCoalgebrasS23\9src:1926advalg.texb)ShorwthatEkhasitsimageinx^E2n(Vp).]܍src:1928advalg.texc)ShorwthatImq(Eù)UR=x\^E2nd^(Vp)ifn!isinvertibleinK.rjsrc:1931advalg.texd)$Shorw$thatx^E2n3(Vp)k,!kTƟ2nJ(V)2 p!׼E2n\g(V)$is$anisomorphismifn!isinrvertibleinK$and:'TƟ2nJ(Vp)(!E2n\g(V)0is/therestrictionof:'Tƹ(Vp)(!E(V),&Rwhere/E(V)/is0theexterioralgebra.ц2.5.LeftA-mo`dules.src:1940advalg.texDe nition]`2.17.src:1941advalg.texLet̅AbSeaK-algebra..A~leftA-moffduleis̆aK-moduleM itogetherwithahomomorphismM B:URA M6!M@,sucrhthatthediagramsTƪA MiOMD432fd>`ά-W]YA A MY A Mt{fd'8ά-iid% HǠ*FfeğǠ?ЍMr idHǠ*FfeğǠ?'$kDJsrc:1947advalg.texandM⍒-MP6԰=@K M⍒ %A M,D{fd@ά-iPI{ idHMŸǠ*Ffe1Ǡ?'"tHk|id44ҁ H44ׁ H44܁ H44 H44 H44 H44 H44 H4džH4džjsrc:1954advalg.texcommrute.src:1956advalg.texLetGeA %IMIandGfA %JNbSeGeleftA-modulesGfandletf;+:-M4M!@NIbeaK-linearGfmap.OThemapGefdiscalledahomomorphism35ofleftA-moffdules꨹ifthediagramR`ά-i/X.MHRǠ*FfeNǠ?`i1 fH RǠ*Ffe@Ǡ?`#f|src:1961advalg.texcommrutes.цProblem2.6.src:1965advalg.texShorwthatanAbSeliangroupM:nisaleftmoSduleover@theringAifandonlyifM+isaK-moSduleandaleftA-moduleinthesenseofDe nition2.17.ч2.6.Coalgebras.src:1974advalg.texDe nition2.18.src:1975advalg.texA0K-cffoalgebraisaK-moSduleCtogetherwithacffomultiplicationordiagonalUR:C1K{!CF CF(K-moSdulehomomorphism)thatiscoassociativre:SuPuCF CJCF C C$32fd)@ά-`VidM^ YHCYBuCF Cğ{fdAά-ˍ[HBǠ*FfetǠ?`"HBǠ*FfetǠ?Ѝ idҠ7o]24oAdv|rancedTAlgebra{P9areigis[o]src:1981advalg.texandacffounitoraugmentationUR:C1K{!K꨹(K-moSdulehomomorphism):M JYCYCF C\{fdY?ά-ˍnHڟǠ*Ffe Ǡ?`0HZǠ*Ffe(Ǡ?Ѝ id[ " CF C"ڟK CP1԰Jع=ܙCP1԰Jع=CF K:μ32fd&fpά-`b. idHk|Lid*̟ҁ H*ׁ̟ H*̟܁ H*̟ H*̟ H*̟ H*̟ H*̟ H̟džH̟džjލsrc:1990advalg.texAK-coalgebraCFiscffocommutative꨹ifthefollorwingdiagramcommutesHCF CECF Cᙴ32fdά-Í臑bY>NC`tׁ t t t ѡ4>ѡ4> H`Dԟׁ Aԟ Aԟ Aԟ A >A >Usrc:1999advalg.texLetoCL'andDbSeoK-coalgebras.ǃAoghomomorphismXofYcffoalgebrasof~:7C-!e*DisaK-moSdulehomomorphismsucrhthatthefollowingdiagramscommute:N~CF CtD6 D~t32fdyά- f fbYnCbYDτ{fd5氍ά-ifH6Ǡ*FfeiğǠ?j<X.CH Ǡ*Ffe DǠ?X.DO>src:2004advalg.texandFōH""6pK,b(X.CСׁ Aա Aڡ Aߡ A>A>UH,X.D ׁ    L>L> bY&CbYrDS:ԟ̟{fd(ά-iVfRemark2.19.Lsrc:2013advalg.texObrviously2the1compSositionoftrwo2homomorphismsofcoalgebrasisagainahomomorphismofcoalgebras.r'FVurthermoretheidenrtitymapisahomomorphismofcoalge-bras.Problem2.7.fsrc:2021advalg.tex(1)mShorwmthatVH) Vp2 Lisacoalgebraforevrery nitedimensionalvrectorspaceVorverS*aS+ eldKifthecomrultiplicationisde nedby(vt u:vn92.=)UR:=P*n U_i=1 ASv u;v2n9RAiw vi vn92gwherefvidg꨹andfv2n9RAi.=garedualbasesofVresp.Vp2\t.src:2028advalg.tex(2)HShorwthatGthefreeK-moSdulesKX˹withthebasisX˹andthecomrultiplication(x)UR=xG+ xisacoalgebra.8Whatisthecounit?Isthecounitunique?src:2032advalg.tex(3)ShorwthatKVwith(1)UR=1 1,(vn9)UR=v 1+1 vXde nesacoalgebra.src:2036advalg.tex(4)LetC-andDbSecoalgebras. uThenC DisacoalgebrawiththecomrultiplicationC DG:=UR(1C ! |X 1D)(C D)UR:C }Dk!C DU C Dcandcounit"UR="C DG:C }D!nK K1K{!K.8(TheproSofisanalogoustotheproofofLemma2.3.)src:2046advalg.texTVoEdescribSeDthecomrultiplicationofaK-coalgebraintermsofelemenrtsweEintroSduceaEno-tation rstinrtroSducedbySweedlersimilartothenotationr(ak b)w=wabusedforalgebras.Insteadof(c)UR=Pci c20RAiOwrewrite.a(c)UR=Xc(1)$ c(2) \|:ݍsrc:2052advalg.texObservre+thatonlythecompleteexpressionontherighthandsidemakessense, notthecompSonenrts!c(1)8orc(2)8which arenotconsideredasfamiliesofelemenrtsofCܞ.4Thisnotationà7o]AlgebrasTandCoalgebrasS25[o]alonedoSesnothelpmruchinthecalculationswrehavetopSerformlateron.+SowreintroSduceamoregeneralnotation.De nitiony2.20.src:2059advalg.tex(Swreedler/Notation)/LetMpbSeanarbitraryK-module/andC TbeaK-coalgebra.8ThenthereisabijectionbSetrweenallmrultilinearmaps'썒 fQ:URCF:::C1K{!M'src:2064advalg.texandalllinearmapsA$fG 0k:URCF ::: C1K{!M:src:2066advalg.texThesemapsareassoSciatedtoeacrhotherbytheformula|:fG(c1;:::ʜ;cnP)UR=f 08(c1j ::: cnP):src:2069advalg.texFVorcUR2CFwrede neS@%>XzfG(c(1) \|;:::ʜ;c(n) Dȹ)UR:=f 08( n1̹(c));src:2072advalg.texwheren2n18;denotestheon99ܹ1-foldapplicationof,zforexample2n1=UR(9 9ܹ1 :::>W 1):::uH( 1).src:2077advalg.texInparticularwreobtainforthebilinearmap ۏ:C#^FC-3ێ(c;d)7!c d2C#^ C(withassoSciatedidenrtitymap)%aXCc(1)$ c(2)ι=UR(c);src:2081advalg.texandforthemrultilinearmap 22V:URCFCC1K{!C C CލiEX|Zc(1)$ c(2) c(3)ι=UR( 1)(c)=(1 )(c):src:2087advalg.texWiththisnotationonevreri eseasilyݍVȬXjc(1)$ ::: (c(i) R) ::: c(n)=URXc(1)$ ::: c(n+1)ߍsrc:2091advalg.texandAʍB$PPc(1)$ ::: (c(i) R) ::: c(n)=URPc(1)$ ::: 1 ::: c(n1)=URPc(1)$ ::: c(n1) src:2100advalg.texThisnotationanditsapplicationtomrultilinearmapswillalsobSeusedinmoregeneralconrtextslikecomoSdules.Prop`osition2.21.usrc:2104advalg.texLffetfoCC beacoalgebrafnandAanalgebrffa.ThenthecffompositionfЙg"g:=rA(f gn9)C de nes35amultiplicffation'덍[TùHomsL(C5;A) Hom$1(C;A)UR3f gË7!fgË2Hom(C5;A);'src:2109advalg.texsuch\thatHom1(C5;A)\bffecomes\analgebrffa.TheunitelementisgivenbyK|$3 7!(c7!n9( (c)))UR2Hom(C5;A).Prffoof.#Rsrc:2115advalg.texThelmrultiplicationoflHom(C5;A)obviouslyisalbilinearmap.,ThemultiplicationisassoSciativresince(fZ`bgn9)hUR=rA((rA(f bgn9)C) h)C t=URrA(rAE 1)((fZa g)a h)(C 1 1)C p=qrA(1 rA)(f (g$I h))(1 C)C p=qrA(f (rA(g$J h)C))C p=qf(g$Ih).FVurthermore;it;isunitarywithunit1HomD(C;A)+w=߮AC [sinceAC f'=߮rA(AC fG)C F=rA(A 1A)(1K c fG)(C @ 1C)C t=URf2andsimilarlyfAC t=URfG.v"De nitionA2.22.Usrc:2131advalg.texThemrultiplicationsT:sSHom(C5;A) Hom(C;A)sT!4Hom/y(C;A)iscalledcffonvolution.Corollary2.23.src:2136advalg.texLffet35CbeaK-coalgebra.fiThenCܞ2=URHom۟K"(C5;K)isanK-algebra.Prffoof.#Rsrc:2141advalg.texUsethatKitselfisaK-algebra.n_ؽ7o]26oAdv|rancedTAlgebra{P9areigis[o]Remark2.24. src:2145advalg.texIfwrewritetheevXaluationasCܞ2W lCYY3|a kc7!ha;ci2KϹthenanelemenrta*2*Cܞ2 ishcompletelydeterminedbrythevXaluesofha;ciforallc2Cܞ.'SotheproSductofaandbinCܞ2 Jisuniquelydeterminedbrytheformula9{Zhab;ciUR=ha b;(c)iUR=Xa(c(1) \|)b(c(2)):9src:2152advalg.texTheunitelemenrtofCܞ2 JisUR2Cܞ2.9~Lemma 2.25.src:2156advalg.texLffet Kbea eldandAbea nitedimensionalK-algebra. ThenA2 [=HomyK2k(A;K)35isaK-cffoalgebra.9Prffoof.#Rsrc:2162advalg.texDe nethecomrultiplicationonA2byQUR:A 2 r-:pVj!(A A) 2Vcanb-:1p Ï!"tA j A :Ksrc:2166advalg.texThejcanonicalimapcan:URA2c _A2V .!5(A A)2Qnisinrvertible,CsincejAis nitedimensional.!ByaSdiagrammaticproSofRorbrycalculationwithelemenrtsitiseasytoshorwthatA2VbSecomesaK-coalgebra.LRemark2.26.src:2173advalg.texIf+Kisanarbitrary+commrutative+ringandAisaK-algebra,{thenA2 7~=HomyK2k(A;K)isaK-coalgebraifAisa nitelygeneratedprojectivreK-moSdule.9Problem2.8.{src:2181advalg.texFindmsucienrtconditionslforanalgebraAresp./acoalgebraC sucrhthatHomy(A;Cܞ)bSecomesacoalgebrawithco-conrvolutionascomrultiplication.2.7.Como`dules.src:2188advalg.tex9De nitiona2.27.dMsrc:2189advalg.texLetCbSeaK-coalgebra.$AleftCܞ-cffomoduleisaK-moduleMtogetherwithaK-moSdulehomomorphismM B:URM6!CF M@,sucrhthatthediagramsKVCF MCF C Mˤ32fd&ά-`P0id8 YMYVCF M{fd>y@ά-,pHBǠ*FfetǠ?k|(LHBǠ*FfetǠ?Ѝ idPsrc:2195advalg.texand@bY M' Ǡ*FfeZ<Ǡ?k|"CF M"۝K MP6԰=@M:̞32fd:]ά-`Ң idHk||idҁ Hׁ H܁ H H H H H HWdžHWdžjsrc:2203advalg.texcommrute.src:2205advalg.texLetn2C M8Qandn2CN8RbSemCܞ-comodulesnandletf:kM~!0N8RbSeaK-moSdulehomomorphism._1Themapf2iscalledahomomorphism35ofcffomodules꨹ifthediagramLw+N&PCF NOt32fd@Z@ά-!X.NY1OMY+CF M?{fd>y@ά-ٌ8X.MHMǠ*FfeğǠ?`%fH?Ǡ*FferğǠ?`%D1 fjsrc:2210advalg.texcommrutes.src:2212advalg.texLetC NbSeC anarbitraryK-moduleandC MbeaCܞ-comodule.ThenthereisC abijectionbetrweenallmrultilinearmapsd/fQ:URCF:::CM6!N7o]AlgebrasTandCoalgebrasS27[o]src:2216advalg.texandalllinearmaps֍zfG 0k:URCF ::: C M6!N:P src:2218advalg.texThesemapsareassoSciatedtoeacrhotherbytheformula;> fG(c1;:::ʜ;cnP;m)UR=f 08(c1j ::: cnR m):<src:2221advalg.texFVormUR2M+wrede ne􍍍ɟXPfG(m(1) \|;:::ʜ;m(n) D;m(M") h)UR:=f 08(s2 n(m));\ۍsrc:2224advalg.texwheres22n *denotesthen-foldapplicationofs2,i.e.82n pԹ=UR(1 ::: 1 ):::(1 ).src:2229advalg.texInparticularwreobtainforthebilinearmap UR:CFM6!C M-;Xm(1)$ m(M")u=URs2(m);|src:2231advalg.texandforthemrultilinearmap 22V:URCFCM6!C C MbgXufm(1)$ m(2) m(M")u=UR(1 s2)(c)UR=( 1)s2(m):Problem2.9.ksrc:2239advalg.texShorwxQthatxPa nitedimensionalvrectorspaceVisacomoSduleorverthexPcoalge-braV" Vp2 :asde nedinexercise2.7(1)withthecoactions2(vn9)UR:=PvR v2n9RAiV vi,2(V" Vp2\t) VwhereꨟPSv2n9RAi viOisthedualbasisofVinVp2  Vp.Theorem 2.28.=Jsrc:2247advalg.tex(FundamentalThefforemforComodules)LetKbea eld._LetMkbealeftCܞ-cffomoduleDandCletm2MP'bffegiven.ThenthereCexistsa nitedimensionalsubffcoalgebraCܞ20PCanda nitedimensionalCܞ20-cffomoduleM@20withm2PM@20_MwherffeM@20_MisaK-submoffdule,35suchthatthediagramNDPMCF MD32fd>y@ά-̍*/ M@20/ Cܞ20U M@20^{fd:?ά-L̟-:0HǠ*Ffe6Ǡ?HǠ*Ffe(Ǡ?src:2257advalg.texcffommutes.CorollaryM2.29. src:2261advalg.tex(1)EachelementcUR2Cofacffoalgebraiscffontainedina nitedimensionalsubffcoalgebra35ofCܞ.src:2264advalg.tex(2)EachelementmUR2M^ofacffomoduleiscontainedina nitedimensionalsubffcomoduleofM@.ÍCorollary2.30.@src:2269advalg.tex(1)Each nitedimensionalsubspffaceVofacffoalgebraCiscontainedina nite35dimensionalsubffcoalgebra35Cܞ20 ofCܞ.src:2273advalg.tex(2)(Each nitedimensional'subspfface(V9ofacffomodule(M iscffontainedin(a nitedimensionalsubffcomodule35M@20BRofM@.Corollary2.31.src:2279advalg.tex(1)35Eachcffoalgebra35isaunionof nitedimensionalsubffcoalgebras.src:2282advalg.tex(2)35Eachcffomodule35isaunionof nitedimensionalsubffcomodules.ÍPrffoof.#Rsrc:2286advalg.tex(of#.theThefforem)9WVe8canassumethatmUR6=08forelsewrecanuseM@20do=UR0andCܞ20)=UR0.src:2290advalg.texUndertherepresenrtationsofs2(m)UR2C Mܹas nitesumsofdecompSosabletensorspickoneh9`s2(m)UR= sX ㇍Si=1ci mip7o]28oAdv|rancedTAlgebra{P9areigis[o]src:2293advalg.texofshortestlengths.Thenthefamilies(cidjiH=I1;:::ʜ;s)and(mijiI=H1;:::ʜ;s)arelinearlyindepSendenrt.8Choosecoecientscij 62URCFsuchthat"_(cjf )UR= wtX ㇍Si=1ci cijJ; 8j%=1;:::ʜ;s;"src:2298advalg.texbryIsuitablyHextendingthelinearlyindepSendenrtfamily(cidjiUR=1;:::ʜ;s)toIalinearlyindepSen-denrtfamily(cidjiUR=1;:::ʜ;t)andtURs.src:2302advalg.texWVe3k rstshorw3jthatwecanchoSoset=s. )By3jcoassociativitry3kwehaveP*s U_i=1"lciX ~s2(midڹ)=AP* s U_ jv=11(cjf )) mj\=URP*s U_jv=1!BP*-0t U_-0i=1>.cip cijtz mj. Since.!the."ciandthemj,arelinearlyindepSendenrtcwrecancomparecoSecientsandget j}(3)Js2(midڹ)UR= sX ㇍jv=1cij mjf ; 8i=1;:::ʜ;s$\@src:2312advalg.texand0UR=P*s U_jv=1!Bcij mjPfori>s.8ThelaststatemenrtimpliesӾcij 6=UR0; 8i>s;j%=1;:::ʜ;s:p+src:2315advalg.texHencewregettUR=s꨹and j~_(cjf )UR= sX ㇍Si=1ci cijJ; 8j%=1;:::ʜ;s:"src:2320advalg.texDe neA niteBdimensionalsubspacesCܞ20 `(=PhcijJji;jb$=1;:::ʜ;siQC߹andBM@20 m=hmidji=1;:::ʜ;siUM@. Then{bry(3){weget{s:UM@20 r (!)TCܞ20f M@20. WVeshow{thatmU2VM@20 andthatctherestrictionboftoCܞ20:givresaK-moSdulehomomorphismUR:Cܞ20) !oCܞ20  Cܞ20:socthatthe^requiredpropSertiesofthetheoremaresatis ed.FirstobservrethatmUR=P"(cidڹ)mi,2M@20andcj\=URP"(cidڹ)cij 62URCܞ20׹.8UsingcoassoSciativitryweget9c6GP*os U_oi;jv=1dci (cijJ) mj =URP*s U_k6;jv=1(B(ck#) ck6j D mj =URP*s U_i;j$;k6=1- ci cik  ck6j D mjsrc:2337advalg.texhence.čg(cijJ)UR= sX'؍?k6=1cik  ck6j :qkݍRemark2.32.src:2345advalg.texWVeTgivreTasketchTofasecondproSofofTheorem2.28whicrhissomewhatmoretecrhnical.[Since CoisaK-coalgebra,QthedualCܞ2 sisanalgebra.ThecomoSdulestructureȄ:URM6!C ^M"leadsj>toamoSdulestructurebry=(ev / 1)(1] ^s2):Cܞ2@ M6!Cܞ2A C M!G M@.yConsiderthesubmoSdule Nn`:=-|Cܞ2m.ThenN@is nitedimensional,Egsincec2m-|=P* n U_ i=1hc2;cidimi:for_all`c2V2URCܞ2whereP* n U_ i=1 `cipL  qmi,=s2(m).Observre`thatCܞ2misasubspaceofthe[2space[3generatedbrythemidڹ. ButitdoSesnotdepSendonthecrhoiceofthemidڹ. FVurthermoreif"wretake#s2(m)X=PciwR xmiwith"ashortestrepresentation#thenthemiareinCܞ2msincec2mUR=Phc2;cidimi,=miOforc2anelemenrtofadualbasisoftheci.src:2362advalg.texN+isaCܞ-comoSdulesinces2(c2m)UR=Phc2;cidis2(mi)UR=Phc2;ci(1)AVici(2) mi,2CF Cܞ2m.src:2366advalg.texNorwweconstructasubScoalgebraDofCsuchthatNعisaDS-comodulewiththeinducedcoaction.$LetD:=URNq 0(N@2.By2.9NisacomoSduleorverthecoalgebraNq 0(N@2.$Constructa `K-moSdulehomomorphism _:DD!Cbryn n2N7!P9an(1) \|hn2;n(N")i.By _de nition `ofthe k7o]AlgebrasTandCoalgebrasS29[o]dualbasiswrehavenUR=Pnidhn2RAi;ni.8Thusweget.jʍ;5( )D(n n2)=UR( )(Pn n2RAij ni n2)=URPn(1) \|hn2RAi;n(N")i ni(1)AVhn2;ni(N")i=URPn(1)$ ni(1)AVhn2;ni(N")ihn2RAi;n(N")i=URPn(1)$ n(2) \|hn2;n(N")iUR=PC(n(1) \|)hn2;n(N")i=URC(n n2):src:2388advalg.texFVurthermore"C(no nn2)WA=WB"(Pn(1) \|hn2;n(N")i)=hn2;P"(n(1) \|)n(N")iWA=hn2;ni=WA"(no nn2).Hence":!Dد:!w]C\isahomomorphismofcoalgebras, DZMis nitedimensionalandtheimageCܞ20):=UR(DS)misam nitedimensionalsubcoalgebraofmCܞ.LClearlyNιismalsoamC20׹-comoSdule,sinceitisaDS-comodule.src:2397advalg.texFinallywreshowthattheDS-comodulestructureonNPifliftedtotheCܞ-comoSdulestructurecoincideswiththeonede nedonM@.8WVeharve$eʍ!IC(c2m)N=URC(P hc2;m(1) \|im(M") h)=Phc2;m(1) \|im(2)$ m(M")N=URPhc2;m(1) \|im(2)$ midhm2RAi;m(M") hiUR=Phc2;m(1) \|im(2)hm2RAi;m(M") hi miN=UR( 1)(P hc2;m(1) \|im(M") m2RAij midڹ)=( 1)(Pc2m m2RAij midڹ)N=UR( 1)D(c2m):$N7o]30oAdv|rancedTAlgebra{P9areigis[o]qs3.wzProjectiveModulesandGenera32tors3.1.Pro`ductsandcoproducts.src:2443advalg.texDe nition3.1.(1)Let(Midji2I)bSeafamilyofRJ-moSdules.yAnR-moSduleQXMi޹togetherwithafamilyofrDhomomorphismsrE(pj/:<&Q{Mi !Mjf jj2<%I)iscalleda(dirffect)productofrEtheMiandthehomomorphisms,pj [:QJMiZ^ s!ioMj E6arecalledprffojections,Nif,foreacrhRJ-moSduleN andforeacrhfamilyofhomomorphisms(fj\:URN6!Mjf jj%2I)thereisauniquehomomorphismfQ:URN!nߟQ4MiOsucrhthatEҀ†QMiҀgMj32fdά-{SpjH!6$fjׁ @ @ @ @>@>RbY=N_rǠ*FfeВǠ?`}fksrc:2455advalg.texcommruteforallj%2URI.src:2458advalg.tex(2)r\ThedualnotionisqcalledcoproSduct":tLet(Midji2I)rbeafamilyqofRJ-modules.>AnRJ-moSdulex`YMi޹togetherxwithafamilyofhomomorphismsx(j _:Mj _ yA!r`+EMidjj2I)xiscalledhaicffoproductorhdirectcsumofhtheMi5Bandthehomomorphismsj Bl:cMj Bm [!8]`)Mi5Barecalledinjeffctions,ʤifforeacrhRJ-moSduleN޼andforeacrhfamilyofhomomorphisms(fj\:RMj!nN@jj%2URI)thereisauniquehomomorphismfQ:`Mj\ !*;N+sucrhthatQ@Mj@@`MiT,{fdά-jH!*fj(|ׁ @(| @(| @(| @n>@n>RHRNt*Ǡ*Ffe\Ǡ?` Yfsrc:2469advalg.texcommruteforallj%2URI.Remark 3.2.src:2474advalg.texAnCanalogousde nitioncanCbSegivrenforalgebras,@󪤟>RH33MiҟǠ*FfeǠ?  gisrc:2519advalg.texcommrute(R*isthefreeRJ-moSduleoverthesetf1g).Thenthereisauniquegع:Rt!bQ"iMiwith?VҀ†QMiҀgMj32fdά-{SpjH36$gjׁ @ @ @ @>@>RbY :R_rǠ*FfeВǠ? єgssrc:2522advalg.texfor.all/j%2URI.Thehomomorphismggiscompletelyanduniquelydeterminedbrygn9(1)UR=:a.andbrythecommutativediagram~=f1gRXܟIJfd&H`ά-8@HaQMi @@@@e@eRltMj0'jlAlAlAlAlAlAlAlAlAlALPALPUHjJ`*Ffe|`?WˍlgHjJǠ*Ffe|Ǡ?3Epjՠܟ@\@\Rՠʟ胀Fufe胀?0|gj`,耎\ܟܟ g=src:2535advalg.texwherepjf (a)='j(1)=aj.Sowrehavefounda2QMi|with(pidڹ(a))=(ai).Hencethemapgivrenin thepropSositionissurjectivre.GivenaandbinQ"]Mi1with(pidڹ(a))֤=֥(pi(b))then'jf (1)UR:=pj(a)*and j(1)UR:=pj(b))de ne*equalmaps'j\=UR j,hencethe)inducedmapsgj\:URR!nMjFandhj\:URRn'!{MjareequalsothatgË=URhandhenceaUR=gn9(1)=h(1)=b.5HencethemapgivreninthepropSositionisbijective.src:2545advalg.texSinceKaisuniquelydeterminedbrythepjf (a)UR=ajUweJhaveKpjf (aEչ+Eb)=pj(a)E+pj(b)UR=aj߹+Ebjandpjf (rSa)UR=rpjf (a)=rajsrc:2549advalg.texThelaststatemenrtispidfQ=URfi./Remark$;3.5.src:2553advalg.texObservreXithatthisconstructioncanalwaysbSeperformedXhifthereisafreeobject(algebra,coalgebra,comoSdule,group,Abeliangroup,etc.)8Rorverthesetf1gi.e.ifH5f1gH R2ԟ{fd&H`ά-H`ׁ @ @ @ @?>@?>RH5T DBǠ*Ffe wtǠ?Zsrc:2558advalg.texhasaunivrersalsolution.Prop`osition43.6.4src:2562advalg.tex(RulesnofmcomputationinacoproSductofRJ-modules)NLffetN(` UUMid;(jf ))bea35cffoproductofthefamilyofRJ-modules(Midڹ)i2I ^.(1)%src:2566advalg.texThe35homomorphismsj\:URMj !!w`&pMiarffe35injective.(2)%src:2568advalg.texForeffachelementa^2_`,Mihthereare nitelymanyai<92^Mihwitha_=P* n U_ i=1!_idڹ(ai).%The35ai,2URMiarffeuniquelydeterminedbya. A7o]32oAdv|rancedTAlgebra{P9areigis[o]Prffoof.#Rsrc:2575advalg.tex(1)TVoshorwtheinjectivityofiOde nefi,:URMiӷ!) MjPbyv\$Rfi,:=URqʍUSid;(ciUR=j;US0;(celse.psrc:2581advalg.texThenthediagramH@dMi@``Mj6{fdά-6ZiH`Mfiѡ4ׁ @ۡ4 @4 @4 @紟>@紟>RHl> MjǠ*Ffe Ǡ?`ҔfXsrc:2583advalg.texde nesauniquelydeterminedhomomorphismfG.hFVorip=jtthisimpliesfGi[=pid ;&M8:if,henceiisinjectivre.]ݍsrc:2587advalg.tex(2)De newڮfM̹:=URPjf (Mj)UR`Mjf .ThenthefollorwingdiagramcommruteswithbSoth0andE@4Mi@w0`̅Mjd{fdά-6>ZiHV7`Mjf =wJfM=Հh0ׁͅ @ׅ @ @ @˄>@˄>RHǠ*Ffe$ğǠ?=H0HҟǠ*FfeǠ?9 Nsrc:2596advalg.texHenceF=30and`QMj =w}fMi. FLeta=P*jf (aj). FDe nefasin(1).ThenwrehavefG(a)D=Cf(Pjf (aj))=CPfGj(aj)=CPfj(aj)=aidڹ,LIhence8theaiare8uniquelydeterminedbrya.dsrc:2603advalg.texPropSositions3.4and3.6givrealreadyanindicationofhowtoconstructproSductsandco-proSducts.vProp`osition 3.7.src:2608advalg.texLffetnB(Midji2I)nCbeanCfamilyofRJ-moffdules. ThentherenBexistaproduct(Q UUMid;(pj\:URQMi, !uMjf jj%2URI))35andacffoproduct35(`Mid;(j\:URMj !!wQ&pMijj%2URI)).vPrffoof.#Rsrc:2615advalg.tex1.8De neڍcҷYu( Mi,:=URfa:IF``![i2I ^Midj8j%2IFչ:a(jӹ)=aj\2Mjf gNsrc:2618advalg.texandoppj Q:Q@hMi O ix!TMjf ,Тpj(a):=a(jӹ)=aj2Mjf . 8ItoqisopeasytocrheckopthatQMiJisanRJ-moSduleqwithqcomponenrtwiseopSerationsqandthatthepj שarehomomorphisms. If(fj Tչ:No 8!Mjf )hishafamilyofhomomorphismsthenthereisauniquemapf&:ދNo 8!Q(lMidڹ,fG(n)5=5(fidڹ(n)ji2I)sucrhthatpjf f}ǹ=5fj jforallj2I.Thefollorwingfamiliesareequal:(pjf fG(nY+n209))N=N(fj(n+n209))N=(fj(n)Y+Xfj(n209))=(pjfG(n)Y+pjfG(n209))N=N(pj(f(n)Y+fG(n209))),hence4fG(n+n209)K=Kf(n)+f(n209). qAnalogously4oneshorws5f(rSn)K=Krf(n). qThrus4f4isahomomorphismandQ?MiOisaproSduct.src:2631advalg.tex2.8De ne荍;aaLsMi,:=URfa:IF``![i2I ^Midj8j%2IFչ:a(jӹ)2Mjf ;a꨹with nitesuppSortdDgNsrc:2634advalg.tex(thenotionwith@ nitesuppffortmeansthatallbuta nitenrumbSerofthea(jӹ)'sarezero)andCj x:Mj x /!̟`)!Midڹ,1jf (aj)(i):=ijJaidڹ.IThenC`EMiwuQgMiUisCasubmoSduleandthejareLhomomorphisms.]Givren(fja%:Mj z!uN@jj2I).]De neLfG(a)=f(Pidai)=PfGi(ai)=Pfidڹ(ai).tThencf bisanRJ-moSdulehomomorphismanddwrehavefGidڹ(ai)UR=fidڹ(ai)chencefi,=URfidڹ.Ifgn9i,=URfiOforalli2I+thengn9(a)=g(Pidai)=Pgn9iai,=Pfi(ai)hencefQ=gn9.3Prop`osition 3.8.src:2648advalg.texLffetn(MidjiUR2I)obenafamilyoofsubmodulesoofM@.G|Thefollowingstatementsarffe35equivalent:!Tڠ7o]Projectiv9eTMoAdulesandGeneratorsWG33[o](1)%src:2651advalg.tex(M;(i,:URMi !uM@))35isacffoproduct35ofRJ-moffdules.(2)%src:2653advalg.texM6=URPi2I^Miand35(Pmi,=UR0=)8i2IFչ:mi,=0).R(3)%src:2655advalg.texM6=URPi2I^Miand35(Pmi,=URPm20RAi=)8iUR2IFչ:mi,=m20RAidڹ).(4)%src:2657advalg.texM6=URPi2I^Miand358iUR2IFչ:Mi\PUUjv6=i;j2I/y*Mj\=0.De nitionF#3.9.msrc:2663advalg.texIsKoneLoftheequivXalenrtconditionsofPropSosition3.8issatis edthenM/iscalledaninternal35dirffectsum꨹oftheMiOandwrewriteM6=URi2I ^Midڹ.Prffoof35ofPrffoposition353.8:bsrc:2670advalg.tex(1)=)(2):8UsethecommrutativediagramOqǴ׍LMjǴ׍_MВ{fd%ά-jHC*M=PMi=bX0fׁ @f @f @f @t>@t>RHӂǠ*FfeǠ?=t80HŸǠ*FfeǠ?9 wtGsrc:2677advalg.textoconclude=UR0andM6=PMidڹ.8IfPSmi,=0thenusethediagramOqǴ׍MjǴ׍ M {fd%ά-rjH!zjvkTׁ @T @T @T @#ԟ>@#ԟ>RH33hMk)Ǡ*Ffe\4Ǡ?  pkڷsrc:2680advalg.textoshorw0UR=pk#(0)=pk(Pmjf )=Pjfpkjf (mj)UR=Pjfjvk (mjf )=mk.>src:2683advalg.tex(2)=)(3):8trivial.src:2685advalg.tex(3)=)(4):8Letmi,=URPjv6=iYmjf .Thenmi,=UR0andmj\=0forallj%6=i:Csrc:2688advalg.tex(4)=)(2):8IfPSmj\=UR0thenmi,=Pjv6=iYmj\=02Mi\PUUjv6=i@ȴ>RHINǠ*FfeǠ?` fsrc:2694advalg.texbry-`fG(Pmidڹ):=Pqfi(mi). Thenfu`isawrellde nedhomomorphism-aandwehavefGjf (mj)=fG(mjf )#="fj(mj). 7FVurthermore^pfoisuniquely^odeterminedsincegn9j 4-=fj z=)gn9(Pmidڹ)=Pgn9(midڹ)UR=Pgidڹ(mi)UR=Pfidڹ(mi)UR=fG(Pmidڹ)=)fQ=g.,Prop`osition=%3.10.Tsrc:2703advalg.texLffetʆ(` UUMid;(jӑ:mMj 9!`&ܟi6=j66Mi))ʅbffeʆacoproductofRJ-modules.,[ThenC` UUMiis35aninternaldirffect35sumofthejf (Mj).Prffoof.#Rsrc:2709advalg.texjPisinjectivre=)MjP\԰D=fjf (Mj)=)K@xMjP\԰D=fjf (Mj)@`Mi {fd"`ά-H`ҁ Hīׁ HΫ܁ Hث H H H H H8džH8džjH`*ܟׁ @*ܟ @*ܟ @ *ܟ @ q\>@ q\>RHTNvǠ*FfeǠ?src:2717advalg.texde nesacoproSduct.8By3.8wrehaveaninternaldirectsum.SDe nition3.11.fsrc:2722advalg.texA!submoSduleM0iLN|iscalledadirffectܺsummandofNifthereisasubmoSduleM@20doURN+sucrhthatN6=MM@20Źisaninrternaldirectsum."k7o]34oAdv|rancedTAlgebra{P9areigis[o]Prop`osition3.12.src:2728advalg.texFor35asubmoffduleM6URNtthefollowingareequivalent:N獍(1)%src:2731advalg.texMtis35adirffect35summandofN@.(2)%src:2732advalg.texTherffe35ispUR2Hom۟R"n(:N;:M@)35with΍Ź(MʶNl32fd% ά-ߍ"M@)=idS+M: 32fd*Ѝά-m{Fp(3)%src:2737advalg.texTherffe35isfQ2URHom۟R"n(:N;:N@)withfG22 ]U=URf{4andfG(N)UR=M:Prffoof.#Rsrc:2743advalg.tex(1)<=)<(2):LetM1:=M}andM2N}with<N ۹=M1!M2.-=WVede nep=p1:N!nM1bryHލ{RMi{qNԁ{fd'Ѝά-6:"iH!Hij\ׁ @\ @\ @\ @2ܟ>@2ܟ>RHl4Mj8 Ǡ*Ffek<Ǡ?3 pjAsrc:2747advalg.texwhereij 6=UR0fori6=j{andij 6=id M8:i޹fori=jӹ.8Thenp11V=11 UZ=id M.src:2751advalg.tex(2)+=)(3):FVorf :=p:Nm!nNkwre+havefG22 =pp=p=fssince+p=id ..FVurthermorefG(N@)UR=p(N)=M+sincepissurjectivre.src:2756advalg.tex(3)%=)(1):LetM@20o=~RKe٩(fG).DWWVe rstshorwN6=~RMbX+!uM@20.TVakren~R2N@.Then%wrehavenpg=pffG(n)J+(nIf(n))'^withf(n)pg2pfM@.Sincef(nIJf(n))pg=pff(n)f22(n)pf=pg0'^wregetnfG(n)|2Ke[(f)=M@20^UsoO7thatNA`=|M0+ M@20.fNorwletO8n2M0\M@20.fThenfG(n)=0andnUR=fG(n209)forn20#2URN+hencen=fG(n209)=f22(n209)=f(n)=0.Problem3.1.src:2766advalg.texDiscussthede nitionandthepropSertiesofproductsofgroups.Problemu3.2.src:2770advalg.texShorwOCthatOBthetensorproSductoftrwoOBcommutativeK-algebrasOCisacoproSduct.Problem3.3.src:2775advalg.texShorwthatthedisjointunionoftwosetsisacoproSduct.3.2.Projectivemo`dules.src:2780advalg.texDe nition3.13.src:2781advalg.texAnRJ-moSdulePiscalledprffojectiveȹifforeacrhepimorphismf:M!Nandforeacrhhomomorphismg:ZPC \!NthereexistsahomomorphismhZ:PC \!MsucrhthatthediagramC_MIN 32fd()`ά- `fH ښ hȴׁ ȴ ȴ ȴ ӂ4>ӂ4> bYfPǠ*FfeǠ?  gwsrc:2786advalg.texcommrutes.Example[3.14.=/src:2790advalg.texAllPvrectorspacesQareprojective.Z=nZ(nUR>1)QisPnotaprojectivreZ-moSdule.Lemma3.15.src:2795advalg.texLffet35P=URi2I ^Pid.fiPisprojectivei allPid,iUR2I$are35projective.Prffoof.#Rsrc:2800advalg.texLet.PȹbSeprojectivre.WVeshowthatPi۹isprojective.Letf:M"e!\NnbSeanepimor-phismandgË:URPi,ӷ!) N+bSeahomomorphism.8ConsiderthediagramNK{ $Pi{Pq{fd)0ά-6ti{󎎍{wPiV{fd)0ά-;zXpiMNl32fd()`ά-]fH褍I>hi\ׁ @\ @ɽ\ @ӽ\ @ܟ>@ܟ>RH Ǡ*Ffe<<Ǡ? hH"JǠ*Ffe"|Ǡ? 'g#Ѡ7o]Projectiv9eTMoAdulesandGeneratorsWG35[o]src:2811advalg.texwherepi*andi*areprojectionsandinjectionsofthedirectsum,inparticularpidi/=DidP8:i.SincepXfWisanepimorphismpWthereisanh8߹:Pڦ1!-M|> !ލsrc:2827advalg.texSincef[issurjectivretherearehi :0Pi !M@,;i12IwithfGhi =gn9idڹ.ASincePisthecoproSductofthePithereisa(unique)hݹ:PJd/! M\withhi =hiforalli2I.ThrusfGhi =fhi =gn9iforalliUR2I+hencefGh=gn9.8SoPnisprojectivre.эProp`osition3.16.src:2836advalg.texLffet35PbeanRJ-module.fiThenthefollowingareequivalent퍍(1)%src:2839advalg.texPis35prffojective.(2)%src:2840advalg.texEachepimorphismfW/:/MP!ŰP9Zsplits,i.e.foreffachRJ-moduleMxandeachepimor-%phism35fQ:URM6!QPtherffeisahomomorphismgË:P]!MtsuchthatfGgË=id P1.(3)%src:2843advalg.texPis35isomorphictoadirffect35summandofafrffee35RJ-moduleRJX.Prffoof.#Rsrc:2849advalg.tex(1)=)(2):8ThediagramQ؍MPϢ$32fd(ά- fH $gaԟׁ aԟ aԟ aԟ T>T> bYPgǠ*Ffe4Ǡ?褍LidXP'ލsrc:2851advalg.teximpliestheexistenceofgXwithfGgË=URid P1¹.src:2853advalg.tex(2)j=)k(3):3LetUR:P!eRJP0bSekthefreemoSduleorverk(theset)P1withamap.5wThenthereisahomomorphismfQ:URRJP!ePnsucrhthatY8RbYUPbYRJP l{fd& @ά- H褍'lidP(|ׁ @(| @(| @(| @n>@n>RH Pt*Ǡ*Ffe\Ǡ?` Yf!ݍsrc:2857advalg.texcommrutes.ObviouslyyKfJissurjectivre.By(2)thereisyJahomomorphismgT:HPl!KRJPwithfGgË=URid P1¹.8By3.12PnisadirectsummandofRJP(uptoanisomorphism).src:2862advalg.tex(3)ފ=)(1): Letfevgo(fGe;p)=%(fGe)(p)=$f(e(p))=ev,V(f;ep)anddb(prr;f)(qn9)%=$(prS)f(q)%=p(rSf(qn9))=dbU(p;rSf)(qn9). WVealsocrheckthatdbAisabimoSdulehomomorphism:dbG(ep! fG)(qn9)_=`e(p)f(q)`=_e(pf(q))`=edb.(p fG)(qn9)anddb(p fe)(qn9)UR=pfe(qn9)=dbc(p f)e(qn9)._Lemma3.18.src:2908advalg.texThe35followingdiagrffamscommutePܔ33?,R R ;PƟ233PƟ2o3,32fdQC@ά-gۍK{-1[PƟ2 r E 0PLn R ;PƟ2{P7PƟ2 r E 0EH̟{fd/Pά-51 dbHUHJǠ*FfeU{|Ǡ?=3zevA 1HʶʟǠ*FfeǠ? Ϝ|33 iE^ E 0P33^P6̞32fdUά-gۍ^4{\ PLn R ;PƟ2 r E 0P{PLn R ;RH,{fd4ά- R¹1 evHǠ*FfeF,Ǡ?O P+db ^Y 1HzǠ*FfeǠ? g,ӍPrffoof.#Rsrc:2918advalg.texTheproSoffollorwsfromtheassoSciativrelaw:(1   dbJ)(fT  p gn9)(q)UR=(fT pgn9)(q)UR=fG(pgn9)(q)UR=f(pgn9(q))UR=f(p)gn9(q)UR=(f(p) gn9)(q)UR=(ev / 1)(fΖ p gn9)(q)[and[(db. 1)(p f qn9)UR=(pf qn9)=pfG(q)=(p fG(q))UR=(1 ev@)(p f q).xProp`osition3.19.src:2927advalg.tex(dualQFbasisLemma)8Lffet9PR bearightRJ-module.7jThenthefollowingareeffquivalent:(1)%src:2931advalg.texPis35 nitelygenerffated35andprffojective,(2)%src:2932advalg.tex(dualbasis)35Therffearef1;:::ʜ;fn2URHom۟R"n(PS:;RJ:)UR=PƟ2 and35p1;:::ʜ;pn2Pso35that~ŚmpUR=Xpidfi(p)<Ӎ%src:2936advalg.texfor35allpUR2P(3)%src:2937advalg.texThe35dualbffasishomomorphismMߍb+dbŭ:URPLn R ;PƟ   !r۹Hom3dR:(PS:;P:)M%src:2939advalg.texis35anisomorphism.%7o]Projectiv9eTMoAdulesandGeneratorsWG37[o]Prffoof.#Rsrc:2944advalg.tex(1)T=)T(2): LetPFbSegeneratedbryfp1;:::ʜ;pnPg.vfLetRJXFbSeafreerighrtRJ-moSduleorverthesetXFչ=URfx1;:::ʜ;xnPg.8Leti,:RJXF``!RbSetheprojectionsinducedbryM؍bYyXbYbRJXѡ4{fd$pά- WHSOӾi ׁ @ @ @ @Qd>@Qd>RHZRVǠ*FfeğǠ?SOW1޺(id ʤPp)isadualbasis,;bSecausePbpidfi(p)t=tdb?(Ppi fi)(p)t=id ʤPp(p)UR=p.src:2974advalg.tex(2)=)(1):8ThepigenerateP7sinceP@pidfi(p),k=pԹforallp,k2Pƹ. :dThrusPis nitelygenerated.FVurthermorethehomomorphismgx:>RJXM!=PVwithgn9(xidڹ)=piwjissurjectivre.LethUR:P!eRJXNbSe]'de nedbryh(p)UR=Pxidfi(p). Then]'gn9h(p)UR=p,yuhencePisadirectsummandofRJX,andconsequenrtlyPnisprojective.Remark-3.20.#src:2984advalg.texObservre0qthat0panalogousstatements0pholdforleftRJ-moSdules. 9TheproblemthatinthatsituationtrworingsR4andEndܩR<(:Pƹ)opSeratefromtheleftonPqisbSesthandledbry(considering(PcasarighrtEnd5\R((:Pƹ)2op-moSdulewhereEnd5\R((:Pƹ)2op hastheoppSositemrultipli-cation55givrenby5f7gË:=URgfG.WWVelearve5ittothereadertovrerifythedetails.XTheevXaluationanddualbasishomomorphismsareinthiscaseev:URR HP Er؟op8P2bR3URp fQ7!fG(p)2R HRR;anddbc:URPƟ2 r R ;P!eHom.߀R5(:PS;:Pƹ).Prop`ositionŒ3.21.src:2997advalg.texLffetRbeacommutativeringandP=jbeanRJ-module.3Thenthefollowingarffe35equivalent܍(1)%src:3000advalg.texRPis35 nitelygenerffated35andprffojective,(2)%src:3001advalg.textherffeg.ThenF=PY QasRJ-moSdules.So0Pɹand0Qareprojectivre.FVurthermore0QP_԰G=RJ.HenceFPm%԰ =P{R.SuppSoseP3wrere mfree./EvXaluatingallelementsofP3inagivenpSointp2Sן22 Hwegetthetangentplaneat@?>RH(wM DǠ*Ffe wǠ? *lhesrc:3094advalg.texde nesauniquehomomorphismhwithhf =fforallfǟ2I.FLetN=Imi(h).FConsider:M2]! MM=XN@.If7dN6=MxHthen6=0.SinceGisageneratorthereexistsanfcsucrhthatf6=0.'This$implies$h6=0$aconrtradictiontoN=Im(h).'HenceN=Mdsothathisanepimorphism.src:3101advalg.tex(2))=)(3):Let`t~G;!vR8rbSeanepimorphism.bSinceR8sisafree(modulehenceprojectivre,3.16impliesthatRisadirectsummandof`?Guptoisomorphism.src:3106advalg.tex(3)V=)(4):SinceRocis(isomorphicto)Vadirectsummandof`pIzGthereisp 3: 2`aI0G%!1Rwith.pt+=id>ϟR2b.2sLetp((gidڹ))=1andfi=pi:G!RJ.2sThen.1=p((gidڹ))=p(Pi(gi))=Ppidڹ(gi)UR=Pfidڹ(qi).src:3112advalg.tex(4)d~=)(1):,Assume(g:$Me"!N@)$6=0.bThend~thereisanm$2Mbwithd~gn9(m)6=0.De nefp:qRF!M[4bryQfG(1)=m,&;f(rS)r=rm.LetQfidڹ,&;qi+bePgivrenwithPfidڹ(qi)q=1.ThenQwehave0UR6=gn9(m)=gfG(1)=PgfGfidڹ(qi),sowrehavetheexistenceofahomomorphismfGfi,:URGn!1Mwithgn9fGfi,6=UR0.f(7o]40oAdv|rancedTAlgebra{P9areigis[o]h4.Ca32tegoriesandFunctors4.1.Categories.src:3136advalg.texIntheprecedingsectionswresawthatcertainconstructionslikreproSductscanbSeperformedfordi erenrtkindsofmathematicalstructures,DWVeharve:d(fG21 {)21FS=fandA(gn9fG)21FT=f21 {g21 ʵ.>TheArelationAofbSeingisomorphicbetrweenobjectsisanequivXalencerelation.Example 4.5.src:3233advalg.texInDtheDcategoriesSet,RJ-MoSd,kg-QVVec,Grer,Ab,Mon,cMon!,Ri,Field$ҋtheisomorphismsareexactlythosemorphismswhicrharebijectiveassetmaps.)7o]CategoriesTandfunctors41[o]src:3238advalg.texInTVoppthesetM}C=<`fa;bgwith0%n eufm10T1 d=<_f;;fag;fbg;fa;bggandwithT2 c=<`f;;M@gde nestrwo7di erenttopSological7spaces. ThemapfԱ=id :(M;T1)>!2(M;T2)7is7bijectivreandconrtinuous.Therinversermap,however,isrnotrcontinuous,hencerfisnorisomorphism(home-omorphism).֍src:3247advalg.texManry!owellknown!pconceptscanbSede nedforarbitrarycategories.WVearegoingtoinrtroSducesomeofthem.8Herearetrwoexamples.׍De nition4.6. src:3252advalg.tex(1)oAmorphismfQ:URAn!1BIuispcalledoamonomorphismif8C12URObC5;m8gn9;hUR2Mor5C(C5;A)UR:'x|0fGgË=URfh꨹=)g=URh (f2isleftcancellable)aW:src:3258advalg.tex(2)AmorphismfQ:URAn!1Biscalledanepimorphismif8C12URObC5;8gn9;hUR2MorOC(B;Cܞ): ݍxEgn9fQ=URhf2=)@gË=h (f2isrighrtcancellable)iv:De nitionh4.7.src:3266advalg.texGivrenO6A;BX2URC5.AnO7objectAm)m(B=inCltogetherwithmorphismspA 36:URAm)B!BsAfandfpB w$:(AJIBw!_Biscalleda(categorical)prffoductfofAandBifforevrery(test)objectT|2rC!andevrerypairofmorphismsf:T|.!Aandg:rT|.!Bthereexistsauniquemorphism(f;gn9)UR:T!eABsucrhthatthediagramH=bYq(T`f{,ׁ {, {, {, 4>4> H'W gׁ @ @ @ @2l>@2l>RA2AB|32fdpά!7NpX.A B,32fd@ά-!7DpX.BHZǠ*Ffe곌Ǡ?`f (fh;gI{) ލsrc:3280advalg.texcommrutes.src:3282advalg.texAnYobjectXE i2URCyiscalleda nalobjeffctifforevrery(test)objectT2URCythereexistsauniquemorphismeUR:T!eE(i.e.8MornݟCܜ(T;E)consistsofexactlyoneelemenrt).src:3287advalg.texAcategoryCJ);0of;1trwoobjectsAand;0B7inCfexiststhenitisuniqueuptoisomorphism.src:3297advalg.texIfthe nalobjectEinCݹexiststhenitisuniqueuptoisomorphism.ProblemZ4.1.Psrc:3302advalg.texLetnC}bSeamcategorywith niteproSducts..!Givreade nitionofaproSductofafamilyA1;:::ʜ;An (nUR0).8ShorwthatproSductsofsuchfamiliesexistinC5.׍De nitionJ:andJ9Remark4.9.˨src:3309advalg.texLetyzC,bSey{acategoryV. WThenC52op̹withthefollorwingdataObC52op %:=1ObpC5,Mor”Cmrop'7(A;B)1:=1MorgϟCՎ(B;A),andl4fJop g :=1gqf3de nesal3newcategoryV,thedual35cffategoryofC5.Remarko4.10.rjsrc:3316advalg.texAnryʷnotionexpressedincategoricalterms(withobjects,morphisms,andtheircompSosition)hasadual35notion,i.e.8thegivrennotioninthedualcategoryV.src:3320advalg.texMonomorphismsOfӹintheOdualcategoryC52op&areepimorphismsintheoriginalcategoryCandNconrverselyV.c,AM nalNobjectI?inthedualcategoryC52op iisaninitialobjeffctintheoriginalcategoryC5.׍De nitionLk4.11.ݝsrc:3327advalg.texThecffoproductùoftrwoobjectsinthecategoryCpisde nedtobSeaproductoftheobjectsinthedualcategoryC52op R.Remark4.12.src:3333advalg.texEquivXalenrttotheprecedingde nitionisthefollowingde nition.src:3336advalg.texGivrensA;BX2URC5.IAnobjectAqBinsC'togetherswithmorphismsjA 36:An!1AqBandsjB :B!nAcfqBb˹isa(categorical)coproSductofAĹandBifforevrery(test)objectT2URCzandevery*7o]42oAdv|rancedTAlgebra{P9areigis[o]pairofmorphismsfQ:URAn!1TandgË:BX !_7Tthereexistsauniquemorphism[f;gn9]UR:AqB!nTnsucrhthatthediagramHHq(T`f4ׁ @4 @4 @4 @{,>@{,>RH'W g2lׁ  2l 2l 2l >> bYAbY][AqB|{fdά-e jX.AbYbY B{fdPYάejX.BHZǠ*Ffe곌Ǡ?`f [€fh;gI{]isrc:3349advalg.texcommrutes.src:3351advalg.texThe UcategoryCis Vsaidtoharve U nitePScffoproductsifC52op isacategorywith nite VproSducts.InparticularcoproSductsareuniqueuptoisomorphism.D4.2.Functors.src:3358advalg.texDe nition4.13.src:3359advalg.texLetCݹandD?bSecategories.8LetFconsistof͍(1)%src:3361advalg.texamapObFC3URA7!F1(A)2ObDUV;(2)%src:3363advalg.texafamilyofmaps1=s(FA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(A);F(B))jA;BX2URC5)|ԍ1 [orFl(FA;B]:URMorOC(A;B)UR3fQ7!FA;B(fG)2MorODڲ(F1(B);F(A))jA;BX2URC5)]src:3370advalg.texFiscalledacffovariant[contravariant$D]functorif΍(1)%src:3373advalg.texFA;A<(1A)UR=1F((A)C@forallAUR2ObC5,(2)%src:3375advalg.texFA;C(gn9fG)UR=FBd;CJJ(g)FA;B(fG)forallA;B;C12URObC5:%src:3378advalg.tex[FA;C(gn9fG)UR=FA;B(f)FBd;CJJ(gn9)forallA;B;C12URObCݹ].src:3382advalg.texNotation:8WVewrite͍src:3384advalg.texʍAUR2C-~BinsteadofyAUR2ObC!fQ2URC-~Binsteadofi~fQ2URMorOC(A;B);F1(fG)-~Binsteadof{DFA;B(fG).#Examples4.14.src:3394advalg.texThefollorwingde nefunctors(1)%src:3396advalg.texId:URSet`!(k?Set8;(2)%src:3397advalg.texFVorget:8RJ-MoSd1!,CSet<;(3)%src:3398advalg.texFVorget:8Riwi!%HAb5MorS著CY]h(A;B)UR3fQ7!MorOC(XJg;fG)2MorOSet$ٓ(Mor5C(XJg;A);Mor5C(X;B));%src:3435advalg.texwithvMorCӹ(XJg;fG)-:Mor*Ct(X;A),3g?f7!fGg?e2Mor*Ct(X;B)vorMorCӹ(X;fG)(gn9)-=fgPisva%cffovariant35functorMori2C(XJg;-33).(2)%src:3438advalg.texLffet35XF2URC5.fiThen獍ݎOb,C3URA7!MorOC(A;X)2ObSeth?VMorTSCZX(A;B)UR3fQ7!MorOC(f;X)2MorOSet$ٓ(Mor5C(B;X);Mor5C(A;X))%src:3443advalg.texwithgMorޟC (f;X):MorٟCZ(B;X)3g%7!gn9f2Mor؟CZ(A;X)gorMorޟC (f;X)(gn9)=gfisga%cffontravariant35functorMori2C(-;X).Prffoof.#Rsrc:3450advalg.tex(1)UMorPRC!(XJg;1A)(gn9)=1Ag /=g=idW(gn9);Mor5C(XJg;fG)Mor5C(XJg;gn9)(h)=fGgh=Mor5C(XJg;fGgn9)(h).src:3454advalg.tex(2)analogouslyV.{iRemark_ 4.16.src:3458advalg.texTheprecedinglemmashorwsthatMorCq(-;-)isafunctorinbSothargumenrts.Afunctorintrwoargumentsiscalledabifunctor.3WVecanregardthebifunctorMor8Cչ(-;-)asacorvXariantfunctor+Mor(C$(-;-)UR:C5 op @C"!wfSet' :src:3463advalg.texTheuseofthedualcategoryremorves thefactthatthebifunctorMor8CĹ(-;-)isconrtravXariantinthe rstvXariable.src:3467advalg.texObrviouslyRthecompSositionoftwofunctorsisagainRafunctorandthiscompSositionisasso-ciativre.8FVurthermoreforeachcategoryCݹthereisanidentityfunctorIdC.src:3472advalg.texFVunctors&oftheformMor\Cb(XJg;-)resp.Mor-CF(-;X)arecalledrffepresentable functors&(corvXariantresp.8conrtravXariant)andX+iscalledtherffepresenting35object꨹(seealsosection5).4.3.NaturalTransformations.src:3481advalg.texDe nitiony4.17.usrc:3482advalg.texLet6 Fc:URC"!wfD^andG :C"!wfD^bSetrwo6functors.A5naturffal/transformationorWaWfunctorialemorphism'UR:Fc!BG+isaWfamilyofmorphisms('(A)UR:F1(A)n!1G.(A)jA2C5)sucrhthatthediagramMnF1(B)fG.(B)ԁ32fd/pά- Vp)Y'(B)HF1(A)HG.(A)'{fd0Ѝά-P߅'(A)H“Ǡ*FfeLǠ?` F1(fG)HڟǠ*Ffe Ǡ?`G.(fG)Tsrc:3489advalg.texcommrutesforallfQ:URAn!1BinC5,i.e.8G.(fG)'(A)UR='(B)F1(f).Lemma4.18.src:3494advalg.texGiven35cffovariantfunctorsFc=URIdSet%:URSet`b!(Set==and捒|G =URMorOSet$ٓ(Mor5Set!A(;A);A)UR:Set`b!(Setsrc:3498advalg.texfor35asetA.fiThen'UR:Fc!B"Gcwith['(B)UR:BX3b7!(Mor5Set!A(B;A)3fQ7!fG(b)2A)2G.(B)src:3501advalg.texis35anaturffaltransformation.,17o]44oAdv|rancedTAlgebra{P9areigis[o]Prffoof.#Rsrc:3505advalg.texGivrengË:URBX !_7Cܞ.8ThenthefollowingdiagramcommutesR )Hr BHacMorŗ`Set夹(Mor5Set!A(B;A);A){fd,Eά-PA'(B)rC֯Mor SetZ(Mor5Set!A(C5;A);A)32fd,ά-'(Cܞ)Hw>Ǡ*Ffewq4Ǡ? l$gH쬂Ǡ*FfeߴǠ?`4Mor1Setu(Mor5Set!A(gn9;A);A)src:3515advalg.texsinceِʍ^*'(Cܞ)F1(gn9)(b)(fG)UR='(C)gn9(b)(fG)=fgn9(b)='(B)(b)(fgn9)?=UR['(B)(b)Mor5Set#?(gn9;A)](fG)=[Mor5Set!A(Mor5Set(g;A);A)'(B)(b)](fG):f[k|QLemmaP4.19.@src:3526advalg.texLffets;f:A l!foB@beamorphisminC5. &{ThenMor8C!(f;-33):MorC"#(B;-)fk!W-Mor)*C.(A;-33)givenbyMorўC ?](f;Cܞ):Mor&C |(B;Cܞ)3g^7!gn9f82Mor&C |(A;Cܞ)isanaturffaltrffansformation35ofcovariantfunctors.src:3532advalg.texLffet;fV:At!B2AbeamorphisminC5.{ThenMor8C:(-35;fG):MorDCG(-;A)t!Mor+C1'@(-;B);givenbyMor5C(C5;fG)p:MorCd(C;A)3g7!fGg2MorCd(C5;B)Aisanaturffaltransformationofcontravari-ant35functors.|RPrffoof.#Rsrc:3540advalg.texLethUR:C1K{!Cܞ20bSeamorphisminC5.8ThenthediagramsN" Mor9CK(B;Cܞ20׹)έMorC"ri(A;Cܞ20׹)32fd0Cά-knկxMorX.Cߙ(fh;C-:0B})HjMorCg(B;Cܞ)H 5MorkǟC#ن(A;Cܞ)Ѱ<{fd30ά-`\Mor`X.C8}(fh;C)H#JǠ*FfeV|Ǡ?`MorWX.C(Bd;h)H%ʟǠ*Ffe%Ǡ?`*w|Mor9X.C>(A;h)jɍsrc:3544advalg.texandFOMorCde(C5;A)҈MorC#vD(C5;B)С32fd3ά-W`SMor嬓X.C(C;f)HMor;C(Cܞ20;A)HMorM۟C!(Cܞ20;B)d{fd0Cά-nԾMor=X.C(C-:0B};f)HǠ*FfeLǠ?`Mor5X.C (h;A)H%7Ǡ*Ffe%j̟Ǡ?`*LMor9uX.C>Mm(h;Bd)ۍsrc:3548advalg.texcommrute.|Remark(G4.20.Ņsrc:3552advalg.texTheOcompSositionofNtrwoOnaturaltransformationsisagainanaturaltransfor-mation.8Theidenrtityid LF(A)UR:=1F((A)C@isalsoanaturaltransformation.De nition+4.21.Dsrc:3558advalg.texAqnaturalqtransformation';1:;2FlB!G'iscalledanaturffalWisomorphismifthere~existsanaturaltransformation :QHGv!!rKFsucrhthat'pq =QHidGand }p'QH=idFz(.Thenaturaltransformation Xisuniquelydeterminedbry'.8WVewrite'21ι:=UR n9.src:3565advalg.texA\functorFissaidtobSeisomorphictoafunctorG@ifthereexistsanaturalisomorphism'UR:Fc!BG..|QRemarkJ4.22.1src:3570advalg.texTheݨisomorphismsݧgivreninTheorem1.22forR ;MS,tS 1VNT,uandT ~PU arenaturalisomorphisms:(1)%src:3574advalg.texAssoffciativity0Law: >:(M*~ S }N@) T PPv԰^=|M* S |(N TPƹ)Gwith ((m n) p):=%m (n p);(2)%src:3578advalg.texLffaw35oftheLeftUnit:8UR:R R ;MP6԰=@M+with(r6 m):=rSm;(3)%src:3580advalg.texLffaw35oftheR2ightUnit:8UR:M SȊSP)԰!=M+with(m rS)UR:=mr;(4)%src:3582advalg.texSymmetryCLffaw:o:URMO jNP6԰=@N M&forK-moSdulesMandNwithW(mk jn)UR:=n jm;-F7o]CategoriesTandfunctors45[o](5)%src:3585advalg.texInner35Hom-Functors:eQ}UR:Hom۟Sr}-]%T+|(:PLn R ;M:;:N:)PUR԰n:=Hom(ySr}-]%R8(:PS:;:HomyT!(M:;N:):)%src:3588advalg.texwith(fG)(p)(m)UR:=f(p m)andP. Ë:URHom۟Sr}-]%T+|(:PLn R ;M:;:N:)PUR԰n:=Hom(yR -^T9xֹ(:M:;:HomyS i(:PS;:N@):)%src:3591advalg.texwith n9(fG)(m)(p)UR:=f(p m)forbimoSdulesR ;MT,S NT,andS PR.Probleml4.2.src:3597advalg.tex(1)dLetF1;G :URC"!wfDSbSedfunctors. RShorwthatanaturaltransformation'UR:F!nGֹisanaturalisomorphismifandonlyif'(A)isanisomorphismforallobjectsAUR2C5.src:3602advalg.tex(2)I(Let(A``B;pA;pBN>)I(bSetheproductofAandI)B.inC5. ThenthereisanaturalisomorphismeMor(-;AB)PUR԰n:=Mor%5C*(-;A)Morय़CNd(-;B):src:3607advalg.tex(3)zLetC2bSeacategorywith{ niteproducts.&FVoreacrhobjectA{inC2showthat{thereexistsamorphismA 36:URAn!1AK]Asatisfyingp1A 36=UR1A=p2A.)SShorwthatthisde nesanaturaltransformation.8Whatarethefunctors?src:3613advalg.tex(4)LetCMObSeacategorywith niteproSducts.G5Shorwthatthereisabifunctor- "- :CQ"C!ZCsucrhthat(-@-)(A;B)istheobjectofaproSductofAandB. WVedenoteelemenrtsintheimageofthisfunctorbryABX:=UR(-P-)(A;B)andsimilarlyfgn9.src:3620advalg.tex(5)Withthenotationoftheprecedingproblemshorwthatthereisanaturaltransformation (A;B;Cܞ)UR:(AB)CP1԰Jع=ܙA(BCܞ).Shorw@that?thediagram(cffoherenceorconstraints)B4卍((AB)Cܞ)D(A(BECܞ))Dp)T:2fdEHά-̯{- (A;Bd;C)1i A((BECܞ)DS) O:2fdEHά-̯) (A;BdC;D@󹬟>RHT_M:ڟǠ*Ffe Ǡ?` f src:3750advalg.tex(2)Givren~moSdulesMR andRN@.|De neFc:URAb0!'oSet:bryF1(A)UR:=Bil.R"(M;N;A).}Then~F0isaLcorvXariantLfunctor.CALrepresentingobjectLforF}isgivrenbyLthetensorproSduct(M$ R[N; UR:M#O?N1K:!<M R BN@)fwiththeepropSertrythatforall(A;f8ʹ:M#O?N1K:!<A)thereexistsauniquegË2URHom(M R ;N;A)sucrhthatF1(gn9)( )UR=Bil.R"(M;N@;gn9)( )UR=g =fN{ȀMN{M R ;N0{fd Uά- H`޵ftׁ @t @t @t @@>@@>RHpA:F"Ǡ*FfeyTǠ?' +g/t͠7o]w~Represen9tableTandAdjointF:unctors,theYonedaLemman>47[o]src:3760advalg.tex(3)GivrenaK-moSduleVp.kDe neFW :%K-Alg!.BSetCbryF1(A):=Hom(V;A).kThenF,isaxcorvXariantyfunctor.QABrepresentingobjectxforF특isgivrenbyxthetensoralgebra(Tƹ(Vp);s:V Ç!Tƹ(Vp))withthepropSertrythatforall(A;fU: V Ç!A)thereexistsauniqueg{2Mor5Alg"Β(Tƹ(Vp);A)sucrhthatF1(gn9)()UR=Hom(V;gn9)()=g=fNHbVHETƹ(Vp)V {fd!wЍά-H`:MfV ׁ @V @V @V @񜌟>@񜌟>RH˱A:Ǡ*FfeǠ?'lg5src:3769advalg.tex(4)^Givren^aK-moSduleVp.De neFK:K-cAlg0!J0!1d޹SetF4by^F1(A):=Hom5(V;A).Then^FisacorvXariantfunctor.ArepresentingobjectforFJisgivrenbythesymmetricalgebra(S׹(Vp);?:V X!S׹(Vp)) withthepropSertrythatforall(A;f`\:^V X!A)thereexistsauniqueg2Mor5cAlg&(S׹(Vp);A)sucrhthatF1(gn9)()UR=Hom(V;gn9)()=g=fHďVHS׹(Vp)у${fd!ά-H`gefу$ׁ @ۃ$ @$ @$ @ɤ>@ɤ>RHA:ҟǠ*FfeǠ?'g5src:3778advalg.tex(5)GivrenaK-moSduleVp.8De neFc:URK-Alg2kK!*JSet>ݮbyM=F1(A)UR:=ffQ2Hom(V;A)j8vn9;v 02URV¹:fG(v)f(v 0@ɤ>RHA:ҟǠ*FfeǠ?'gsrc:3790advalg.tex(6)GivrenaK-moSduleVp.8De neFc:URK-Alg2kK!*JSet>ݮby"osxF1(A)UR:=ffQ2Hom(V;A)j8vË2V¹:fG(vn9) 2V=0g:5src:3794advalg.texThenLF5]isacorvXariantLfunctor.AFrepresentingobjectforF5]isgivenbytheexterioralgebra(E(Vp);{:zVv!E(V))amwiththepropSertrythatalforall(A;fgz:zVv!A)suchthatfG(vn9)22~={0forallvË2URVqthereexistsauniqueg2URMorOAlg&#(E(Vp);A)sucrhthatF1(gn9)()UR=Hom(V;gn9)()=gn9UR=fH)H&VHE(Vp){fd ά- {H`-fׁ @ @ @ @`l>@`l>RHA:eǠ*Ffe̟Ǡ?'KLgsrc:3802advalg.tex(7)LetKbSeacommrutativering.LetX-q2;Set6beaset.Fl:;K-cAlg Q"kr!3cSetC,PF1(A):=Map+Q(XJg;A)isacorvXariantfunctor. ArepresentingobjectforFisgivenbythepSolynomialring(K[X];R:QX`!KK[X])withthepropSertryV,ݼthatforall(A;fP:RX_!KA)thereexistsa0-7o]48oAdv|rancedTAlgebra{P9areigis[o]uniquegË2URMorOcAlg)(K[X];A)sucrhthatF1(gn9)()UR=Map(XJg;gn9)(x)=g=fIڍH-XHK[X]U{fd!ά-c3H`ڣfѿDׁ @ۿD @D @D @ğ>@ğ>RH4A: Ǡ*Ffe>$Ǡ?'gesrc:3810advalg.tex(8)6aLet6bKbSeacommrutativering.  Let6aX{x2SetbSeaset.  F:K-Algg !/ Set?\K,PF1(A):=Map+Q(XJg;A) is acorvXariantfunctor.!UArepresenting objectforFisgivrenby thenoncommuta-tivrepSolynomialring(KhXi;y:yXk!1KhXi)withthepropertryV,Fthatforall(A;f:yXk!1A)thereexistsauniquegË2URMorOAlg&#(KhXi;A)sucrhthatF1(gn9)()UR=Map(XJg;gn9)(x)=g=fHyXHvKhXiѡ4{fdά-sH`%f ׁ @ @ @ @Qd>@Qd>RHA:VǠ*FfeğǠ?'49[o]morphism ZAg :AX i! OAY \(withFX(Ag)(aX)=Fg(AYP)(aY)) Zandthe Yfollowingidentitieshold35A1X.X6=UR1AX.X ;Ahg A=AheAg.fiSowegetacffovariantfunctorD3XF@!AX r2C5.{Prffoof.#Rsrc:3882advalg.texChoSoseg;arepresenrtingobjectg<(AX;aX)g;forFX ĹforeachXΞ2D(bytheaxiomofcrhoice).8ThenthereisauniquemorphismAg*P:URAX r f!AY ;ewithFX(Ag)(aX)UR=Fg(AYP)(aY)UR2FX(AYP);src:3885advalg.texforL2eacrhgË:URXF``!Y袹bSecauseFg(AYP):FY(AY)n!1FX(AY)L1isL2givren.WVehaveFX(A1)(aX)UR=F1(AX)(aX) =aX( =FX(1)(aX)NhenceMA1 ʅ= 1,andFX(Ahg )(aX)=Fhg (AZ8)(aZ)=Fg(AZ8)Fhe(AZ)(aZ)5=5Fg(AZ)FYP(Ahe)(aY)5=FX(Ah)Fg(AYP)(aY)5=5FX(Ah)FX(Ag)(aX)5=FX(AheAg)(aX)henceAhAg*P=URAhg forgË:XF``!Yandh:Y M!`ZFinDUV.\d{~Corollaryzu5.6.Ysrc:3897advalg.tex(1),Map}(XJg;M@)PA԰)=WHom*aR1(RJX;M@),is-anaturffaltransformation,inM(andin35X!).fiInpffarticularSetC3URXF7!RJX2RJ-LMoSdis35afunctor.src:3901advalg.tex(2)vvBilORC>(M;N@;A)P԰Ĺ=Hom)r(M\ R N;A)vvisvuanaturffaltransformationvuinA(andin(M;N@)2MoSd~-RRJ-LMoSd^).fiIn35pffarticularMoSd-IRRJ-LMoSd 3URM;N67!M r= N2Abވis35afunctor.src:3907advalg.tex(3)35RJ-LMoSd^-S]S- MoSdd- T3UR(M;N@)7!M SȊN62RJ-LMoSd^-Tis35afunctor.5.2.TheYonedaLemma.src:3913advalg.tex{~Theorem=5.7.src:3914advalg.tex(YoneffdazLemma)zLetC-bezacategory. )= n9(B)(fG)"|=UR n9(B)Mor5C(A;fG)(1A)=F1(f) n9(A)(1A)=F1(f)n9( ):KRemarkC5.10.src:4005advalg.texBythepreviouscorollarytherepresenrtingobjectAisuniquelydetermineduptoisomorphismbrytheisomorphismclassofthefunctorMor Cd(A;-).Prop`ositionq5.11.src:4011advalg.texLffet9G:;C ~mID;!3Set.?bea9covariantbifunctor9suchthatthefunctorG.(C5;-33)l:Da.!Set*is^vrffepresentable^wforallC 2lC5..Thentherffeexistsacffontravariant^wfunctorF5:C,!"D+suchthatGP԰Ӣ=Mor'JD.(F1-dF;-33)holds.IFurthermorffeFisuniquelydeterminedbyGup35toisomorphism.Prffoof.#Rsrc:4020advalg.texFVor6eacrh6Cg2C*choSosean6objectF1(Cܞ)2DKandan6isomorphismC :G.(C5;-)P԰׹=Mor5D`(F1(Cܞ);-).Givrenf:C;!hOC20finCn޹thenletF1(fG):F(Cܞ20׹)Ҝ!F(Cܞ)bSetheuniquelydeterminedmorphism(brytheYVonedaLemma)inD?suchthatthediagramOW/G.(Cܞ20;-)Mor4D`(F1(Cܞ20׹);-)ƈL32fdpά-ΧLCjPG0HG.(C5;-)HfMorD}(F1(Cܞ);-)l{fd ά-iX.CHǠ*FfeLǠ?`eGv(fh;-)H Ǡ*Ffe LǠ?`Mor!c(F((f);-)xsrc:4028advalg.texcommrutes.BBecauseoftheuniquenessofF1(fG)andbSecauseofthefunctorialityofG6itiseasy)toseethatF1(fGgn9)z=F(gn9)F(fG))andF(1C)z=1F((C)uhold)and*thatFJ:isaconrtravXariantfunctor.src:4033advalg.texIfOF120:iC(!ϓDisOgivrenwithGP԰=WMor'DTD.(F120J-;-)then:Mor7fD ɹ(F1-;-)Ph԰P=X(Mor&%D-݈(F120J-;-).h/Hencebry3Lthe3MYVonedaLemma n9(Cܞ):F1(C)P԰߹=EF120J(C)3Mis3LanisomorphismforallC2C5.Withtheseisomorphismsinducedbrythediagram3β7o]w~Represen9tableTandAdjointF:unctors,theYonedaLemman>51[o]src:4039advalg.texFӍeMor{'DwU(F120J(Cܞ20׹);-)$=Mor9s D@p(F1(Cܞ20׹);-)32fde ά-knMorFu( I{(C-:0B});-)HgYMor|Dq(F120J(Cܞ);-)H%-Mor:*DB)(F1(Cܞ);-){fdhά-`FMorY( I{(C);-)H*Ǡ*FfeH\Ǡ?nX6Morg(F(-:0~(f);-)HI*Ǡ*FfeJ,\Ǡ?`NMor^7s(F((f);-)src:4048advalg.texcommrutes.8HencethediagramMΌF1(Cܞ)^F120J(Cܞ)ڜ32fd!ά-W` I{(C)H{F1(Cܞ20׹)HBF120J(Cܞ20׹){fdά-n4 I{(C-:0B})HjǠ*FfeǠ?n)F(-:0~(f)HAǠ*FfeuǠ?`'F((f)src:4052advalg.texcommrutes.8Thus Ë:URFc!BF120isanaturalisomorphism.e '5.3.Adjointfunctors.src:4058advalg.texLDe nition5.12."src:4059advalg.texLet.`Cᖹand.aDbSecategoriesandF:ȗC{W!]DandG~Ź:D7x!CᕹbSecorvXariantfunctors.F_isOcalledNleftadjointtoGT}andGT|rightadjointtoF`ifthereisanaturalisomorphismofbifunctorsUR:MorODڲ(F1-;-)n!1Mor).C/g(-;G.-)fromC52op @D?toSet{NatR$$(Mor5D`(F1-dF;-33);Mor5C(-35;G.-c))UR37!(-;F1-dF)(1F(- q)2Nat(Id =C1;G.F1):src:4099advalg.texFurthermorffeRNatf(F1G.;Id ;C1)UR3 7! -35F-2Nat(Mor5C(-35;G.-c);Mor5D^(F-dF;-33))src:4102advalg.texis35abijeffctivemapwithinversemap?oNatRJ(Mor5C(-35;G.-c);Mor5D^(F1-dF;-33))UR3 Ë7! n9(G.-;-33)(1Gv- 5)2Nat(F1G.;Id ;C1):LPrffoof.#Rsrc:4108advalg.texThenaturaltransformationG.--visde nedasfollorws.fQGivenCw2"C5,=Dvg2DO$andfI2JKMorHD ϫ(F1(Cܞ);DS)zthenzlet(G.--)(C5;DS)(fG)JJ:=JKG.(f)(Cܞ):JKC&@s!G.F1(C)c!"G.(DS).ItziseasytocrheckthepropSertiesofanaturaltransformation.src:4114advalg.texGivrenthenoneobtainsbryapplyingthetwomapsG.(1F((C)L)(Cܞ)UR=GF1(1C)(Cܞ)=(C).Givrenoneobtains̍ʍQG.(fG)((C5;F1(Cܞ))(1F((C)L)UR=MorOC(C;G.(fG))(C;F1(Cܞ))(1F((C)L)Q=UR(C5;DS)Mor5D^(F1(Cܞ);fG)(1F((C)L)=(C5;DS)(fG):4x7o]52oAdv|rancedTAlgebra{P9areigis[o]src:4122advalg.texSothetrwomapsareinrversesofeacrhother.src:4124advalg.texThesecondpartofthelemmaisprorvedsimilarlyV. M׍Prop`osition5.16.src:4128advalg.texLffete4oUR:MorODڲ(F1-dF;-33)!Mor*GC/͹(-35;G.-c)3=and3 Ë:MorOC(-;G.-c)!Mor*GD1q(F1-dF;-33)src:4132advalg.texbffe5naturaltransformationswithassociatednatural6transformations(byLemma5.15)UR:IdCfk!G.FdFrffesp.fi UR:F1G q!?Id |D'.nsrc:4136advalg.tex(1)35Thenwehave Ë=URid Morx(-;Gv-R)8 uifandonlyif(G25Gp `!-G.F1G2Gv p `!G)UR=id G"S.Usrc:4141advalg.tex(2)35Furthermorffewehave n9UR=id Morx(F(-];-35)9iTif35andonlyif(F2)F(pc ۼ!1F1G.F2s Fpc ۼ!F)UR=id F~2.M֍Prffoof.#Rsrc:4148advalg.texWVeget0UTʍmyG. (DS)G(D)UR=G. (D)(G(D);F1G(D))(1F(Gv(D S<-6:URS s4MURn!1RMisleftadjoinrttoHomd1R#WĹ(M;-)UR:R HMn!1SM.8DeterminetheassoSciatedunitandcounit.src:4190advalg.tex(2)h7Shorwthatthereh8isanaturalisomorphismMap(A%B;Cܞ)P+ ԰C=jMap(ֻ(B;Map+O(A;Cܞ)).De-terminetheassoSciatedunitandcounit.src:4194advalg.tex(3)ShorwthatthereisanaturalisomorphismK-Alg(KG;A)P԰=Gr\(G;U@(A))whereU(A)isthe!group!ofunitsofthealgebraAandKGisthegroupring(seeSection12).2DeterminetheassoSciatedunitandcounit.src:4203advalg.tex(4)UseSection12toshorwthatthereisanaturalisomorphismRC8?K-Alg(U@(g);A)PUR԰n:=Lie-Alg5(g;A LGع):esrc:4209advalg.texDeterminethecorrespSondingleftadjoinrtfunctorandtheassociatedunitandcounit.5.4.Universalproblems.src:4215advalg.texM׍De nition+p5.19.src:4216advalg.texLet^G:ΎD#=n! CbSeacorvXariant^functor. Gݹgeneratesa^(cffo-)universalprffoblem꨹afollorws:src:4219advalg.texGivren C,#2OC5.FindanobjectF1(Cܞ)2Dicandamorphism:C,#E!5G.(F1(Cܞ))inCBsucrhthatforeacrhobjectD2^Dandforeacrhmorphismf:^C;U!G.(DS)inC9thereisauniquemorphismgË:URF1(Cܞ)n!1D>6inD?sucrhthatthediagramIՍHCHIG.(F1(Cܞ))̣{fdά-XH`եfׁ @ @ @ @>@>RHgG.(DS) JǠ*Ffe@|Ǡ?`Gv(gI{)57o]w~Represen9tableTandAdjointF:unctors,theYonedaLemman>53[o]src:4225advalg.texcommrutes.src:4227advalg.texAdpair(F1(Cܞ);)thatsatis estheabSorveconditionsiscalledauniversal՛solutionofthe(co-)univrersalproblemde nedbyGֹandCܞ.src:4231advalg.texLetFc:URC"!wfD?bSeacorvXariantfunctor.8Fgeneratesauniversal35prffoblemafollorws:src:4234advalg.texGivrenD@$2DUV.FindanobjectG.(DS)2C"andamorphism]:F1(G.(DS))!!D-{inD/Csucrhthat{foreacrhobjectC'2K C.9andforeachmorphismf :K F1(Cܞ)d!DΒin{DZthere{isauniquemorphismgË:URC1K{!G.(DS)inCݹsucrhthatthediagramJQF1G.(DS) hD:\32fdMά-H`,f} ׁ @} @} @} @Ì>@Ì>RH( F1(Cܞ)zǠ*FfeDǠ?`zF((gI{)src:4240advalg.texcommrutes.src:4242advalg.texAKpair(G.(DS);ǹ)thatsatis estheaborveconditionsiscalledauniversalZsolutionoftheunivrersalproblemde nedbyFandDS."Prop`osition5.20.src:4248advalg.texLffetQFy:iC@!4sDbeleftadjointtoQ~GC:hDI*!֔C5.GThenF1(Cܞ)andtheunitu=(Cܞ):CQ!-G.F1(C)DSformaDTuniversalsolutionforthe(cffo-)universalproblemDSde nedbyGcand35Cܞ.src:4253advalg.texFurthermorffeȭG.(DS)andthecounit+=j (DS)j:F1G.(D)І!:D;formȭauniversalsolutionforthe35universalprffoblemde nedbyFdFandDS.Prffoof.#Rsrc:4259advalg.texBy1Theorem5.161themorphisms^:_Mor\D S(F1-;-)!JMor*GC0Z(-;G.-)and1 <:Mor[Cr(-;G.-)!Mor*QD1(F1-;-)SareTinrversesofTeachother. UsingunitTandcounittheyarede nedas(C5;DS)(gn9)3=3G.(g)(Cܞ)resp. (C;DS)(fG)3= (D)F1(fG).Henceforeacrhf{:3CN)!]G.(D)thereisauniquegË:URF1(Cܞ)n!1D>6sucrhthatfQ=UR(C5;DS)(gn9)=G.(g)(Cܞ)=G.(g).src:4268advalg.texThesecondstatemenrtfollowsanalogouslyV.:Remark=5.21.8src:4272advalg.texIf GX:D !OCUandCT2CUaregivrenthentheunivrersalsolution(F1(Cܞ);:C!nG.(DS))+canbeconsideredasthebest(co-)approrximation+oftheobjectC2inCɹbyanobjectD>6inD?withthehelpofafunctorG..8TheobjectD2URDturnsouttobSeF1(Cܞ).src:4279advalg.texIf=Fc:URC"!wfDandD2Daret(C5;G.(DS20!ǹ))#Ğ32fdc:Ѝά-W`,5Gv(-)Hf6Mor{J3D(F1(Cܞ);DS)H%8OMor:nLC? (C5;G.(DS)){fdf Pά-`,5Gv(-)H~bǠ*FfeǠ?`R7(MoraX.Dh(F((C);h)HJbbǠ*FfeJǠ?`OHMor^X.Ccx5(C;Gv(h))67o]54oAdv|rancedTAlgebra{P9areigis[o]src:4309advalg.texcommrutesforeachhUR2MorOD @(DS;D20!ǹ).8Infactwehavee8 MorM< CRȹ(C5;G.(h))(G(gn9))UR=G.(h)G(g)UR=G.(hg)=G.(Mor5C(F1(Cܞ);h)(g)):src:4312advalg.texHence9forallC2B+C7the9functorMornC ܾ(C5;G.(-)):D !:Set3}induced9brythebifunctorMor5C(-;G.(-))B1:C52op  hD!ESet+isurepresenrtable.]ByuTheorem5.11thereisafunctorFsB:B1C!nD?sucrhthatMor Cd(-;G.(-))PUR԰n:=Mor%5D,[(F1(-);-).src:4318advalg.texThesecondstatemenrtfollowsanalogouslyV.:Remark[05.23.ajsrc:4322advalg.texOnencanmcrharacterizethepropSertiesthatGP7: D_!C\(resp.u1F:CM>f!DUV)mrust2haveinorderto2pSossessaleft(right)adjointfunctor.OneoftheessentialpropSertiesforbthisbisthatG׹preservreslimits(andthruspreservesbdirectproSductsanddi erencekrernels).Prop`osition<5.24.Tsrc:4330advalg.texTheT2cffonstructionoftensoralgebrasTƹ(Vp)de nesafunctorT4.:iK-35MoSdfk!K-35AlgXthat35isleftadjointtotheunderlyingfunctorU6:URK-Algzc!+6K-MoSd.Prffoof.#Rsrc:4336advalg.texFVollorwsfromtheuniversalpropSertyand5.22.3=Prop`osition55.25.src:4340advalg.texThecffonstructionofsymmetricalgebrasS׹(Vp)de nesafunctorSD:K-35MoSdf!m!0%K-35cAlgthat35isleftadjointtotheunderlyingfunctorU6:URK-cAlg!C!0oK-MoSd.Prffoof.#Rsrc:4346advalg.texFVollorwsfromtheuniversalpropSertyand5.22.3=7*7o]LimitsTandColimits,ProAductsandEqualizersϪ55[o]U&ǹ6.d*KLimitsandColimits,ProductsandEqualizers6.1.Limitsofdiagrams.src:4364advalg.texLimitconstructionsareavreryimpSortanttoolincategorytheoryV.WVewillinrtroSducethebasicfactsonlimitsandcolimitsinthissection.De nition-6.1.Csrc:4369advalg.texA diagrffamscheme2Disasmallcategory1(i.e.!theclassofobjectsisaset).LethCSbSeanarbitraryhcategoryV.BAgdiagrffaminCorverhthediagramhscrhemeDtisacovXariantfunctorFc:URD3!C5.Example6.2.src:4375advalg.tex(fordiagramscrhemes)(1)%src:4377advalg.texTheemptrycategoryDUV.(2)%src:4379advalg.texThecategorywithpreciselyoneobjectD>6andpreciselyonemorphism1D.(3)%src:4382advalg.texTheMcategorywithtrwoMobjectsD1;D2QandonemorphismNfi:kD1fo !&gD2(apartfrom%thetrwoidentities).(4)%src:4385advalg.texThe@categorywithtrwo@objectsD1;D2 andtrwo@morphismsf;gU:qD1v !tD2 bSetrween%them.(5)%src:4388advalg.texThecategorywithafamilyofobjects(DidjiUR2I)andtheassoSciatedidenrtities.(6)%src:4391advalg.texThe&ZcategorywithfourobjectsD1;:::ʜ;D4 ]andmorphismsf;gn9;h;kwsucrhthatthe%diagramO338 D333D4(|32fd1 ά-̍@h{8 D1{D2(|{fd1 ά-i7fHEǠ*Ffex̟Ǡ?'qNgH Ǡ*Ffe LǠ?k|k%src:4396advalg.texcommrutes,i.e.8kgfQ=URhgn9.De nition~6.3.%src:4401advalg.texLet3SDbSea3TdiagramscrhemeandC批acategoryV.Each3TobjectC2C戹de nesaHcffonstant3diagramGKC :tVDɭ8!WCF}withKC(DS)tW:=CoforallHD2tWD蝹andK,`(fG):=tW1C ߹forallmorphisms}inDUV.Eacrh}morphismfQ:URC1K{!Cܞ20([inC0de nesacffonstantnaturaltransformationKfq:URKC t u!KC0 ,withZKfw(DS)=fG.Thisde nesZacffonstantfunctorK:C"!wfFVunct3(DUV;C5)fromthecategoryCݹinrtothecategoryofdiagramsFVunct(DUV;C5).src:4412advalg.texLet*.F:qD0S!CcbSeadiagram.qAnobjectC̹togetherwithanaturaltransformation/:rKC!nFښiscalledalimitoraprffojectiveZlimitofthediagramFwiththeprffojection¹ifforeacrhobjectCܞ20)2URCMHandforeacrhnaturaltransformation'UR:KC0 &@@ԟ>R{ɎKC0ŸǠ*Ffe Ǡ?&#Ki?fsrc:4421advalg.texcommrutes,thismeansinparticularthatthediagramsQ뚍HCHBF1(Didڹ)̲{fdά-+8:iH+эߚ8:jФׁ @Ф @Ф @Ф @$>@$>RH}wF1(Djf )RǠ*FfeOǠ?`F((gI{)84l7o]56oAdv|rancedTAlgebra{P9areigis[o]src:4424advalg.texcommruteUforUallmorphismsg-:UDi $0 =!Dj ȹinD(isanaturaltransformation)andthediagramsBi΍CF1(Didڹ):ѡ432fdyά-6H8:iH'*d'8:iѿDׁ @ۿD @D @D @ğ>@ğ>Rɯ }rCܞ20SǠ*FfeɆǠ?`Efsrc:4428advalg.texcommruteforallobjectsDiOinDUV.src:4430advalg.texA~category~'C1]haslimitsfordiagrffamsoveradiagrffamscheme~'D~if~(foreacrhdiagramFs:PaD!cCdorver"/Dwthere"0isalimitinC5.vA" categoryCdiscalledcffompleteifeacrhdiagraminCdhasalimit.ύExampleJ6.4.vsrc:4437advalg.tex(1)LetDM/bSeadiagramscrhemeconsistingoftrwoobjectsD1;D2 ݹandtheidenrtities. uA_Bdiagram_eFc:URD3!Cisde ned_fbygiving_ftwoobjects_fC1iandC2inC5. tAnobjectC1 {IwC2{togetherwithtrwomorphisms1V:URC1 {IwC2 .!5C1{and2:URC1 {IwC2 .!5C2{iscalledaprffoductƹofthetrwoobjectsifC1vrC2;Ë:URKCq1*Cq2 g!0SF׹isalimit,i.e.ifforeacrhobjectCܞ20DinCPand#foranrytwomorphisms'1ul:gCܞ20`? y!/3C1and'2:gCܞ20`> y!/3C2thereisauniquemorphismfQ:URCܞ20) !oC1jC2sucrhthatBiύH''q1بDׁ ΨD ĨD D ağ>ağ> H'$'q2ׁ @ @ @ @_>@_>Rɯ 2Cܞ20rǠ*FfeटǠ?`$f33[C133c?C1jC2BĞ32fd Vά;9q13333C2d32fd ά-; q2src:4456advalg.texcommrutes.TheXPtwoXQmorphisms1:C1UQC2 !C1 TandXQ2:C1UQC2 !C2 UareXPcalledXQtheprffojections꨹fromtheproSducttothetrwofactors.src:4460advalg.tex(2)LetDladiagramscrhemeconsistingofa nite(nonempty)setofobjectsD1;:::ʜ;Dn 5fandthefassoSciatedfidenrtities.AfvlimitofadiagramFYS:(CD}$!iC˹iscalleda niteprffoductofftheobjectsC1V:=URF1(D1);:::ʜ;Cn:=F(DnP)andisdenotedbryC1j:::Cn=URQ*n U_i=1Cidڹ.src:4467advalg.tex(3)DADlimitorverDaDdiscretediagram(i.e.GED hasonlytheidenrtitiesasmorphisms)iscalledprffoduct꨹oftheCi,:=URF1(Didڹ),i2I+andisdenotedbryQ?IJCidڹ.src:4472advalg.tex(4)}LetD8bSe}theemptrydiagramschemeandFc:URD3!C1the}(onlypSossible)emptydiagram.TheblimitC5;ڹ:!KC A8 Z!|fFofFiscalledthe nalobjeffct.IthasthepropSertrythatforeachobjectCܞ20S˹inC\((theuniquelydeterminednaturaltransformation'UR:KC0 &@morphismsgn9;hUR:C1V .!5C2.hThelimit>ofsucrhadiagramiscalledeffqualizerof thetrwo morphismsandisgivrenby anobjectKee8(gn9;h)andamorphism1V:URKe(g;h)URn!1C1.The2second3morphismtoC2|6arisesfromthecompSosition2V=URgn91=h1.)cThe3equalizer2hasthe:follorwinguniversal9propSertyV.CFor9each:objectCܞ20and9eachmorphism'1i:[fCܞ20< !{.C1>withgn9'1 =h'1(='2)^9thereis^:auniquemorphismfb:Cܞ20 j!tKe(S(gn9;h)with1f='1 >(andthrus2fQ=UR'2),i.e.8thediagramJT UˍCܞ20K `f ' ' ' '|l|l :n`*Ffeln`?<獒'q1yKeA(gn9;h)gcC1ϓfdNά-<%qq1̟fd&ά-n ,g󎎍&C2M3333̞32fd&ά-̍ h src:4504advalg.texcommrutes.9F7o]LimitsTandColimits,ProAductsandEqualizersϪ57[o]Problem?6.1.\src:4508advalg.tex(1)LetFc:URD3!Set(bSeadiscretediagram.ShorwthattheCartesianproSductorverFcoincideswiththecategoricalproSduct.src:4512advalg.tex(2)LiLetDbSeapairofLhmorphismsasin6.4(5)andletFc:URD3!Set(bSeadiagram. Shorwthatthesetfxw2xF1(D1)jF(fG)(x)=F(gn9)(x)gwiththeinclusionmapinrtoF1(D1)isanequalizerofFc:URD3!Set%kC.src:4517advalg.tex(3)LetFc:URD3!Set)UbSeadiagram.8Shorwthatthesete;Df(xDjD2URObDS;xD 2URF1(D))j8(fQ:Dk!D 0!ǹ)2D:F1(fG)(xD)=xDp1 r!MM f6 @!Nesrc:4528advalg.texisexact. 6.2.Colimitsofdiagrams.src:4533advalg.tex De nition6.5.src:4534advalg.texLet"F<:,D $ !:C,bSeadiagram.AnobjectCand"anaturaltransformation:F!:!*KC eisEcalledcffolimitorEinductivelimitofthediagramFvwiththeinjeffctionifforeacrh"objectCܞ20`29CWandforeachnaturaltransformation'9:9Fj%!KC0؁thereisauniquemorphismfQ:URC1K{!Cܞ20sucrhthatC{F{-KCV{fd&ά-5H'؉'8tׁ @8t @8t @8t @~>@~>RH33@>RHdOF1(Didڹ)>Ǡ*FfeqğǠ?`F((gI{)orsrc:4546advalg.texcommrutes?forallmorphismsgX:Di !Dj inD7(is?anaturaltransformation)andthediagramD4H?F1(Didڹ)HŞC/{fdά-8:iH''8:iׁ @ @ @ @/>@/>RHfKCܞ20: 4ŸǠ*Ffe gǠ?`tfs2src:4550advalg.texcommrutesforallobjectsDiOinDUV.src:4552advalg.texTheIlspSecialcolimitsthatImcanbeformedorverIlthediagramsasinImExample6.4arecalledcffoproduct,35initialobjeffct,꨹resp.8coequalizer.Example)6.6.Ksrc:4559advalg.texIn]ZK-VVectheobject]Y0isaninitialobject. InK-Alg:stheobject]ZKisaninitial?object.8InK-Algtheobject?faa2bAjfG(a)=gn9(a)g?istheequalizerof?thetrwo?algebrahomomorphismsf:A!?Bandg>:A!?B.InK-AlgtheCartesian(setofpairs)andthecategoricalproSductscoincide.RemarkU6.7._src:4568advalg.texAcolimitofadiagramCXisalimitofthecorrespSonding(dual)diagraminthe[!dualcategoryC52op R. ThrustheoremsabSoutlimits[ inarbitrarycategoriesautomaticallyalsoproSduce&(dual)&theoremsabout&colimits.Horwever,5observe&thattheoremsabSoutlimitsina:`77o]58oAdv|rancedTAlgebra{P9areigis[o]particular&category(forexamplethecategoryof'vrectorspaces)translateonlyintotheoremsabSoutcolimitsinthedualcategoryV,whicrhmostoftenisnottoouseful.B Prop`osition6.8.src:4579advalg.texLimits35andcffolimitsofdiagramsareuniqueuptoisomorphism.ōPrffoof.#Rsrc:4584advalg.texLetV-F=d: TDa{5!C bbSeV,adiagramandletC5;fandx.~C ;C~ nbSelimitsV-ofF1.{mThenthereareTuniquekmorphismskfyQ:xS~1QCM!#+CHandg:1QC 'z!xй~XC% `withn9Kf p=͖~and-~ mKg O=n9.Thisimpliesn9K1X.CM=URn7id 8۟KX.C=Ë=~ mKg*P=KfwKg*P=Kfg andanalogouslyS8~ xK1Kɍ~FCM=~ mKgI{f .'Becauseoftheuniquenessthisimplies1C t=URfGgXand1;U ~C=URgn9fG.<Remark[6.9.aVsrc:4598advalg.texNorwFthatweGhavetheuniquenessoftheGlimitresp.tcolimit(uptoisomor-phism)wrecanintroSduceauni ednotation.EThelimitofadiagramFAj:ZDe;!C%willbSedenotedbrylim c⡎morphism6fQ:URDS20w!DS200 W6in7D@}>RHlF1(DS200p)BǠ*FfetǠ?` gF((f);sC7o]LimitsTandColimits,ProAductsandEqualizersϪ59[o]src:4669advalg.texisBcommrutativeBbSecauseofF1(fG)(DS20!ǹ)=F(f)F((DBbSeK-algebras.bLetA MbSealeftA-moSdule.bThenitisalsoaK-moSdulebrymUR:=(ZwZv1A)m.AnalogouslyFaFrighrtB-moSduleisalsoaK-moSdule.WVerede nethenotionofabimoSduleasfollorws:De nition=7.3.src:4835advalg.texAK-bimoffduleA MB DOisanA-B-bimoSdulesatisfying(`1A)`m|=m=mUR=m(1B )i.e.8theinducedrighrtandleftstructuresofaK-moSdulecoincide.De nition7.4.src:4842advalg.texAMoritacffontextpconsistsofa6-tuple(A;B;APBN>;B N,B 8QAandhomomorphismsofK-bimoSdulesenfQ:URA 36PLn B QA L!A"AA; gË:URB Q A PB !oB#`BBN>;src:4848advalg.texsucrhthat:s2(1)%src:4850advalg.texqn9fG(p q20;:AA)Ag}fG(p qn9)UR:=(p)q(p209)[gn9(q p)]UR:=(p20)qn9p:YPrffoof.#Rsrc:4874advalg.texasin3.18.t9(De nition 7.7.C!src:4878advalg.texAmK-effquivalence of K-categoriesnC!RandnDsconsistsofapairofK-functorsFc:URC"!wfDUV,G :D3!CݹsucrhthatIdDPS԰l=%CF1GֹandIdCPq԰޹=$G.F.Theorem7.8.src:4884advalg.tex(MoritaI)Lffet(A;B;PS;Q;f;gn9)bffeaMoritacontext._Letffandg%beepimorphisms._Thenthefollowingstatements35holds2(1)%src:4888advalg.texPis35a nitelygenerffated35projectivegeneratorinA-MoSdIandinMoSd-B.%Q35isa nitelygenerffated35projectivegeneratorinMoSd-IAandinB-;MoSdM.(2)%src:4892advalg.texf{4and35gnarffeisomorphisms.(3)%src:4893advalg.texQPUR԰n:=Hom(yA/Wh(:PS;:A)PUR԰n:=Hom(yB/¹(P:;B:)%PP԰=Hom*JB1i(:Q;:B)PUR԰n:=Hom(yA/Wh(Q:;A:)%as35bimoffdules.(4)%src:4896advalg.texAPUR԰n:=Hom(yB/¹(:Q;:Q)PUR԰n:=Hom(yB(PS:;P:)%BPX԰ @=Hom)A/n(:PS;:Pƹ)PUR԰n:=Hom(yA/Wh(Q:;Q:)%as35K-algebrffasandasbimodules.=7o]?6TheTMoritaTheorems:61[o](5)%src:4899advalg.texPd B ~-VJ:8B-;MoSd S#W!3A-35MoSd #andQ0 A -:9A-35MoSd VM"!3`B-;MoSd!5)arffemutuallyinverse%K-effquivalences.oSymmetricallyJ7- A W6Pù:YMoSd-! AYh!gMoSd0F-4{B=andJ7- B ǐQY:MoSd-B%(I!7~MoSdOg]-SAZarffeZmutuallyinverseK-effquivalences.FurthermoretheZfollowingfunctors%arffe35naturallyisomorphic:'ǺʍPLn B -P5԰N=HomBRʹ(:Q;:-35);+Q A -P5԰N=HomAp(:PS;:-35);EQ-#. A PP5԰N=HomAp(Q:;-33:);;- B QP5԰N=HomBRʹ(PS:;-33:):(rf(6)%src:4911advalg.texWe35havethefollowingisomorphismsoflatticffes(orderedsets):!rgʍpVV(APƹ)P԰ȹ=%V(BN>B);V(PBN>)P+L԰+d=AsV(AA);pV(BN>Q)P԰ȹ=cV(AA);V(QA)P+L԰+d=Ac*V(BBN>);lV(BN>QA)P԰ȹ=7V(AAA)P}԰ e=GV(BN>BB)P+L԰+d=>^V(APBN>):!rf(7)%src:4921advalg.texThe35followingcffentersareisomorphicCenrt(A)PUR԰n:=Cenrt(z(B).PPrffoof.#Rsrc:4926advalg.tex(1)%TheisomorphismsfromTheorem1.22(5)map%g۴2m{HomBd-OB/((:Q A _PS:;:B:)tohomomorphismsofbimoSdulesg1 'V:gRP "!Hom1B8Q(:Q;:B)andg2:gQQgR!1Hom/aB6(PS:;B:).FVur-thermoretfrinduceshomomorphismsofsbimoSdulesf1 ':]$P t!uHom4#A;(Q:;A:)andf2 (:]#Q!n߹Hom)hA0L(:PS;:A).src:4933advalg.texIfwgisanepimorphismthenvthereisanelemenrtP@"qi pi2xQ A P7=withwgn9(Pqi pidڹ)x=1B =:EidP.gHenceq+p:E=:FPpqidpi=:FP(p)[f2(qi)]piforeacrhq*p:F2:EPƹ.gBythedualbasisLemma3.19ꨟA ȌPnis nitelygeneratedandprojectivre.src:4940advalg.texIfX9f8isX:anepimorphismthenthereisanelemenrtPxiZ Ayit2P B CQwithfG(Pxi Ayidڹ)=1A W= sP (xidڹ)[f2(yi)].Bya3.24aA ?PĹisaagenerator.TheaclaimsforPBN>,ԟB Q,andQA ?follorwbysymmetryV.src:4946advalg.tex(2)IffG(Pai CGbidڹ)=0thenџPu~iVai! CFbi7=P}i;j$ai CGbidfG(xjP yjf )=P}ai CFgn9(bi! xjf )yj 8=CPaidgn9(bi] @xjf )A yj}Y=PPfG(ai bidڹ)xj^J yj}Z=O0.Hence\fisinjectivre.By\symmetrywregetthatgXisanisomorphism.src:4954advalg.tex(3)Thehomomorphismf2 ):B%Q[!عHom3aA9E(:PS;:A)de nedasin(1)satis es(p)[f2(qn9)]B%=fG(pG qn9)_=`pq. aLetM'2`Hom*A%͹(:PS;:A). aThen(p)'`=(pPqidpi)'`=P\ (pqi)(pi)'Mhence'=PXqidڹ(pi)'=PXf2(qi(pi)').4Thrusf2ɹisanepimorphism.5Let(p)[f2(qn9)]=pq>=0forallpUR2Pƹ.8ThenwregetqË=UR1BN>q=PqidpiqË=0.8Hencef2isanisomorphism.src:4963advalg.tex(4)UTheUstructureofaB-moSduleonPinducesBZ_s!3EHom2ΟA9(:P;:Pƹ). z&LetpbY=0UforUallp|2Pƹ. Thenb=1B ]b=P3'qidpib|=0.If'2HomA"(:PS;:Pƹ)thenwrehave(p)'|=(p1BN>)'=(Pp(qidpi))'߹=P0(pqidڹ)(pi)'޹=ߟP0p(qi(pi)')andthrus'߹=Pqidڹ(pi)'.QThisshorwsthatweharveanisomorphismBX !_7Hom-A4(:PS;:Pƹ)ofK-algebrasandbimodules.src:4973advalg.tex(5)J"A (Pp B 9Q A ɍXPY԰A=6AiA AXPY԰A=6AiX;isJ#naturalJ"inXandJ#B aQ A ɍPp B 9YPF԰.=sB/B BYPF԰.=BN>YisEnaturalFinYthrusFweEgettheclaim.FVurthermoreB eQwT A U9UP԰͹=@=ByHom7A=(:PS;:A) AUPT԰<=B!WHom9A@fĹ(:PS;:A A eU@)Pp԰X=7BsHom8GA?%(:P;:U@)ӇisnaturalinUksincethehomomorphism'S8:HomA%(:PS;:A) A U !Hom4zjA;XN(:P;:A A U@)with(p)['(f% u)]S8:=((p)fG) uisanisomorphism.8Moregenerallywreshow:PLemma 7.9.>src:4986advalg.texIfA rPFTis nitelygenerffatedprojectiveandA rVB andBUrarffe(bi-)modulesthenthe35naturffaltransformation(inUtandVp)rfr'UR:Hom۟A"(:PS;:Vp) B U6!QHom.~A5b(:P;:VG B U@)src:4991advalg.texis35anisomorphism.Prffoof.#Rsrc:4995advalg.texLetꨟPSfi pi,2URHom۟A"(:PS;:A) A PnbeadualbasisforPƹ.8Thenm)M' 1ι:URHom۟A"(:PS;:VG B U@)URn!1Hom-=A4(:P;:Vp) B U>k7o]62oAdv|rancedTAlgebra{P9areigis[o]src:5000advalg.texde nedbry'21 \|(gn9)f;=f:Pi;j?()fidvij b uij with(pi)gt=:f;Pjvvij b uij isinrverseto'de nedbry>(p)['(f'a bu)]UR=(p)f'b u: ͹Since 'isahomomorphism(p)['(fGbb u)]UR=(p)fbb cu=(p)f'a bu=(p)['(ft ubu)]itsucestoshorwthat'21N1isamap.NNorwwehave(pidڹ)'(ft uu)aR=aQ(pi)f uuhence D'21 \|'(f mu)=+==*P׹()fidڹ(pi)f mu=*=Pع(()fipi)f mu=*=f mu.FVurthermore Ewre Dhave''21 \|(gn9)UR='(P ()fidڹ(pi)g)UR=P()fidڹ(pi)gË=UR(P ()fipi)g=URgn9.zsrc:5016advalg.texPrffoof35of7.8:8(conrtinued)src:5018advalg.tex(6)\Under[theequivXalenceofcategoriesA ?P"ismappSedtoHomrA#Pȹ(:PS;:Pƹ)PnX԰@=2BEB.dThisimpliesV(APƹ)PG԰/=kV(BN>B).[-Infact,HasubmoSduleofA ~PBisanisomorphismclassofmonomorphismsAU6!APƹ,trwo-suchisomorphisms.bSeingcalledisomorphic,ifthereisa(necessarilyunique)isomorphismUP6԰=@U@20,sucrhthatDDbYɛUVVPު$ҁ H$ׁ Hk]Hk]j`fK4cU@20R0ު$Ǡ$Ǡcc*H2Ǡ*Ffe9dǠ?Jsrc:5029advalg.texcommrutes.ObviouslyasuchsubSobjectsbarebeingpreservredunderanequivXalenceofcategories.FVorsubSobjectsofA ȌPB 8wrehavefurthermorethatKǹc$?UPot32fd*ά-Y@bYc$?UbYPot{fd*ά-HgǠ*Ffeg$Ǡ? ZdbHF2Ǡ*FfeydǠ? +bjandHom An(:PS;:U@)i;BA32fd#ά-Y@HHom An(:PS;:U@)Hi;BA{fd#ά-H2Ǡ*FfedǠ? ĤbHmǠ*Ffem$Ǡ? rb$src:5036advalg.texcommrute.8HenceꨟA ȌUB 2URV(APBN>)i Homd1A#B(:PS;:U@)2V(BN>BB).src:5039advalg.tex(7).TheproSofofthispart-willconsistoftrwo.steps.-WVeusethealgebraEndfunkt-+U(Id =A-UTMo(M@)Awrehavezamڹ=azmBhencez7=z2EndA}(M@).֭Thrusz꨹de nesanendomorphismofIdA-UTMoM@)STƹ(BN>M@)YT32fdAά- VphS'Tƹ(M@){󍍒2ПBM{󍍒PB"M{fdZ@ά-Pڿ'209(M@)H2Ǡ*Ffe)dǠ?`h O(M@)H%dǠ*Ffe%Ǡ?`*Jd O(M@)hsrc:5084advalg.texwhere^S':PA-MoSd!b!1B-MoSd,|T:B-MoSd "9h!2TA-MoSdȁarethemrutuallyinverseequivXalencesfromr(5),and OK:;IdA-UTMoŐڒ'200r(N@)v\=TS׹(N@)v"&]TS׹(N@)sH2fdg ά-mӀT'209S׹(N@)Tƹ(ST)S(N@)[Tƹ(ST)S(N@)䟢A2fdMά-̯KTS'TS(N@)\=TS׹(N@)"&]TS׹(N@):2fdg ά-˰TS'(N@)&N.NJD32fd,ά- '(N@)DsHbjܠ@fe{jܠ?^o. (N@)Ds4jܠ@fe4Ejܠ?^84 (N@)sHbՠ@fe{ՠ?q=T OS׹(N@)s4ՠ@fe4Eՠ?84T OS׹(N@)HbΠ@fe{F`6pTS (N@)4Π@fe4EF`684TS (N@)HbǠ@fe{?`6냀o. (N@)4Ǡ@fe4E?`6냀84 (N@)Asrc:5112advalg.texThrusthemap'UR7!'200 isaninnerautomorphismofE(A),henceitisbijective.7-2QTheorem7.10.src:5116advalg.tex(MoritaISI)LffetYSX:7A-35MoSd"P!2B-;MoSd sandTG:B-;MoSd "!3"A-35MoSd _nbffemutuallyinverseK-equivalences.LffetڟA cPB <:=\Tƹ(B)andڟB QA @:=[S׹(A).^WThenthereareisomorphismsf6[:[A ?P B 6QA 2!!A#AAand35gË:URB Q A PB !_OB#BBN>,suchthat(A;B;PS;Q;f;gn9)isaMoritacffontext.src:5125advalg.texFurthermorffe35thefollowingholdSP)԰!=Q A -andTP԰=PLn B -,:@n7o]64oAdv|rancedTAlgebra{P9areigis[o]Theorem7.11.src:5130advalg.tex(MoritaISII)LffetHPeq2êA-35MoSd!5\bea nitelyIgeneratedprojectivegenerator(=XIprogenerator). VThentheMoritaPcffontext(A;HomyA!Wk(:PS;:Pƹ);P;Q;f߹=ev-;g=db)Pisstrict, i.e.pf)NandgOarffeepimor-phisms./ۍPrffoof.#Rsrc:5137advalg.texSinceA Pis nitelygeneratedprojectivre,ugË=URdbFjisanisomorphism(3.19).6KSinceA Pisagenerator,fQ=URev+isanepimorphism(3.24).՜>!Prffoof35of7.10:Ksrc:5144advalg.tex1.=WGivrenAzS׹,W/Tƹ.=VThenS:HombA#@(:M;:N@)3f17!S(fG)2HombB#(:SM;:SN@)Azisanisomorphism.8Let h:URTSP)԰!=IdwA-UTMo{fdά-XISTf  ԟ"fd`ά-ZoxSgbYbYo+SM˻˻ԟ2fd`ά-B[ShMN32fd()`ά-]̑fHRǠ*FfeFǠ?`G HʒǠ*FfeğǠ?`D &src:5184advalg.texwithSgSTfQ=URShSTfG.8Sincef2isanepimorphismthisimpliesSgË=URSh,henceg=URh.src:5187advalg.tex5.3IfP2URA-MoSd\'isprojectivre,$;thenSP2URB-MoSd.isprojectivre.3Infactgivenanepimorphismf:ϝM * !N3in_NB-MoSd!candahomomorphism_Og=ֹ:ϞSPqd !ZN@. ThenTf:TM * !TNislzanlyepimorphismandTgTA:TSP Y!bTN^isinlzA-MoSdi. TSince :TSPP԰=,P,thereisanvgh:P >!,TMKwithTfh=Tg&R 21rorvfTfh  {=Tgn9. WVevfapplyvgS)>andgetSTf S(h )-=STgn9,?where.S(h )-2HomBB#(:STSPS;:STM@).Since. s|:-STMP ԰"=(M@,?wreharveBanAisomorphismHom( O21 ; O)L:Hom՟B#L(:STSPS;:STM@)M!"&Hom-B4(:SPS;:M@)AwithBinrverseHomy( O; 21 ˹). [FVor_ko:URSP!eMwith_k= 7S׹(h ) O21ewre_thenhave_ 7STƹ(kg)UR=k = `oS׹(hn ) O219 =r `S׹(h ),FhenceSTƹ(kg)r=S(hno )andTƹ(kg)=ho .lSowregetSTfSTko=URSTgË=URST(fkg)andthrusgË=URfkg.8SoSPnisprojectivre.A O7o]?6TheTMoritaTheorems:65[o]src:5209advalg.tex6.ٍSA̰is nitelygeneratedasaB-moSdule:SinceSAisprojectivre,wḛhaveSAbbXPF԰_=~LF֟i2I-B.RBys(3)sappliedtoTswregetATXPF԰_=~TSATXPF԰_=LF֟i2I-TB.7SincesAsis nitelygenerated,theimageofAinLi2I 4TB̹isalreadyadirectsummandina nitedirectsubsumLi2E"-TB,soJAmnYP ԰$=EL"%i2E3TB.)HenceSASYP ԰$=FL"%i2E3STBP E԰#-=۟L $3i2E1CB+PandJthrusSAis nitelygenerated.src:5212advalg.tex7.ʏIfpG9929:A-MoSdisapgeneratorthenSG2B-MoSd uisalsoagenerator.ʐInfactlet(f9:M!TN@);Z6=03in4B-MoSd.ThenTfY6=;Z0,Ohencethereisag:;ZGT!ATMIwith3Tfmg6=;Z0.ConsequenrtlySTfSgË6=UR0andf( 7Sgn9)UR= STfSgË6=UR0.src:5234advalg.tex8.8ThisshorwsthatS׹(A)isa nitelygeneratedprojectivegenerator.src:5245advalg.tex(Remark:HAnequivXalenceShalwraysmaps nitelygeneratedmoSdulesto nitelygeneratedmoSdules.8WVewillgivretheprooffurtherdorwninProposition7.12.)xsrc:5249advalg.tex9.8APUR԰n:=Hom(yB/¹(:SA;:SA)asalgebras,sinceAPUR԰n:=Hom(yA/Wh(:A;:A)2(SpURn!1Hom-=B4(:SA;:SA).src:5252advalg.tex10.XTBPX԰ @=Hom)B0bȹ(:SA;:B),0sinceHom-B"{ع(:SA;:B)2QTpURn!1Hom-=A4(:TSA;:TB)PUR԰n:=Hom(yA/Wh(:A;:TB)PUR԰n:=TB.src:5256advalg.tex11.8(B;A;SA;TB;f;gn9)de nesastrictMoritaconrtextbyMoritaISII.src:5261advalg.tex12. LhTheFfunctorSVisisomorphictoSA| A u`.InfactwrehaveHom B%G(:SA| A u`M;:N@)Pd԰L=HomyAWm(:M;:HomyB!Ź(:SA;:N@))P԰ۅ=mFHomϟAij(:M;:HomyA!Wk(:A;:TN@))P԰ۅ=mFHomϟAij(:M;:TN@)P԰ۅ=mFHomϟB5 (:SM;:N@):TherepresenrtingobjectB 8SMP6԰=@ߟBSA A M+depSendsfunctoriallyonMbry5.5.4۽Prop`osition=7.12.T\src:5274advalg.texAM is nitelygenerffatedi ineffachsetofsubmodulesfAidjin2IgwithAiYT~M$andG?Pi2I!PAi=M$therffeG?isG@a nitesubsetfAidjiT~2TI0g(I0 I8 nite)G@suchthatRP i2Iq0WAi,=URM@.Prffoof.#Rsrc:5282advalg.texLetuM<=BWAm1{+ w:::ݎ+ wAmnP.ڠEacrhmjiscontainedina nitesumuoftheAidڹ,henceallofthemj eandhencethemoSduleM?itself.uConrverselyconsiderfAmjm+2M@g.uThenM6=URPAm,_hence=JM~-isasum=Iof nitelymanryoftheAmandthrusis nitelygenerated. Corollary~7.13.4bsrc:5292advalg.texUnder>aneffquivalence>ofcffategories>T:URA-35MoSdf!m!0%B-;MoSd nitelygenerffatedmoffdules35aremappedinto nitelygeneratedmodules.Prffoof.#Rsrc:5298advalg.texThelatticeofsubmoSdulesV(M@)isisomorphictothelatticeofsubmoSdulesV(TM@).kProblem87.1.ksrc:5303advalg.texLetA-MoSdbSeequivXalenrttoB-Mod.$+ShorwthatMod+j-AandMod+i-BGarealsoequivXalenrt.ProblemnX7.2.;src:5308advalg.texShorwJthatKanequivXalenceofarbitrarycategoriespreservresmonomorphisms.Problem7.3.esrc:5312advalg.texShorwsthatsanequivXalenceofmoSdulecategoriespreservresprojectivesmoSdules,butnotfreemoSdules.B%}7o]66oAdv|rancedTAlgebra{P9areigis[o]c칹8.r=SimpleandSemisimpleringsandModules8.1.SimpleandSemisimplerings.src:5387advalg.texDe nition8.1.src:5388advalg.texAnidealR ;IFURR HRiscallednilpffotent,ifthereisnUR1sucrhthatI2n %=UR0.src:5391advalg.texAlmoSduleR RM棹iscalledA2rtinian(EmilArtin,1898-1962),ifeacrhnonemptysetofsubmoSdulesofM+conrtainsaminimalelement.src:5395advalg.texAmoSduleR GM㗹iscalledNoffetherian(EmmryNoSether,з1882-1935),жifeacrhnonemptrysetofsubmoSdulesofM+conrtainsamaximalelement.src:5399advalg.texA'ring'RA is'calledsimple,7 ifR VRA asamoSduleisArtinianandifRA doSesnotharve'nontrivial(6=UR0;RJ)trwosidedideals.src:5403advalg.texAringRdiscalledsemisimple,vifR RcisArtinianandifRcdoSesnotharvenontrivial(6=40)nilpSotenrtleftideals.Lemma8.2.src:5409advalg.texEach35simpleringissemisimple.獍Prffoof.#Rsrc:5413advalg.texCZ:=~[P)(IjRIo~[R qRunilpSotenrt)+is*atwosided*ideal.DhInfacttake*a~\2Iand+r2RJ.Then0֍U#?(r1arS)(r2ar):::ʜ(rnPar)UR=(r1a)(rr2a):::ʜ(rrnPa)r2URI nRn=0:src:5417advalg.texHenceywrehave(RJarS)2n=UR0=)RarCܞ,Osoyar2CandyCisatrwosidedideal.#{ThusC1=UR0or^C=RJ.IfC=0thenthereare_nonontrivialnilpSotenrtideals.IfC=R㨹thentherearevidealsandelemenrtsait2CIiۀsuchthat1C=Ca1+ :::ޑ+ anP.TheidealI1+ I2 6isnilpSotenrtsince&(a1͹+Qb1)(a2+b2):::ʜ(a2n :+b2n T)%consists&ofmonomialseitherinI2nRA1QRoorinI2nRA2QRJ.*_ButI2nRA1 %=UR0=I2nRA2=)(I1j+I2)22n ==0.8Hence1isnilpSotenrt.Contradiction.h!De nition+8.3.L src:5431advalg.texAkmoSduleR s$Muiscalledsimplei M6=R0andMhasonlythemoSdules0andM+assubmoSdules.8AnidealR ;I+iscalledsimpleorminimal,ifitissimpleasamodule.Lemma8.4.src:5438advalg.texLffet35RLbesemisimple.fiTheneachleftidealofRLisadirectsummandofRJ.獍Prffoof.#Rsrc:5443advalg.texLetI"bSeanidealinRJ,Cthatisnotadirectsummand,CandletI#bSeminimalwithrespSecttothispropertryV.8Suchanidealexists,sinceRArtinian.src:5447advalg.texCase71:LetIFURRbSeanidealthatisnotminimal(simple),i.e.thereisanidealJqIwith0m6=mJ6=I.&ThenWJɹisadirectsummandofRJ,i.e.thereisahomomorphismf:mRv!JaXwith=(J!EYIy0!iRh2.fJ!B1Jr)=id RQJ7O.This>impliesIy0=J?K۹for>KdJ:=Ke(I!jRh2-fJ!B0Jr).SinceK16=URI,=there"isalsoagË:Rn'!{K{with(K1K{!IF``!RhPgJn'!Kܞ)UR=id K[.The"mapfXd+fg~gn9fQ:URI!nRn'!{Isatis es_(fg+Zigȡgn9fG)(jӹ)UR=f(jӹ)Zi+Zhgn9(j)gn9fG(j)UR=j;+gn9(jӹ)Zhg(jӹ)UR=jp1for_all^j%2JandQd(f+qggn9fG)(kg)UR=f(kg)q+qgn9(k)gn9fG(k)UR=0+kطq0=kforQdallko2URKܞ,p henceQc(f+gqgn9fQ:I!nRn'!{I)UR=id IC.8ThrusI+isadirectsummandofRJ.Conrtradiction.src:5461advalg.texCase"p2:pLetIbSea"qminimalorsimpleideal.8SinceIisnotnilpSotenrtand0E6=DI22 eIholds,wre@!get@ I22 ٹ=URI. InparticularthereexistsanaUR2I1withIa=I,b;since@!Iais@ alsoanideal.ThrusaUR:IF``!I*isħanepimorphismĨandevrenanisomorphism,AforKe(a)mustbSeĨzeroasanideal(seeLemmaofScrhur8.5.)SothereisaneUR2I,Fe6=0witheaUR=a:=)(e22Ȣe)a=eeaea=ar r aUR=0=)e222 e=02IFչ=)e22V=e2I./FVromI=RJewregetRn=URRer r R(1e),{sinceRn=URRJe[(+R(1e)üandrSeUR=s(1[(['e)2Re[(\R(1e)UR=)rSe=re22V=s(1[(e)eUR=0:üThrusIisadirectsummandofRJ.8Conrtradiction.tLemma8.5.src:5476advalg.tex(Scrhur)35LffetR &M@,RNtbffesimplemodules.fiThenthefollowinghold:(1)%src:5480advalg.texIf35M66P԰= N@,thenHomR#Q(:M;:N)UR=0:(2)%src:5481advalg.texHomyRm(:M;:M@)35isaskew- eld(=divisionalgebrffa=noncommutative eld).Prffoof.#Rsrc:5487advalg.texLet7fwu:/vMpZ!]NBbSeahomomorphismwithf6=/v0.|ThenIm (fG)=N@,FsinceNBissimpleGandKex(fG)_:=0,isinceHM^+isGsimple,hencefeFisanisomorphism.оThisimplies(1).C:7o]8SimpleTandSemisimpleringsandMoAdules67[o]FVurthermorewrehave(2),ŹsinceeachendomorphismfǞ:M!YM͹withfǟ6=0isinvertibleunderXJtheXImrultiplicationofHomӟR#f(:M;:M@).ObservethatXIaskew- eldisaring,swhosenonzeroelemenrtsformagroupunderthemultiplication.z,Remark8.6.wQsrc:5498advalg.texLet87R +MybSe87simple.eThen86EndDR8(:M@)UR=DŹisaskrew- eld.dHencetheRJ-moSdulestructureofM+canbSecrharacterizedbyRn'!{End):D1޹(M:)UR=MnP(DS):Theorem8.7.src:5503advalg.tex(Artin-WVedderburn)35Thefollowingarffeequivalent:R(1)%src:5506advalg.texRLis35simple.(2)%src:5507advalg.texRLpffossesses35asimpleidealthatisanRJ-progenerator.(3)%src:5509advalg.texRP<԰$=MnP(DS)risafullmatrixringoveraskew- eldD.$(nisunique,DHisuniqueup%to35isomorphism.)(4)%src:5512advalg.texRn=URI1j:::In ۅwith35isomorphicsimpleleftideffalsI1;:::ʜ;InP.Prffoof.#Rsrc:5518advalg.tex(1)=)Թ(2):?7SinceRisArtinianthereisasimpleideal0Z6=ZIL:RJ.BbLetJw):=PdfI20jI20ideal[inRu.andI20P԰=A1Ig.ThenJxVisatrwo[sidedideal,x4sinceI20|ri6=0=)r:I20 V!gRu.withKe[W(rS)=0,)henceKrpٹisinjectivreLandtheimageI20"risisomorphicLtoI20resp.I,)henceisinJr.fSincew:Rissimplewrew;havew:Rn=URJqĹ=PIidڹ.Sincew:12I1~+:::Hf+InP,Pthereisw;anepimorphismI1+k:::)kIn i !2R(exteriorJdirectsum),M2thatsplitssinceRisprojectivre.HenceRisadirectƒsummandƓofI1 `:::`In nuptoisomorphism,andthrusIisagenerator.,FVurthermoreI濹is=:::P%԰%ٹ=2InP԰ =KRVG D `LDP԰ȹ=SRGVp.src:5547advalg.tex(4)=)(2):8I1isobrviouslyanRJ-progenerator.src:5550advalg.tex(2)+=)(1):RJ-MoSdPF԰_=,_DS->6MoSdwithDPa԰0I=0End&=JR-0ݹ(I).HenceV(RRJ)P԰ܻ=V(D Hom"/+D)Ϲ(I:;D DS:))+isArtinian,andwrehaveV(RRR)PUR԰n:=V(DDD)UR=f0;DSg.8ThrusRissimple._Corollary8.8..src:5557advalg.texLffetUVRnbeaUWsimpleringandletR HMz6=n0be nitelygenerated.Thenthefollowing35holdR(1)%src:5560advalg.texRMtis35anRJ-prffogenerator.(2)%src:5561advalg.texS):=UREndbRU(:M@)35isasimplering.(3)%src:5562advalg.texCenrtz(RJ)PUR԰n:=Cenrt(z(End RR(:M@)).(4)%src:5563advalg.texRPn԰=EEnd%&S+C(M:).Prffoof.#Rsrc:5568advalg.tex(1)TheclaimfollorwsfromthefactthatRJ-MoSdP E԰ -=/5DS->6MoSd andsinceeacrh nitelygeneratedDS-moduleisaprogenerator.src:5572advalg.tex(2)M8S׹-MoSdP!3԰!=0_RJ-MoSdP!3԰!L=09DS->6MoSd! MimpliesM9thatV(SS׹)P԰ɽ=V(DPƹ)isM8Artinian. `FVurthermoreV(SSS)PUR԰n:=V(DDD),henceSisasimplering.src:5576advalg.tex(3)+(4)follorwfromtheMoritatheorems.(~8.2.InjectiveMo`dules.src:5585advalg.texDe nitionQZandQ[Remark8.9.] src:5586advalg.texAnRJ-moSduleߟR rJPiscalledinjeffctive,mifforeacrhmonomor-phism>@f+:M$> !!N$andforeacrhhomomorphismgQչ:M$> !JZthereexistsahomomorphismDVG7o]68oAdv|rancedTAlgebra{P9areigis[o]hUR:N6!JwithhfQ=gQgEbYG9MbY#NUޟ{fd()`ά-:fH,Jc|Ǡ*FfeΖǠ? ՞gH :.hׁ    >> :src:5591advalg.texVVectorspacesareinjectivre.Z_BZisnotinjectivre.TheinjectiveZ-moSdulesareexactlythedivisibleAbSeliangroups.8Z 4Qisinjectivre.Theorem8.10.src:5598advalg.tex(The35Baercriterion):fiThefollowingarffeequivalentforQUR2RJ-LMoSd^:(1)%src:5601advalg.texQ35isinjeffctive.(2)%src:5602advalg.tex8RIFURR HRJ;338gË:URI@!Q359h:Rn!*[Qwithh=gQUVbYIbY1RՓf{fd,'ά- 铙HXQ̴Ǡ*FfeǠ? >gH fh~Ɵׁ ~Ɵ ~Ɵ ~Ɵ 8F>8F> -:(3)%src:5605advalg.texEachmonomorphismf|:4Qh;fJ4!ϕMsplits,i.e.therffeisanepimorphismg͹:4Muy!zQ%with35gn9fQ=UR1Q/.Prffoof.#Rsrc:5612advalg.tex(1)=)(2):8follorwsimmediatelyfromthede nition.src:5615advalg.tex(1)=)(3):8ThediagramR񍍍7ΞQ7M`d{fd(ά-EfHΞQǠ*FfeFǠ?Ի1QH gׁ ~t>~t> csrc:5617advalg.texde nestherequiredgn9.src:5619advalg.tex(3)=)(1):8InthediagrambY6MbYND{fd()`ά-fPQnPiiԟ̟Rfd*ά-盍'tώtώԟ̟fd*}`ά bHRJǠ*Ffe̅|Ǡ? lgH Ǡ*Ffe<Ǡ?` < src:5630advalg.texassume@thatfF?is?amonomorphismandPr:=vNQ=f(fG(m);gn9(m))jmv2M@g@with'resp.f` are>lcanonical>kmapstotheleftresp.4*therighrtcompSonents:g'(qn9):=`z p(0;q); (n):=`zJ p(n;0).|.SinceO n9fG(m)=`z.R p(f(m);0)7˻=`z-; p(0;g(m))6Ǥ='g(m)Owrehave n9f˳='g.MLetO'(q)=`z p(0;q)"]=0.Thenthereexistsanm|{2MBwithfG(m)=0andgn9(m)||=q.}SincefIisaninjectivremap,jweharvem6=0andthrus'injective.eBy(3)thereisawith'5=61Q/.eThen n9f4='go=g,andthrusQisinjective.src:5643advalg.tex(2)j=)(1):8Givrenamonomorphismf*:M# =[! INandahomomorphismjgQ%:N# =[! IQ.ConsiderR#theR$setS;:=URf(Nid;'i)g,pwhereNi,URNisR#asubmoSdulewithImY(fG)URNiandR#'i,:NiEp7o]8SimpleTandSemisimpleringsandMoAdules69[o]!nQ꨹isahomomorphismsucrhthatRk{_M{-NNi{fd'Ѝά-̑f{󎎍{"NqT{fd( Pά-H5QǠ*FfeԟǠ? gH ׀T'i[ׁ [ [ [ 4>4> !src:5654advalg.texcommrutes. =WVe^have^S;6=UR;,zsince(Im(fG);gn9f21 {)UR2S׹. =FVurthermore^SE!isorderedbry(Nid;'i)UR(Njf ;'j)_ifNi)ONjŚand'jf jN8:i='idڹ.Letf(Ni;'i)jiN2OJrgbSeacrhaininSb.Then[Ni)NisZasubmoSdule. *,:[Ni :X!MQwith n9(nidڹ)='i(ni)Zis[awrellde nedhomomorphismand([Nid; n9)yt2Sb.;FVurthermoreIwrehave(Njf ;'j)yt([Nid; n9)Iforallj&G2ysJr.;ByZorn'sLemmathereexistsamaximalelemenrt(N@20;'209)inSb.WVeshowthatN@20do=URN@,forthenthecontinuationof gz:to NLexists.Letx2NAn]N@20.Then N@203$N@20z+\RJx.LetI:=fr2RJjrSx2N@20g.Then Iisanidealandwrehaveacommutativediagram򍍍nIRZRIJfd,'ά-VE7ɯ Mɯ dN@20(t{fd&ά-pf99]N@20Ź+RJxW{fd@`ά-H ntgׁ @ @ġ @Ρ @$>@$>RHyXQRǠ*Ffe Ǡ?U'20H 񻔟컔绔X> src:5681advalg.texwithd(rS)$:=$rQ1x.Thenwrehave(I)$$N@20.Thusdby(2)thereisahomomorphism:$R!nQwithn9UR='20`'(x).WVede neo:URN@20D+RJxn!1QbryW(n20`+rSx)UR:='209(n20)(+n9(rS).Thisisawrellde nedmap,öforifn20p+G7rSxUR=n20RA1<+r1xthen(rG8r1)xUR=n20RA1;n20#2N@20hencerr1V2I.Thrusn9(r[r1)='209((rr1)x)='209(n20RA1n20)and'20(n20)+n9(rS)='20(n20RA1)+n9(r1).ƄItiseasytoseethatM2isalsoahomomorphism.[(SinceWjN"03й=h'20Nholdswrehave(N@20+pRJx;W)h2Swand(N@20;'209)UR(N@20Ź+RJx;W)aconrtradictiontothemaximalityof(N@20;'209).8ThusN@20do=URN@.ݮuCorollary8.11.src:5698advalg.texIf35RLisasemisimpleringtheneffachRJ-moduleisprojectiveandinjective.uPrffoof.#Rsrc:5702advalg.texLetFDQFEbSeanRJ-moSdule.KBy8.4eacrhidealisadirectsummandofRJ.KThefollorwingdiagramtogetherwiththeBaercriterionshorwsthatQisinjective:MҍbY(IbYjRyylfd,'ά-i99l҄fd,'؀u`HʑQ:Ǡ*Ffe8Ǡ?H`ׁ̟ ̟ ̟ ٷ̟ qL>qL> src:5715advalg.texLetf>:N7QX!HBPùbSesurjectivre.SinceKe;U(fG)N isasubmoSduleandinjectivrethereisag%:N[!lIKe$Ǡ(fG)withgn9(n)=nforalln2KeC(fG).WVede nek :P*D=!*NIݹbrykg(p)=nLgn9(n)forhn+2NwithfG(n)+=p.IfalsofG(n209)=pthenfG(nmnn209)=0hencenmnn202Ke(fG)andgn9(n(n209)UR=n(n20.This1impliesn(gn9(n)UR=n20}a(g(n209). So1kYNisawrellde nedmap.FVurthermorefGkg(p)U=f(ngn9(n))U=f(n)fgn9(n)=p0,4hence&fk!r=U1P̹.In&orderto&shorwthatkisahomomorphismletfG(n)UR=p,f(n209)=p20.5OThenwregetfG(rSnι+r20!n209)UR=rpι+r20!p209.5OThisimplieskg(rSpLY+r20!p209)UR=rnLX+LYr20!n20gn9(rnLX+r20!n209)UR=r(nLYgn9(n))LX+r20!ǹ(n20gn9(n209))UR=rkg(p)LX+LYr20!k(p209).)|ThrusPnisprojectivre.}xF~7o]70oAdv|rancedTAlgebra{P9areigis_{aLemmah8.12.src:5731advalg.texLffetA#0I,!Mh AfJ|!9Nh gJ|!P Q^!0bffeA"ashortexactsequence.2MandA"PareA2rtiniancifcandonlyifNoisA2rtinian.jInpffarticularifMoandNnarffeA2rtinianthenMZuNis35A2rtinian.Prffoof.#Rsrc:5738advalg.texLet /NLbSeArtinian.vThisimpliesimmediately 0thatMLisArtinian.vIffLidgisasetofsubmoSdules-ofPKthenfgn921 ʵ(Lidڹ)gisasetofsubmodules,ofN@.woLetgn921 ʵ(L0)beminimalinthisset.8Sincegn9g21 ʵ(Lidڹ)UR=LiOwrehavethatL0isminimalinfLidg.src:5744advalg.texLetsMandP׹bSeArtinian.LetfLidgbeassetofsubmodulesofN@.LetL0 3becrhosensuchthatugn9(L0)isuminimalinthesetfg(Lidڹ)g.:LetLubSecrhosensuchthatfG21 {(L)uisminimalinthevksetvjffG21 {(Ljf )jLj@2C5fLidg꨹and|gn9(Lj)C6=gn9(L0)g.(WVevkshorwthatvjLisminimalinfLidg.(LetL20k22fLidg/RwithLL209.Thengn9(L0)=g(L)g(L209),@|hence/Sg(L20)2=gn9(L0).FVurthermore/RwreharvefG21 {(L)URfG21(L209),henceLUR=L20. ]8.3.SimpleandSemisimpleMo`dules.src:5757advalg.texLemmať8.13.src:5758advalg.texLffetR1;:::ʜ;Rn Fpbesemisimplerings.4ThenR1a_^:::[_^Rn Foisa semisimplering.Prffoof.#Rsrc:5762advalg.tex(OnlyforthecaseR1-mR2)ByLemma8.12R1mR2 !isArtinian.?LetI,;4R!gbSenilpSotenrt.FVrom~I2n =Q0we~getforeach~aQ2QIptheequation(RJa)2n j=Q0.FVromaQ=(a1;a2)follorwsU0a=(RJa)2n =(R1a1;R2a2)2nP.HenceTR1a1 !=a0andTR2a2 !=a0,i.e.RJa=a0andUthrusIFչ=UR0.ZLemmaK8.14.3>src:5772advalg.texEachqprffopersubmodulerNUofa nitelygenerffatedmoduleMUisrcontainedina35maximalsubmoffduleofM@.fiInparticularMtpossessesasimplequotientmodule.Prffoof.#Rsrc:5778advalg.texLet,N $%MmչbSe,apropersubmoduleofM@.LetMbethesetof,submodulesUmչwithN6URU$M@.M is orderedbryinclusion.Let (Uidڹ)bSeacrhaininMandU@20do:=UR[Uidڹ.ThenU@20isJagainJ~asubmoSduleandN9VjsU@20.XdIfU@20=MbthenJallgeneratingelemenrtsm1;:::ʜ;mtIarein(TU@20,7hencethere(UisamoSduleUi.withm1;:::ʜ;mtP2LUidڹ.ThrusUi#%=M@.Thisis(TimpSossible.So6U@206=MwandthrusinM.-FVurthermoreU@20E߹isanuppSer6bound6of(Uidڹ).ByZorn'sLemmathereisamaximalsubmoSduleofM+(inM),thatconrtainsN@.Lemma8.15.(1)vxsrc:5793advalg.texIfSXixFZ MQTisasetofgenerffatingelementsofQoverZandxxF2X%then35X+nfxgisalsoasetofgenerffatingelementsofQ.(2)%src:5797advalg.texZTQ35pffossessesnomaximalsubmodules.Prffoof.#Rsrc:5803advalg.tex(1)LetBX=URhXo`n}fxgi.ThenQ=Zx}ݹ+B.ThereisayË2Qwith2yË=x.WVerepresenrtyas)[yË=URnxӹ+b)Zwithn2Z,Pb2B.qThis)Zimpliesx=2yË=2nxӹ+2b)Zandthrus(1Թ2n)xUR=2b2B.FVurthermore8thereisaz52URQ9with(1+,2n)z=URx,#since9obrviously1+,2n6=0.WVerepresenrtz[aswz5=URmx+b209.ThisimpliesxxUR=(12n)z5=UR(12n)mx+(12n)b20#=UR2mb+(12n)b20#2URB.ThrusBX=URQandwecanomitxfromthesetofgeneratingelements.src:5814advalg.tex(2)kLetkN6URQbSeamaximalsubmoSduleandxUR2QnN@.ThenkN[fxgisasetkofgeneratingelemenrtsofQ,hencealsoN@.8Contradiction. ]LemmaS8.16.]src:5821advalg.texLffetR ϤMbeamoffduleinwhicheachsubmoduleisadirectsummand.`Theneffachbosubmodulebn06=NǹMScffontainsabosimplesubmoffdule.FurthermoreMSisboasumofsimple35submoffdules.Prffoof.#Rsrc:5828advalg.texLetxz2N@,x6=0.zItsucestoshorwthatRJxhasasimplesubmoSdule.zSinceRJxis nitelyoYgeneratedoXRJxpSossessesamaximalsubmoSduleL.SinceLisadirectsummandofM@,aXthere?isfQ:URM6!Lwith>(Ln!1RJxn!Mh 'mfJ6!L)=1LGع,!hence?L1O1NIFչ=RJx,where?IFչ=Ke(RJx!M}!L):`If06= J$Ithen`L$L.+J$RJx`inconrtradictiontoLmaximalinRx.HenceI+issimplewithIFURRJxN@.G@7o]8SimpleTandSemisimpleringsandMoAdules71[o]src:5838advalg.texLet~UN:=PPYIj_bSethesumofallsimplesubmodulesofM@.ThenM=PNP4Kܞ.IfK-L6=P0thenl.KH͹conrtainsal/simplesubmoSduleI]andwrehavel/IFURN7\TKܞ.Contradiction.ThusK1=UR0andM6=URPIjf .}oLemmaO8.17.\psrc:5846advalg.texLffetٚR -M}beٙasumٚofsimplesubmoffdules:2MV=rP4jv2X$`Ijf .YLetٙNrM~beٙaCsubmoffdule.!ThenFthereisasetYReX7withMI=RdN-vILˡjv2Y#Ij 'andFasetZ/X7withNP6԰=@ߟL7jv2Z0'Ijf .fiIn35pffarticulareachsubmoduleNtofMisadirffect35sumofsimplesubmoffdules.}䍍Prffoof.#Rsrc:5855advalg.texLet*Sw̹=ifZnjXjN6ɹ+(P jv2ZIjf )=N6(L UXjv2ZHIjf )g. eThe*setSis)orderedbry>inclusionandnotemptrysince;b2bSb.Let(Zidڹ)bSeacrhaininSb.ThenZܞ20 ǹ:=b[Zi2S.Inordertoshorwthisletn+PZjv2Z0$aj=V0.Thenatmost nitelymanryaj2VIjaredi erenrtcfrom90. &Hence9thereisaZiϹinthecrhainwithj<2 Ziforall9aj 6= 0inthesum. &FVromN+}(P jv2Z8:i!YIjf )UR=N}(L UXjv2Z8:i"UIjf )WwreWgetnUR=0=ajforWallj%2Zܞ20׹.ByWZorn'sLemmathereisCamaximalelemenrtZܞ200 Nb2URSb,3andwehaveP:=URNw+M(P jv2Z00#+Ijf )=NM(L UXjv2Z00$Ijf ).)LetIkਹbSesimplej}withko2URXanZܞ200.'IfPF+Ikx=URPFIk#,thenNù+(P jv2Z00#+Ijf )+Ikx=URN߹(L UXjv2Z00$Ij)IRԍin6conrtradictiontothemaximalityof6Zܞ200.(Hence06=P=\wIkiIk#,Ior6IkPƹ.(This6impliesP=URN댹+PUUjv2X#Ij\=M@.src:5875advalg.texNorw:weapply9the rstclaimtoLjv2Y$Ij CandobtainNsl2(L UXjv2Y 6Ijf )=M=(L UXjv2YIjf )2(L UXjv2ZHIjf ).8ThisimpliesNP6԰=@M=(L UXjv2Y 6Ij)PUR԰n:=LUSjv2Z/CIj.ŋ}ߍTheoremdV8.18.e|src:5883advalg.tex(StructurffeTheoremforSemisimpleModules):ForR tM*thefollowingareeffquivalentߍ(1)%src:5886advalg.texEach35submoffduleofMtisasumofsimplesubmodules.(2)%src:5888advalg.texMtis35asumofsimplesubmoffdules.(3)%src:5889advalg.texMtis35adirffect35sumofsimplesubmoffdules.(4)%src:5890advalg.texEach35submoffduleofMtisadirectsummand.oPrffoof.#Rsrc:5895advalg.tex(1)=)(2):8trivial.src:5897advalg.tex(2)=)(3):8Lemma8.17.src:5899advalg.tex(3)=)(1):8Lemma8.17.src:5901advalg.tex(2)=)(4):8Lemma8.17.src:5903advalg.tex(4)=)(2):8Lemma8.16.HToDe nitionŁ8.19.src:5907advalg.texAmoSduleR MF۹iscalledsemisimple,Lifitsatis esoneoftheequivXalenrtconditionsofTheorem8.18.Corollary8.20.(1)ʞsrc:5914advalg.texEach35submoffduleofasemisimplemoduleissemisimple.(2)%src:5916advalg.texEach35quotient(rffesidueclass)moduleofasemisimplemoduleissemisimple.(3)%src:5918advalg.texEach35sumofsemisimplemoffdulesissemisimple.Prffoof.#Rsrc:5923advalg.tex(1)trivial.src:5925advalg.tex(2)@LetN6URM@.1ThenMP԰=@N~M=XN@,وinparticularM=XN$isAisomorphictoasubmoSduleofM@.src:5929advalg.tex(3)trivial.8Remark8.21.%src:5933advalg.texWiththenotionofasemisimplemoSdulewrehaveobtainedaparticularlysuitablegeneralizationofthenotionofavrectorspace.,ImpSortantthefforemsRofQlinearalgebraharvebSeengeneralizedinTheorem8.18.#=Thesimplemodulesorvera eldareexactlytheonedimensionalbvrectorspaces.nCondition(2)cofTheorem8.18istriviallysatis edsinceeachvrectorpspaceisthepsumofsimple(onedimensional)vectorpspaces,4onesimplyphastoformV¹=URPvI{2Vnf0g4nKܞv(orV=URPvI{2E#EBKv(foranarbitrarysetofgeneratingelemenrtsEofVp.ThruseacrhCvectorCspaceVissemisimple.7Socondition(3)holds.7ItsarysthateachCsetofgeneratingH7o]72oAdv|rancedTAlgebra{P9areigis[o]elemenrtsJE`containsabasis.(4)isItheimpSortantstatementIthateachsubspaceofIavectorspacehasadirectcomplemenrt.#Lemma8.17alsoconrtainsclaimsabSoutthedimensionofvrectorspaces,subspacesandquotientspaces.Theorem8.22.src:5951advalg.tex(WVedderburn)35ThefollowingarffeequivalentforRJ:(1)%src:5954advalg.texRRLis35semisimple(asaring).(2)%src:5955advalg.texEach35RJ-moffduleisprojective.(3)%src:5956advalg.texEach35RJ-moffduleisinjective.(4)%src:5957advalg.texEach35RJ-moffduleissemisimple.(5)%src:5958advalg.texRRLis35semisimple(asanRJ-moffdule).(6)%src:5959advalg.texRLis35adirffect35sumofsimpleleftideffals.(7)%src:5960advalg.texRPn԰=ER1j:::Rn ۅwith35simpleringsRi(iUR=1;:::ʜ;n).(8)%src:5962advalg.texRPn԰=EB1j:::BnP,35wherffetheBiareminimaltwosidedidealsandR &RLisA2rtinian.(9)%src:5965advalg.texRR &is35semisimple(asaring).Prffoof.#Rsrc:5970advalg.tex(1)=)(3):8Corollary8.11.src:5972advalg.tex(3)=)(4):8Theorem8.18(4)andTheorem8.10(3).src:5975advalg.tex(4)=)(5):8SpSecialization.src:5977advalg.tex(5)=)(6):8Theorem8.18(3).src:5979advalg.tex(6)=)(3):8Theorem8.18(4)and8.11.src:5982advalg.tex(6)=)(2):8Theorem8.18(4)and8.11.src:5985advalg.tex(2)=)(4):LetN.IMbSeasubmodule.MThenM=XNisprojectivre,jsothereisfI:IM=XN!TM1withqM(M=XN{d!rM{d!M=XN@):=:idvrorqM(M{e!M=XN{e!M@):=:pqNwithqMp22=p.HenceM6=URKe(p)Imq(p)andKeE(p)UR=N@.src:5991advalg.tex(6)k=)k(8):~LetRn=URI11 :::I1iq1 wCI21:::I2iq2 wC:::In1 :::IninGbSekadirectksumofsimple.ideals,? nitelymanryV,?since.RGis nitely.generated,andlet.IijP ԰ ,=2Iik 7forall.i;j;kandzGIi1 06P԰= Ijv1 foriUR6=jӹ.8LetBkx:=L*ii?k U_jv=1!.Ik6j .csrc:5999advalg.texLetIFURRBbSesimple./Letpkx:Rn'!{BkbSetheprojectiononrtoBkw.r.t./Rn=URB14*t%:::t&BnP.Thenthereisatleastonekwithpk#(I)^6=0. ThenIPPQ԰i9=upk#(I)=J{?^Bk 2isasimpleideal.BecauseMof8.17MwregetIy(L* UXm U_ UXjv=risan;?RJ-ideal.*SinceBiCX=}I1:::In ㎹isadirectsumofsimpleRJ-idealsresp.*Bidڹ-idealsandsinceIjPV԰ >=Ik vholds,BiisasimpleringbryTheorem8.7.IInparticularBihasnotrwosidednonrtrivialideals,i.e.#thetwosidedidealsBiR!areminimal.$8.12impliesthatR!isArtinian.src:6030advalg.tex(8)W=)(7):?SinceBidBj 5Bi6\?Bj=0XtheWBi 1aresimpleringsasabSorve,henceWRO=R1Ι:::Rn %with}lRi=O#Bidڹ,bSecause}madditionandmrultiplicationarepSerformedintheBi(compSonenrtwise).src:6036advalg.tex(7)=)(1):8Lemma8.12.src:6038advalg.tex(7)U=)(9):~InorderUtoharveUcondition(7)symmetricinthesides,pitsucestoshorwthata_simple_ringRxԹisrighrtArtinian. ButRPL԰4=MnP(DS)P԰=[Hom-nD5$(Vp2\t:;V:2)isleftandrighrtArtinian.I7o]8SimpleTandSemisimpleringsandMoAdules73[o]8.4.No`etherianModules.src:6046advalg.tex{ЍDe nitionL8.23.[src:6047advalg.texAmoSduleF PMiscalledNoffetherian(EmmryNoether1882-1935),ɗifeacrhnonemptrysetofsubmoSdulesofM+hasamaximalelement.{эTheorem8.24.src:6052advalg.texFor35R &Mtthe35followingarffeequivalent:߈(1)%src:6054advalg.texMtis35Noffetherian.(2)%src:6055advalg.texEach ascffending chainMi,URMi+1AV;i2N ofsubmoffdulesofMKgbecomesstationary,i.e.%therffe35isannUR2N35withMn=URMn+i\foralli2N.(3)%src:6059advalg.texEach35submoffduleofMtis nitelygenerated.Prffoof.#Rsrc:6064advalg.tex(2)}q=)}r(1):^rLetMbSeanonemptrysetofsubmoSduleswithoutamaximalelemenrt.UsingV^theaxiomV]ofcrhoicewechoSoseforV]eachN62URManN@20do2MV]withN6$N@20.rFVorN62MwrethenhaveanascendingchainM1V=URN;Mi+1=M2@0RAiŹwithM1V$URM2$:::uD$Mi,$Mi+1$:::src:6071advalg.texThisisimpSossiblebry(2).src:6073advalg.tex(1)Y=)X(3):>BLetM@20  M@.@ThenfNjNN M20;N+ nitelygenerated^Eg 6=;YhasXamaximalelemenrt%FN@20.IfN@20 {6=lM@20,sthenthere%Gisanml2lM@20nN@20.SoN@20+RJmllM@20 4dis nitelygeneratedandN@20do$URN@20+RJminconrtradictiontothemaximalityofN@20.HenceN@20do=URM@20,}i.e.M@20Źis nitelygenerated.src:6081advalg.tex(3)"I=)"H(2):"LetM1tM2t:::2Mn \Q:::Mc-bSe"Ian"HascendingcrhainofsubmoSdulesofM@.NLetN6:=URS UTi2N+Midڹ.Nֹisa nitelygeneratedsubmoSduleofM@,i.e.NN6=URRJa1׹+:::xG+RanP.Then2thereisanMrewitha1;:::ʜ;an w2Mrb.ThisimpliesMra*=N=MrIf>RWisleftNoffetherianthenRJ[x]isleftNoffetherian.{ЍPrffoof.#Rsrc:6137advalg.texLetJqURRJ[x]bSeanideal..WVeharvetoshorwthatJ搹 nitelygenerated..LetJ0V:=URfr2RJj9p(x)UR2JwithhighestcoSecienrt)DrSg.5^(The"highest!coecienrtof!thezeropolynomial!is0bryde nition.)xnJ09fyaR!isanideal,#henceJ0=yahr1;:::ʜ;rnPi.xnFVortheridcrhoSosepidڹ(x)2ybJIwithhighestcoSecienrtsridڹ.Letmdegnh(pi(x))fori=1;:::ʜ;n.Letg2J withdegyi(gn9)m.Then gҹ=msx2ty\+瑟P>itXsidx2i. GSincesm2mJ0 |$wrehavesm=mP*Fn U_Fjv=1$Zjf rj. GThisimpliesg1 -:=i gZR!P*n U_jv=1!?Rjf pj(x)x2tdeg E(p8:j(x))682Janddeg](g1)tR!1.Byinductionwrehavegf9=g0%+R!dzKgJ7o]74oAdv|rancedTAlgebra{P9areigis[o]withg0V2URP*n U_jv=1!BRJ[x]pjf (x)anddeg(dzKg)URconsider1RJ20:=Z[rijJ]RJ,7the2subringofR|generatedbrytherijJ.X|SinceZisNoSetherianandRJ20uis nitelygeneratedasaZ-algebraRJ20isNoSetherian.3LetM@20do:=URP*n U_i=1 ASRJ20yi,URMչandN@20h|=Y_P* n U_ i=0!E`RJ20xi9Y_N@.7ThenpN20Y_M20ispanoRJ20烹-submoSdule,M20aspanoRJ20烹-moSduleis nitelygenerated,henceNoSetherian, andthefG(xidڹ)=Ԁyi;f(x0)=0generateahomomorphismofRJ20烹-moSdulesfG20:N@20 V!\M@20.\SincefG20) issurjectivrefG20isinjectivreandthusx0Y=0sothatfZйisinjectivre.fProblemyg8.1.nsrc:6219advalg.texWhereúdoSesûthecommrutativityúofRenterúthesecondpartoftheproSofofPropSosition8.29?Corollary8.30.!src:6224advalg.texLffetoR5beocommutativeoroR c~MbeNoetherian.LetoM=RJy1+ס:::y+סRym.Lffet\,N-Mbea\+freesubmodulewith\+thefree\+generatingelementsx1;:::ʜ;xnP.LThenn-m.If35nUR=mthenMtisfrffeeovery1;:::ʜ;ym.iPrffoof.#Rsrc:6232advalg.texSince;Nisfree:thereisahomomorphismfX):+NQ j!zMwithfG(xidڹ)+=*yifori=1;:::ʜ;minF(m;n)andfG(xidڹ)=0else.d|Ifnmܹthenf۹issurjectivre,)hencebijective.d|ThuswrehavenWm.úIfn=mthenf`isbijectivreandMYչfreewiththegeneratingelemenrtsy1;:::ʜ;ynP.CorollaryG8.31.C#src:6241advalg.texLffetRXbecommutativeorNoetherian.LetMbefreeoverx1;:::ʜ;xn T^andfrffee35overy1;:::ʜ;ym.fiThenwehavemUR=n.iPrffoof.#Rsrc:6247advalg.texIfORisNoSetherianthenM'3isPalsoNoetherian.7mThrustheclaimfollowsfrom8.30. De nitionf8.32.f~src:6252advalg.texLet_R̩bSecommrutative^or_Noetherian.Therffankofa^ nitelygeneratedfreemoSduleR ;M+isthenrumberoffreegeneratingelemenrtsuniquelydeterminedby8.31.Kf7o]8SimpleTandSemisimpleringsandMoAdules75[o]ExampleC8.33.esrc:6258advalg.texTheendomorphismringofacounrtablyin nitedimensionalvrectorspaceisneitherleftnorrighrtNoSetherian.Prffoof.#Rsrc:6263advalg.texFVrom6Bapn+mbq=1;pa=1;qn9b=1;pb=0;qn9a=06Bwreget6C(asintheexercise1.4)RRn=URR HRJpR ;RqXfreeandRR H=URaRR ;bRR ;free. De nition8.34.Dsrc:6270advalg.texAn welemenrt xrl2R%inaringR%iscalledaleftRHunit(rightunit),ifrSR(=R(RJr=URR).8r2URRiscalledaunit,ifRJr=URRn=rSRJ.Lemma 8.35.^fG(x)=f(u)+vimplies>^f(xu)=vn9;xu2fG21 {(Vp),andthrusxUR2Ut+33fG21(Vp),so,fG21(fG(U@)34+Vp)URUt+33f21 {(Vp).%aSince,UissmallwregetfG21 {(Vp) =M@.ThisbKimpliesfG(f21(Vp)) =fG(M@)V,4hencebKfG(U@)VandVe=N@.SowreharvefG(U@)smallinM.8ThisshorwsfG(Rad (M))UR=PUzsmall2-fG(U)PVsmall2jV¹=Radb(N).src:6565advalg.tex(2)ͯLetU6URMbSesimple.ThenfG(U@)Nissimpleor0.SowrehavefG(PUidڹ)URSoScH(N@). ~Corollary9.14.src:6571advalg.texRad?and35SoScYarffe35covariantsubfunctorsofIdL:URRJ-LMoSd !!0oRJ-LMoSd^.Corollary9.15.(1)ʞsrc:6577advalg.texLffetUMbesmallandf2HomjR#(M;N@).ZIThenfG(U)Nis%small.(2)%src:6579advalg.texLffet35U6URNtbelargeandfQ2URHom۟R"n(M;N@).fiThenfG21 {(U)URMtislarffge.Prffoof.#Rsrc:6585advalg.tex(1)wrasprovedinPropSosition9.13(1).src:6587advalg.tex(2)LetV#_MHandfG21 {(U@)\V=_0:ThenfG(f21 {(U@)\Vp)=0=fGf21 {(U@)\fG(Vp),jbSecauseifb(xv2wfGf21 {(U@)P\fG(Vp)b'withxv=wf(vn9),thenf(vn9)v2wU bryff21 {(U@)vwU. _Thisb'impliesv2jfG21 {(U@)\Vp,reso$ x2jfG(f21(U@)\Vp)j=j0. Norw$ this$ implies0=jfGf21 {(U@)\f(Vp)=Uu\4Im<(fG)4\f(Vp)=Uu\4f(Vp)andthrusf(Vp)=0,kbSecauseUislargeinN@.SowrehaveVURKe(fG)URf21 {(U@).-FVromf21(U@)e\eV¹=UR0wregetV=UR0.-ThrusfG21 {(U@)islargeinM@. Corollary9.16.(1)ʞsrc:6602advalg.texRad (RRJ)M6URRadb(M@).(2)%src:6603advalg.texSoSc(RRJ)M6URSoScH(M@).~Prffoof.#Rsrc:6608advalg.texLet='m2M@.0\Then(R3r5K7!rSm2M@)2Hom[FR#Nٹ(RJ;M@).0\This='implies=&RadI(RR)mRad (M@),SoSc*(RRJ)mURSoScH(M)andthatimpliestheclaim.VlCorollary9.17.src:6615advalg.texRad (RRJ)35andSoSc&(RRJ)arffetwosidedideals.Prop`osition9.18.src:6619advalg.texLffet35fQ2URHom۟R"n(M;N@)andKe(fG)URRadb(M).fiThen35wehaveōRhfG(Rad (M@))UR=Radb(f(M@)):Prffoof.#Rsrc:6625advalg.tex:8follorwsfrom9.13.src:6627advalg.tex:ZLetfG(m)UR2Radb(f(M@)).2IfRJmM0issmallthenm2Radb(M@)andfG(m)2f(Rad (M@)).IfRJmURMlisnotsmallthenbry9.6thereisamaximalsubmoSduleU6$URMlwithm=2U@.WVeharveRJmA"+U6=URMandthrusfG(U@)+RJf(m)UR=f(M@).'FVromf(m)2Radb(f(M@))wregetthatRJfG(m) f(M@)TisTsmall.vThisimpliesfG(U@) =f(M@)TandTthrusU3+KeN((f) =M@.vFVromTtheassumptionKeE(fG)URRadb(M@)U+wregetU6=URM,aconrtradiction.ui?~Corollary9.19.src:6640advalg.texLffet35N6URMtbeasubmodule.fiThenthefollowinghold⍍(1)%src:6643advalg.tex(Rad (M@)+N)=XN6URRadb(M=N).(2)%src:6644advalg.texN6URRadb(M@)UR=)Rad(M@)=XN6=Rad(M=XN@).Prffoof.#Rsrc:6650advalg.tex(1)ǂfQ:URM6!M=XNfimpliesǁfG(Rad (M@))Radb(M=XN@)ǂandfG(Rad (M))UR=(Rad (M)b+N@)=XN.src:6653advalg.tex(2)FVromN6=URKe(fG)URRadb(M@)theclaimfollorws.dCorollary9.20.src:6658advalg.texRad (M@)35isthesmallestsubmoffduleU6URMtwithRad?(M=U)UR=0.~Prffoof.#Rsrc:6663advalg.texWVe eharveRad$(M=Rad (M@))=Rad(M)=Rad(M)=0. If eRad$(M=U@)=0 ethenRad (M@)+U=U6=UR0andthrusRadg(M)+U6=URU+sothatRadg(M)URU.b~Lemma9.21.src:6669advalg.texIf35SoSc&(M@)UR=Mtthen35Rad?(M)=0.O[7o]ƶ RadicalTandSoAclev$79[o]Prffoof.#Rsrc:6673advalg.texIf6SoSc(M@)E=MOholds6then5Missemisimple.Sono5submoSduleissmallandthrusRad (M@)UR=0. kDLemma9.22.src:6678advalg.texLffet35MtbeA2rtinian.fiThenwehaveRad(M@)UR=0()SoScH(M)=M:Prffoof.#Rsrc:6683advalg.texLet׬MbSe׫ArtinianandRadk(M@)UR=0.7Let׬U6MandNbSeminimalwith׫N+xU6=M@.By89.4(2)wre8have8N \U4PMysmallsothatN\U4=P0.#FThrusUyis8adirectsummandofM@,M+issemisimpleandM6=URSoScH(M).|kEProp`osition9.23.src:6691advalg.texThe35followingarffeequivalentforM@: (1)%src:6693advalg.texMtis35 nitelygenerffated35andsemisimple.(2)%src:6694advalg.texMtis35A2rtinianandRad?(M@)UR=0.Prffoof.#Rsrc:6699advalg.texIt 6sucestoshorw 5thefollowing:{IfMMis 5semisimple,thenMMis nitelygeneratedi MisArtinian.Let MbSesemisimple.ThenM6=URUiwithsimplemodules Uidڹ.Mis nitelygeneratedOifandonlyNifthedirectsumhasonly nitelymanrysummands(6=0).IfM#3isArtinian?then@thedirectsumhasonly nitelymanrysummands.ͦIfthedirectsumhasonly nitelyFmanrysummands,mtheneachdescendingchainN1 ]1-N2.:::inEM*canFonlyhave nitelyjmanrydirectcomplementsby8.17.EThussuchachainmustbSecomestationaryV,i.e.M+isArtinian.fpkDProp`osition9.24.src:6712advalg.tex(LemmaofNakXaryama)35ForR &IFURR HRLthefollowingarffeequivalent:(1)%src:6715advalg.texIFURRadb(RRJ).(2)%src:6716advalg.tex1+I$cffontains35onlyrightunits.(3)%src:6717advalg.tex1+I$cffontains35onlyunits.(4)%src:6718advalg.tex1+IRLcffontains35onlyunits.(5)%src:6719advalg.texIM6=URM=)M=035forall nitelygenerffated35modulesR &M@.(6)%src:6721advalg.texIM댹+U6=URM=)U=Mtfor35all nitelygenerffated35modulesR &M@.(7)%src:6723advalg.texIM6URRadb(RM@)35forall nitelygenerffated35modulesR &M@.Prffoof.#Rsrc:6729advalg.tex(1)DP=)(2):0Rad(RJ)R]isDPsmall.EThrusIpRissmall.EFVromRJ(1+i)+Ip=RitfollorwsRJ(1+i)UR=R.8Thrus1+iisarighrtunit.src:6734advalg.tex(2)4=)i (3):5Letkg(1+i)=31. Thisimplieskgi=31kGQ2IZandi thrusk142I. Sokq=zU1u+v(k 1)fisarighrtunit.zSincekgisgalsoaleftunit,weget(1v+i)kq=zT1,softhatg1u+iisaunit.src:6740advalg.tex(3)=)(4):yQGivrenif 2f I|dandr2RJ.Then1+rSiisaunitwithinrverse(1+ri)21 \|.Since(1i(1+rSi)21 \|r)(1+irS)=1+irC'i(1+ri)21 \|(rC'+rir)=1+irC'i(1+ri)21 \|(1+ri)rU;=1[?+[>irir=UR1andsymmetrically(1[?+irS)(1i(1+rSi)21 \|r)UR=1wregetthat1[?+irUisaunit.src:6747advalg.texIfa۹isaunitandiUR2I,r2R$thenag+firhisaunit,sincea(1f+a21 \|irS)UR=(ag+fir)isaproductoftrwounitsbya21 \|iUR2I.src:6752advalg.texIf7P*2n U_2k6=1"2ik#rkx2URIRthen71+PBikrkɹis8a7unit,since1+PBikrkx=UR(((1+i1r1)+i2r2):::5+inPrn)andeacrhofthebracketedtermsisaunit.src:6758advalg.tex(4)=)ǹ(5):LetMڪbSe nitelygeneratedandIMt=3#M@. F;LettbSetheminimallengthofCNaCOsystemofgeneratorsofMع=RJm1UT+O:::?+ORmtʹ. BByIMٹ=M3eacrhCNelementinCNMcan]WbSe]Xrepresenrtedasa nitesumoftheformPi20RAjf m20RAj;them20RAj acanbSe]WrepresenrtedasaFlinearcomrbinationofthemidڹ.@SotherearecoSecienrtsik#rk 2Iǡwithm1 =P*t U_k6=1#ik#rkmk.AThisBimplies(1i1r1)m1 ' =gP*t U_k6=2%ik#rkmk. SinceBalso1i1r1 xFisaunit,+wregetm1 ' =P* t U_ k6=2(1i1r1)21 \|ik#rkmk +2RJm2+:::¹+RmtȹaIconrtradictionItotheminimalitryoft.VSowreharveM6=UR0.src:6771advalg.tex(5)I=)(6):'IMX+dtU6=URM=)URI(M=U@)=(IMW+dtU@)=U6=M=U=)M=U=0=)M=U@.Pv7o]80oAdv|rancedTAlgebra{P9areigis[o]src:6775advalg.tex(6)=)(7):8IM+smallinM6=)URIMRadb(M@).src:6778advalg.tex(7)=)(1):8M6=URRn=)IRnRadb(RRJ). RCorollary9.25.src:6783advalg.texRad (RRJ)UR=Radb(RR).Prffoof.#Rsrc:6787advalg.texLet2It =RadI(RRJ). Then1+I#consistsofunits.SinceI#isarighrtideal,wegetIFURRadb(RR).8BysymmetrywregetRadg(RRJ)UR=Radb(RR).,MLemma9.26.src:6793advalg.texRLleft35A2rtinian=)URRJ=Rad (R)35semisimple.Prffoof.#Rsrc:6798advalg.texBy8.12RJ=Rad (R)isArtinian.By9.20Rad%(RJ=Rad(R))UR=0andbry9.23RJ=Rad(R)issemisimple.Lemma9.27.src:6804advalg.texRLA2rtinian35=)URRadb(RJ)nilpffotent.Prffoof.#Rsrc:6808advalg.texLet!I7:=ѳRadr(RJ).MSinceRkis"Artinian,thecrhainI6ѳI22 ;I23 :Ѵ:::nI2t+11}=:::bSecomesfstationaryV.AssumeI2t,6=(0.SincealsoI2tMIb6=(0thereisfaminimalmoSduleK}Iw.r.t.UI2tMK16=0.SoIethereexistsanIdx2K&withI2tMx6=0,ai.e.UwrehaveK1=RJx.UBecauseofI2tMK;b=^I2t+1_K=^I2t(IKܞ)^6=05and4IK;bKҹwre5get4IK;b=Kܞ.IBy4the5LemmaofNakXaryama4wegetK1=UR0,aconrtradiction,soI2t؟=0. %Theorem9.28.src:6819advalg.tex(Hopkins)35LffetR &RLbeA2rtinian.fiThenR &RLisNoetherian.Prffoof.#Rsrc:6824advalg.texLet IFչ:=URRadb(RJ) andI2n+1ˡ=UR0.6VThenI2iV]=I2i+1isanRJ=I-moSduleanditisArtinianasanRJ-moSdule.SoI2iV]=I2i+1:۹isalsoArtinianasR=I-moSdule.By9.26R=IissemisimplehenceI2iV]=I2i+13isalsosemisimple,/i.e.I2i=I2i+1+=URk62XkEk$kwithsimpleRJ=I-moSdulesEk#.SinceI2iV]=I2i+1isArtinianthedirectsumis nitehenceI2iV]=I2i+1AareNoSetherian(asRJ=I-moSduleandasR-moSdule).1Withmthenexactsequences0P!H-I2i+1*!${I2i 9!I2iV]=I2i+1*!0,withmI2n+1 =P0;I20 Hع=Randwith8.25wregetbyinductionthatRisNoSetherian.LCorollary9.29.src:6838advalg.texIf35R &IFURR HRLis35nilpffotentthenIURRadb(RJ).Prffoof.#Rsrc:6843advalg.texLetSI2n =0Sandi2I.qThenS(1+i)(1i+i22:::i2n+1̹)=1ShenceS(1+i)isSaunit.8BytheLemmaofNakXaryamawegetIFURRadb(RJ).oProp`osition9.30.src:6850advalg.texRMtis35 nitelygenerffated35ifandonlyifs2(1)%src:6852advalg.texRad (M@)URMtis35small,and(2)%src:6853advalg.texM=Rad (M@)35is nitelygenerffated.Prffoof.#Rsrc:6858advalg.tex=):8trivialbry9.12.src:6860advalg.tex(=:sLetKfdzRKxRi =xiQ+Rad(M@)ji=1;:::ʜ;ngKbSeasetofgeneratingelemenrtsofM=Rad (M@).Then'M6=URRJx1+*::: +*RxnN+*Rad7(M@)(whicrhimpliesby((1)thatM6=URRJx1+*::: +*RxnP. Corollary9.31.csrc:6867advalg.texM'6iscalledinvertible,ifrisaleftorarighrtunit.src:6895advalg.texRiscalledaloffcal35ring,ifthesumofanrytwononinvertibleelementsisanonunit.Lemma10.2.src:6900advalg.texLffet35rbeanidempotent(rS22h=URrS)inalocalringRJ.fiThenr=UR0orr=UR1.Prffoof.#Rsrc:6905advalg.texWVeJharveJ(1rS)22=12r?#+rS22 5=1rS.VSinceJ1=(1r)+risJaunit,aror1risminrvertible.ƒIfrmisinvertible,e.g.‚bymsr6=41,thenmwehaver7=4srS22 H:=sr6=1.ƒIf1rlisinrvertiblee.g.8bys(1rS)UR=1,thenwrehave1r=UR1,thrusr=UR0.uTLemma/10.3.3src:6913advalg.texLffet'Rrbe(aringwiththeuniqueidempffotents0and1.}ATheneachinvertibleelement35inRLisaunit.Prffoof.#Rsrc:6918advalg.texLetrGbSeinrvertiblee.g.U)bysr=e^1.U*Then(rSs)22%c=e_rsrs=e_rs,qsors2e_f0;1g.U)Ifrs=e_0,thenwrehave1UR=(srS)22V=srsr=0,aconrtradiction.8SowehaverSsUR=1,i.e.8r>6isaunit. wCorollary10.4.src:6925advalg.texIn35aloffcal35ringRLallnonunitsarffenotinvertible.Prop`osition10.5.src:6929advalg.texLffet35RLbealocalring.fiThenthefollowinghold:s2(1)%src:6931advalg.texA2ll35nonunitsarffenotinvertibleandformatwosidedidealN@.(2)%src:6933advalg.texNtis35theonlymaximal(onesideffdandtwosided)andlargestidealofRJ.󆍍Prffoof.#Rsrc:6939advalg.tex(1)LetNbSethesetofthenonunitsofRJ.SinceRisloScal,sononunitsarenotinrvertible,mrN-isSIclosedw.r.t.rtotheaddition.Givrensj2iN-andSJrZ2RJ.rWVeshorwSIthatalsorSsp92p8N;Zholds.hJInvfactifrsՑ=p92rN;Zthenrsisaunit,jsothereisautp92Rwithvtrsp8=1.hJBecauseof10.3sisalsoaunitinconrtradictiontosUR2N@.8ThusN+isatwosidedideal.src:6947advalg.tex(2)weObrviouslywehaveN$DRJ.IfI6b$DRands2DI,thenRJs$DR,sowesiswdanonunitandthrussUR2N@.8SoIFN+holds.8Prop`osition10.6.src:6953advalg.texRTis;Lloffcal,}Qif;KandonlyifRTpffossessesauniquemaximal(larffgest)leftideffal.󆍍Prffoof.#Rsrc:6958advalg.tex=):8follorwsfrom10.5.src:6960advalg.tex(=:JYLetseNIbSesdtheonlymaximalidealofRJ.ThenN~=>RadJ(R)issdatrwosesidedideal.Letr2kRnN@.` ThenNp+RJr=kR.` SinceNp=RadxK(R)issmallinRJ,wrehaveRJr=kR,sothereisIaHtwithtr =[~1.CIftisarighrtunit,1thenalsorAֹisaunitbryLemma8.35.CButiftisnotarighrt3unit,EthenRJtј6=їR,Eso3RtNtand3thrus3t2їN@.SinceNtis3atrwosidedideal3wehavealso1UR=tr2N@,>aconrtradiction.$IThuseachr2URREn,Nɹisaunit.$JSoeacrhnonunitliesinN@.If^x,9y,arenonunits,:thenitfollorwsfromx;yË2URNAthatxP2+P1y2URNBhencexP2+P1y,isanonunitandthrusRisloScal.g`Lemma10.7._bsrc:6975advalg.texLffetrxRberwaloffcalrwringwithmaximalideffalmUR$RJ.&*LetrxM[berwa nitelygeneratedmoffdule.fiIf35M=mM6=UR0thenM=UR0.󆍍Prffoof.#Rsrc:6981advalg.texFVrommUR=Radb(RJ)andmM6=MxitfollorwsthatM6=0brytheLemmaofNakXayama.k10.2.Lo`calization.src:6989advalg.texInthissectionletRbSealwraysacommrutativering.src:6991advalg.texRecallfromBasicAlgebra:8AsetSwith;UR$S)Riscalledmultiplicffatively35closed,ifen8s;s 0#2URS):ss 02SoandJv30=2S :src:6996advalg.texOnRSde neanequivXalencerelationbryɊ(rr;s)UR(rS 0!;s 09):()?9t2S):tsrS 0w=ts 0rr:RM7o]82oAdv|rancedTAlgebra{P9areigis[o]RJ[Sן21 S]~o=Sן21R:=R:!S =6isacommrutative7ring6withunitelemenrt.DTheelementsaredenotedbryōr}rr}[z ΍ sښ:=UR`z p(rr;s)=B:3src:7002advalg.texThemap'UR:Rn3r7!ōsr[z ' ΍ύs92RJ[Sן 1 S]src:7004advalg.texisahomomorphismofrings.hItisindepSendenrtofthecrhoiceofs$2S׹.hIfRԹhasnozerodivisors,then'isinjectivre.7Prop`osition10.8.zsrc:7009advalg.texLffetS'R)Ebeamultiplicativelyclosedset.LetR MPbeanRJ-module.Then35therffelation/㍑{O(m;s)UR(m 09;s 0)UR:()9t2S):tsm 0#=ts 09msrc:7013advalg.texon35MS isaneffquivalence35relation.fiFurthermoreYpSן 1 SM6:=URMS =with35theelementsō~m~ϟ[z G ΍_si:=`z7 p(m;s)&src:7016advalg.texis35anSן21 SRJ-moffdulewiththeoperationsVmō{!m{![z G ΍_s+ōm20۟[z N ΍_s0{=ōs209m+sm20[z36 ΍ss0I+andōm?rm?[z ΍ sōumu[z G ΍_s$0G=ōrSm[z. ΍ss0:Prffoof.#Rsrc:7022advalg.texasinBasicAlgebraforSן21 SRJ.Problem10.1.src:7026advalg.texGivreacompleteproSofofProposition10.8.Lemma10.9.src:7030advalg.texFu33m33zğꍑDs @|=UR035holdsinSן21 SMtifandonlyiftherffeisat2S withtm=0.Prffoof.#Rsrc:7035advalg.tex(m;s)UR(0;s209)()9t20#2URS):t20s20m=0()9t20s20#2URS):t20s20m=0.P=5Lemma10.10.(1)}8src:7041advalg.tex'M B:URM63m7!Fusmz sꍑbs 2Sן21 SMXisuathomomorphismofgrffoupsindepen-%dent35ofsUR2S.(2)%src:7043advalg.tex'MjisinjeffctiveifandonlyifScffontainsnozerffodivisorsforM@,i.e.sm=0=)%mUR=0.(3)%src:7046advalg.tex'M %is35bijeffctiveifandonlyifthemapM63URm7!sm2Mtis35bijectiveforallsUR2S.(4)%src:7048advalg.tex'R &is35ahomomorphismofrings.(5)%src:7049advalg.tex'M B:URM6!QSן21 SMtis35'R-semilineffar,i.e.fi'M (rSm)='R(r)'M (m).8Prffoof.#Rsrc:7056advalg.tex(1)t209(tsmstm)UR=0impliesFusm۟z sꍑbs`=Futmz ꍑbtRF.#Zsrc:7059advalg.tex(2)'M (m)UR=0()Fusmןz sꍑbs\=0()9t2S):tm=0bry10.9.src:7062advalg.tex(3) r'M asurjectivreQ()a8Fu33m33zğꍑDs 2Sן21 SMaU9m202MĹ:Fusm-:0z$ǟꍑFs=FumzğꍑDs()8m2M;s2SI9m202Mù:sm20#=URm()8s2S):(s:M6!M@)surjectivre.j.src:7067advalg.tex(4)+(5)'M (rSm)UR=Fus-:2*rzcsts0t0 s209t20m =Fu \t33zcsts0t0t st209mUR=Fu؇tst-:0zcsts0t0 m20#=Fu؇tz ͟s0t- m20.src:7119advalg.texWVeharve O h=URid ,since (M@) (M)(Fu33r33zbꍐs #p m)UR= O(M@)(Fu33rN_'bSeamonomorphismandBletSן21 SfG(Fu33m33zğꍑDs *)=`0=Gf(m)z[ꍑsߨ.ӰThenthereishat2SĹwithhtfG(m)=0=f(tm),~sohwithtm=0. ThenFu m zğꍑDs4=0,hencehSן21 Sfisamonomorphism.zr!_src:7274advalg.texRecallfromBasicAlgebra:[(1)%src:7276advalg.texAn[idealpqrRiscalledaprime{ideffalZifandonlyifp6=qRand(rSs2qp꨹=)rW2%p_sUR2p).(2)%src:7279advalg.texIfmURRisamaximalideal,thenmisaprimeideal.(3)%src:7281advalg.texpUR2R isaprimeidealifandonlyiftheresidueclassringRJ=pisaninrtegraldomain. Lemma10.18.src:7286advalg.texLffet35pURRLbe35anideal.fiThefollowingareequivalent(1)%src:7289advalg.texp35isaprimeideffal.(2)%src:7290advalg.texRnp35isamultiplicffativelyclosedset. Prffoof.#Rsrc:7295advalg.texfollorwsimmediatelyfromthede nition.y De nition!E10.19.src:7299advalg.texLet5pURRbSea4primeidealandMbeanRJ-module.dThenM1\%eufm8pY:=URSן21 SMwithS)=URRnpiscalledtheloffcalization꨹ofthemoSduleM+atp.src:7304advalg.texThesetSpSec(RJ):=fpRjp꨹primeideal=\Hg㓹iscalledthespffectrum㓹oftheringRJ.#ThesetSpSecm!"(RJ)UR:=fmRjm꨹maximalidealL >g꨹iscalledthemaximal35spffectrum꨹oftheringR.Prop`ositionJ10.20."src:7311advalg.texLffet:M{be:anRJ-moffdule,<suchthatMm AJ=c0forallmc2SpSec6(RJ).|ThenM6=UR0. Prffoof.#Rsrc:7316advalg.texAssume`thereisanm2M{with`m6=0.ThenI:=Keyf(R7X3rq7!rSm2M@)$Ryisanmideal.SinceRKis nitelymgeneratedthereisamaximalidealmwithI$31m$RJ.SinceMm 3=UR0,iwreyhaveFu m zğꍑDsU=0yinyMm5,jhencethereisatUR2R݈n?myٹwithtm=0.FThis,ihorwever,givestUR2IFm,aconrtradiction.Bc>Corollary10.21.src:7326advalg.texLffet35fQ:URM6!QNtbegiven.fiThefollowingareequivalent[(1)%src:7328advalg.texf{4is35amono-(epi-rffesp.fiiso-)morphism.(2)%src:7329advalg.texFor35allmUR2SpSec((RJ)35theloffcalization35fm jisamono-(epi-rffesp.fiiso-)morphism. Prffoof.#Rsrc:7335advalg.tex(1)P=)",(2):8follorwsfrom10.17and10.12. src:7337advalg.tex(2)j#=)!ѹ(1):IThe{sequence0URn!1Ke#(fG)URn!1Mh 'mfJ6!N6!Cok*(f)URn!10zis{exact.&Consequenrtly޻wJ0URn!1Ke#(fG)m 3 M!fMmh f2e{eufm6mJ 3 M!Nm 3 M!Cok02X(f)m 3 M!0Ҷsrc:7341advalg.texis+exact.ThruswegetinparticularKeI(fG)mP Vf԰ oN=#Ke)E(fm5)andCok(f)mP Vf԰ oN=#Cok/(fm5).Norwiffm @is amonomorphismforall m2SpSecm&b(RJ),then wrehaveKeb(fG)m %=0forallm,henceKe[W(fG)UR=0i.e.RpmisloScalandpplisthemaximalideal.CorollaryG10.23.rsrc:7369advalg.texLffetpURRbeaprimeideal.-Thenthequotient eldQ(RJ=p)isisomorphicto35Rp=pp.V 7o]86oAdv|rancedTAlgebra{P9areigis[o]Prffoof.#Rsrc:7374advalg.texAsGintheFprecedingproSof(RJ=p)pPIL԰b4=?Rp=pp.FVurthermoreGRp=ppMisa eld,"nbSecauseppisthemaximalidealofRp.8FVurthermorewrehave濍3(RJ=p)pY=URSן 1 S(R=p)=fō33dzKr33[z ΍ sjdzKrk2R=p;s=2pgP԰n:=fō33dzKr33[z ΍ dzKsjdzKrk2R=p;dzKs 12R=p;dzKs 16=0g=Q(R=p):;<kProp`ositionTs10.24.8Msrc:7384advalg.texLffetCR 7#MubeCa nitelyCgenerffatedCmodule.LetM=mM6=UR0Cforallmaximalideffals35mURRJ.fiThenM6=0.Prffoof.#Rsrc:7389advalg.texM=mMP6԰=@RJ=m7 R +TMP6԰=Rm5=mm  RmjIRm  RMP6԰=Mm5=mmMm.& SincegRm isloScalandMm RTist nitelygenerated,}itfollorwst thatMm =?N0forallmaximalidealsm?MRJ.FSowregetM6=UR0.RCorollary10.25.-src:7397advalg.texLffetfQ:URM6!QNSbeanRJ-homomorphismandletNSbe nitelygenerated.Lffet4fG=mf:XM=mMk!X^NF:=mNubean4epimorphismforallmaximalidealsmXRJ.kThenf|isan35epimorphism.aXPrffoof.#Rsrc:7404advalg.texMh fJ(!N(!QD!0#KisexactandthrusQis nitelygenerated.WVeapplythefunctorRJ=m5 R ǹ-yandggetthefexactsequenceM=mM!!NF:=mN!!Q=mQ[u=!0.DSincefG=mf6eisanepimorphism,wregetQ=mQUR=0,henceQUR=0.8Sof2isanepimorphism.[W%7o]WMonoidalTCategories87[o]11.MonoidalCa32tegoriessrc:7443advalg.texFVorōourŌfurtherinrvestigationsitōisusefultoinrtroSduceageneralizedvrersionofatensorproSduct.TۍB(I+ A) B I+ (A B)t|:2fd0ά-Í  A BꃀN(A) 1X.BiL?`@sL?`@}L?`@~T@~TRꃀQL(A Bd)M?`M?`M?`LL S(A B) ITSA (BE I)9:2fd0ά-ÍBy 4A Bꃀ(A Bd)s?`@(s?`@2s?`@2@2Rꃀc1X.A ^ (Bd)a?`W?`M?`MlMl Ǎsrc:7602advalg.texand35(I)UR=(I)35holds.@Prffoof.#Rsrc:7606advalg.texWVe_ rst_observrethattheidenrtity_functorId$9CandthefunctorI h- :areisomorphicbrythenaturalisomorphism.8InparticularwrehaveI+ fQ=URI gË=)URfQ=gn9.8Inthediagram v/n((I+ I) A) Bvqn(I+ (I A)) B4sH2fd)ά-omB 1vvI\nI+ ((I A) B)4sH2fd)ά-pÍ.X sT( 1) 1jJ4 QtJ4Q~J4M^QJ4Q0Q0ss (1 ) 1Ԍ ʌM^00+s+*1 ( 1)aw WwMwM^Cw=0=0+(I+ A) BI+ (A B)KA2fdG)ά-Í% s]rǠ|z@fe]Ǡ?CcS͍ swrǠ|z@fewˤǠ?|~$1 7Π@fek$Π?Cc 1"Π@fe1V$Π?i(c1I+ (A B)I+ (A B)K:2fdG)ά-Y1/n(I+ I) (A B)I\nI+ (I (A B))432fdά-Í% ꃀT (1 1)jJ4ǠtJ4~J4rLJ4Ǣzz3=Y1 = QG攴QQ?^Q[Qaw0Qaw0sYC7o]WMonoidalTCategories89[o]src:7640advalg.texallsubSdiagramscommrute,MexceptfortherighrthandtrapSezoid.cSincethemorphismsareisomorphisms,*alsom therighrthandtrapSezoidcommutes,*hencethewholediagramcommutes.src:7645advalg.texThecommrutativityoftheseconddiagramfollowsbyanalogousconclusions.src:7647advalg.texFVurthermorethefollorwingdiagramcommuteso?`pI+ (I I)s(I+ I) IߤA2fdάÍ, ,eI+ (I I) ѤA2fdά-Í Ԡr%I+ IWMu1 'F`@'F`@'F`@D@DR|GڬبDF`ΨDF`ĨDF`   Ԡ d%I+ IWMu 1F`@F`@F`@D@DR81 6DF`,DF`"DF`"" ЎIuG ?`@ ?`@ ?`@بD@بDRuG D?`D?`D?` ܍src:7661advalg.texHerethelefthandtrianglecommrutesbSecauseofthepropSertryshownbSefore,therighthandtriangleisgivrenthroughtheaxiom.!Finallythelorwersquarecommrutes,sinceisanaturaltransformation.AInLparticularKwreget(1 )|=|(1  ).ASinceLisanisomorphismandI+ -P ԰ =KId!YC&G,itfollorwsUR=.:X‹Problem.Q11.1.Mysrc:7671advalg.texFVor_morphisms`f:WIIb!/MCandgȹ:WIIb!.NDinamonoidalcategoryC5wrede ne(fI 1UR:N6!MB N@):=(f 1IM)(I)21oand(1 gË:URM6!MB N@):=(1 gn9)(I)21 \|.ShorwthatthediagramN 덍NM N\32fdm`ά- f 1bYIbY#I,{fdά-fHzǠ*Ffe$Ǡ? cgH&zǠ*Ffe'Ǡ?W+,1 gsrc:7677advalg.texcommrutes.ŒDe nition11.6.src:7681advalg.texLet(C5; )and(DUV; )bSemonoidalcategories.8AfunctorQ Fc:URC"!wfDۍsrc:7683advalg.textogetherwithanaturaltransformations(M;N@)UR:F1(M) F1(N)URn!1F(M N)src:7685advalg.texandamorphismw 0V:URID @!F1(ICm)src:7687advalg.texiscalledweffakly35monoidal,ifthefollorwingdiagramscommute:src:7689advalg.text> HKB',X.Bϼׁ Aϼ Aϼ Aϼ A>A>U̞32fd3- 7f[l7o]WMonoidalTCategories91[o]RemarkM11.8.3fsrc:7791advalg.texObrviouslyߋthecompSositionߊoftwomorphismsofalgebrasisߊagainamorphismof+algebras.jAlsotheidenrtity,morphism+isamorphismofalgebras.iThruswe+obtainthecategoryAlg(C5)ofalgebrasinC.De nitionD11.9.-src:7798advalg.texAcffoalgebra*oracffomonoid)inamonoidalcategoryC_isanobjectCȹtogetherwithacomrultiplicationUR:C1K{!CF Cܞ,thatiscoassoSciative:JYCYCF CF{fdY?ά-ˍ[HǠ*Ffe,4Ǡ?`dXHgǠ*FfeǠ?Ѝ$M4 id5CF C CF C C@32fdAJά-`VidM^ Ysrc:7808advalg.texandacounitUR:C1K{!I,forwhicrhthediagramYCYCF C<{fdY?ά-ˍQH"Ǡ*Ffe"TǠ?`ZxH]Ǡ*FfeԟǠ?ЍCTid!\ UCF CI+ CP1԰Jع=ܙCP1԰Jع=CF I732fd+7ά-`A idHk|]idҁ Hׁ HÓ܁ H͓ Hד H H H H džH džjsrc:7818advalg.texcommrutes.src:7820advalg.texLetC+andDbSecoalgebras.*Amorphism of cffoalgebrasfQ:URC1K{!DisamorphisminC5,-sucrhthatE@bYCbYDqD{fdd߰ά-iVfH#ŸǠ*FfeVǠ?X.CH$BǠ*Ffe$tǠ?)wX.DCF C[D6 Dk32fdLrά- Pf f-src:7829advalg.texandbYȵCbYDr<{fd*(pά-ifHvI,̔X.Ctׁ At At At A\>A\>UH,,X.Dׁ    ϼ>ϼ> s2src:7835advalg.texcommrute.Remark11.10.src:7839advalg.texObrviouslythecompSositionoftwomorphismsofcoalgebrasisagainamor-phism#of#coalgebras.Alsotheidenrtitymorphism#isamorphismofcoalgebras.ThrusweobtainthecategoryCoalg(C5)ofcoalgebrasinC.\}7o]92oAdv|rancedTAlgebra{P9areigis[o]12.BialgebrasandHopfAlgebras12.1.Bialgebras.src:7861advalg.texDe nitionn12.1.Tsrc:7862advalg.tex(1)lAhbialgebrffa(B;r;n9;;)lconsistskofanalgebra(B;r;n9)kandlacoal-gebra(B;;)sucrhthatthediagramstJiX|8BE B4iBE B B B}A2fdLά-/S, ,1 r 1tF`HtF`HtF`H$tF`H.tF`H8tF`H9H9ji&ǠM@feiYǠ?»]rdBCY8BE Bq32fdSЍά- "FԠ+&iBE B B BTǠ@feT6Ǡ?ԏX4r rsrc:7877advalg.texandHt䍍j⍑pրKSaxB'[$ldׁ gd bd ]d \A>\A> HGXI{ y\ׁ A~\ A\ A\ A>A>USaxBPBE B`]32fd.ά- k>YBE BY8Bܟ{fd.ά-ˍb.rH""EK`ǜD Ѱ<ׁ Aְ< A۰< A< A|>A|>UH]CL/ׁ / / /  ܟ> ܟ> j⍒>WKj⍒y KKܟ{fd*ά-,\~idH[B'E$Gׁ AL AQ AV AWA>AWA>UH]Cv5dz\Ǡiz\Ǡnz\Ǡsz\Ǡtׂtׂpsrc:7892advalg.texcommrute,9i.e.)and)arehomomorphismsofalgebrasresp.randarehomomorphismsofcoalgebras.src:7896advalg.tex(2)GivrenbialgebrasAandB.*AmapfQ:URAn!1BZҹiscalledahomomorphism ofbialgebrffas̹ifitisahomomorphismofalgebrasandahomomorphismofcoalgebras.src:7900advalg.tex(3)ThecategoryofbialgebrasisdenotedbryK-Bialg .Problem%12.1.òsrc:7904advalg.tex(1)Let(B;r;n9)bSeanalgebraand(B;;")bSeacoalgebra.ThefollorwingareequivXalenrt:􍍍qa)%src:7908advalg.tex(B;r;n9;;")isabialgebra.ʤb)%src:7910advalg.texUR:BX !_7BE Band":BX !K꨹arehomomorphismsofK-algebras.c)%src:7912advalg.texrUR:BE BX !_7BandË:Kn!1BarehomomorphismsofK-coalgebras.src:7916advalg.tex(2)LetMfand%Nbffeleft%B-modules.>Then%M K cNtis35aB-moffdulebythemapV68BE M N2G 1p6 !@B B M N261 r 1p!vq!"VB M B Nh՛ J6 !@M N:鍍(2)%src:7935advalg.texLffet&Bbe&abialgebra.A2LetMgandNgbeleftB-comodules.A2ThenM K NgisaB-%cffomodule35bythemap68M N2L p6 !@BE M BE N261 r 1p!vq!"VB B M N2Gr 1p6 !@B M N:荍(3)%src:7945advalg.texK35isaB-moffdulebythemapBE KPUR԰n:=B2 "pX E!K.(4)%src:7948advalg.texK35isaB-cffomodule35bythemapKh LJUR !BPX԰ @=BE K.]7o][BialgebrasTandHopfAlgebras93[o]Prffoof.#Rsrc:7954advalg.texWVegivreadiagrammaticproSoffor(1).8TheassociativitrylawisgivenbyVvBE B M Nv&BE B B M NYsH2fd#ά-o/W1  1 1vvgfBE B M B N0sH2fd#:ά-om1 1 r 1vvBE M NdpsH2fd0ά-o>͍Y1  9~BE B B B M N^BE B B M B N`A2fd(eά-m=1 1 1 r 1MBE B M Nz!@A2fd[ά-7͍uK1 1  Ԡ9~BE B B B M NԠ^BE B M B B N`:2fd(eά-έyyЀk1 1 r(Bd B;M") 1ԠԠMBE M B Nz!@:2fd[ά-0͍uK1  1  BE M N׍BE B M NMP32fd;Ѝά-/]  1 1%BE M B N32fd;kά-m#1 r 1"M NL32fd0aά- s*徟Ǡ|z@fe+Ǡ?ۏ/pr 1 1sT>ՠ@fepՠ?⏍9 1 1 1 1sDՠ@feDՠ?⏍Ip 1 1 1 1s~ՠ@feՠ?⏍0 1 1T>Π@fepΠ?91 r 1 1 1DΠ@feDΠ?Ip1 r(Bd;B M") 1 1~Π@feΠ?01 r 1T>Ǡ@fepǠ?ԏ9r r 1 1DǠ@feDǠ?ԏIpr 1 r 1~Ǡ@feǠ?'0 src:8031advalg.texTheunitlarwisthecommutativityofJv"f=M NP6԰=@K M Nv"%iBE M NݛsH2fdDvpά-o>͍I{ 1 1"rK K M N"lBE B M NTA2fd$+@ά-7͍+I{  1 1ԩ"eM NP6԰=@K M K Nԩ"lBE M B NT4:2fd `ά-0͍I{ 1  1sz2ՠ@fedՠ?מ⍍__=z2Π@fedΠ?_1 r 1sDe2ՠ@feDdՠ?⏍IJ 1 1De2Π@feDdΠ?IJ1 r 1De2Ǡ@feDdǠ?'IJ svҟΠM@fewΠ?k =1M Nij 1ۿ`X?`X`XǕ?`Xѕ`Xە?`X`X?`X`X?`X `X?`XXzsrc:8060advalg.texThe7correspSondingpropertiesforcomodulesfollorwsfromthe8dualizeddiagrams.ThemoduleandcomoSdulepropertiesofKareeasilycrhecked.ǙҍProblemb12.2.esrc:8066advalg.tex(1)LetBbSeabialgebraandMB 8?bethecategoryofrighrtB-modules.ShorwthatMB 8isamonoidalcategoryV.src:8070advalg.tex(2)LetBҹabialgebraandM2B ^ bSethecategoryofrighrtB-comoSdules.KShorwthatM2B ^ isamonoidalcategoryV.ӍDe nition12.3.src:8076advalg.tex(1)9[Let9\(B;r;n9;;)bSe9[abialgebra. $LetAbSealeftB-moSdulewithstructureiemapidUR:B= An!1A. Letfurthermore(A;rA;A)iebSeidanalgebrasucrhthatrA GIandA ȌarehomomorphismsofB-moSdules.8Then(A;rA;A;)iscalledaB-moffdule35algebra.src:8084advalg.tex(2)sLetr(B;r;n9;;)bSeabialgebra.#LetCbearleftB-modulewithrstructuremapUR:B] WC!nCܞ. Leta2withasimilarpropSertryV.! ThiswillbSecalledaHopfalgebra.De nition 12.6.src:8165advalg.texAleft՜Hopf՝algebrffaHr*isabialgebraHtogetherwithaleft՜antipffodeS):URH!nHV,i.e.8aK-moSdulehomomorphismSsucrhthatthefollowingdiagramcommutes:@\)"ԙH)"5K:2fd)@ά-Í"  CHf:2fd.bά-(7B 2Ǡ@fe)dǠ??;a*zH HH HŦԞ32fdJsά-׀8Sr} id%dǠ@fe%?`6?;*Jdr 'src:8176advalg.texSymmetricallyޯwrede neޮarightHopfalgebrffaHV.ApHopfalgebrffaޯisaleftandrighrtHopfalgebra.8ThemapSiscalleda(left,righrt,two-sided)antipffode.src:8182advalg.texUsing theSwreedlernotation(2.20)thecommutativediagramabSovecanalsobSeexpressedbrytheequationXXS׹(a(1) \|)a(2)ι=URn9"(a)src:8187advalg.texfor+|allaUR2HV.'Observre+|that+}wedonotrequire+}thatS):URHB\3!Hҹisanalgebrahomomorphism.Problem12.4.{src:8191advalg.tex(1)>"Let>#H+ybSeabialgebraandS)2URHom(HF:;HV).^ThenSisananrtipSode>"forH(andSHA.isaSHopfalgebra)i SisatrwosidedSinverseforSidinthealgebra(Homy(HF:;HV);;n9")(see2.21).8InparticularSisuniquelydetermined.src:8197advalg.tex(2)M LetH:ubSeaHopfalgebra.`GThenSisananrtihomomorphismofMalgebrasandcoalgebrasi.e.8S\inrvertstheorderofthemrultiplicationandthecomultiplication".src:8202advalg.tex(3)=_Let=`H*andKbSeHopfalgebrasandletf*:Hs!KbSeahomomorphismofbialgebras.ThenfGSH n=URSK;f,i.e.8f2iscompatiblewiththeanrtipSode.De nition12.7.+src:8208advalg.texBecauseofProblem12.4(3)evreryhomomorphismofbialgebrasbSetrweenHopfPalgebrasPiscompatiblewiththeanrtipSodes. jSoPwede neaPhomomorphism|3of|4Hopfalgebrffas&Rto(vbSe(uahomomorphismofbialgebras.$ThecategoryofHopfalgebraswillbSedenotedbryK-Hopf._A7o][BialgebrasTandHopfAlgebras95[o]Prop`osition12.8.Usrc:8218advalg.texLffetgHTmbeabialgebrawithganalgebrageneratingsetX."_LetS):URHB!gHV2opbffeanalgebrahomomorphismsuchthatPlS׹(x(1) \|)x(2)=]rn9"(x)forallx]r2X.NThenStisaleft35antipffodeofHV. Prffoof.#Rsrc:8225advalg.texAssumea;bUR2HsucrhthatPSS׹(a(1) \|)a(2)ι=n9"(a)andPSS׹(b(1) \|)b(2)ι=n9"(b).8Then#ʍ?7wPM"S׹((ab)(1) \|)(ab)(2)0=URPS׹(a(1) \|b(1))a(2)b(2)ι=URPS׹(b(1))S(a(1))a(2)b(2)0=URPS׹(b(1) \|)n9"(a)b(2)ι=UR"(a)"(b)="(ab):src:8234advalg.texSinceOevreryelementofOH<عisa nitesumof niteproSductsofelemenrtsinX,hforwhichtheequalitryholds,thisequalityextendstoallofHbyinduction.$Exampled712.9.{src:8240advalg.tex(1)yLetzVnbSeavrectorspaceandTƹ(Vp)thetensoralgebraorverzVp.0WVeharveseen(6inProblem2.2thatTƹ(Vp)isabialgebra(5andthatVĦgeneratesT(Vp)asanalgebra.De neSl:V ˑ!(Tƹ(Vp)2op bryS׹(vn9):=v`harve=seeninProblem2.3thatS׹(Vp)isabialgebraandthatVgeneratesS׹(Vp)asanalgebra.De neS :b5V 1!zhS׹(Vp)bryS(vn9)b6:=b5vAforallvn2Vp.S޹extendstoanalgebrahomomorphismS:S׹(Vp)U!"S(V).hSince(vn9)=vx A>1A?+1 v6wrehave՟PrS׹(v(1) \|)v(2)*G=r(Sv 1)(vn9)r;=vع+vt=0="(v) forallvt2r;Vp,henceS׹(V) isaHopfalgebrabrytheprecedingpropSosition.Example12.10.Esrc:8264advalg.tex(GroupAlgebras)sFVorseacrhalgebraAwrecanformthegrffoupofunitsU@(A)WC:=WBfa2Aj9a212Ag̹withthemrultiplicationofAascompSositionofthegroup.@4>RHU@(A):bǠ*FfeNǠ? g<src:8279advalg.texTheOgroupalgebraOKGis(ifitexists)uniqueuptoisomorphism. gItisgeneratedasanalgebrabrytheimageofG.2ThemapUR:Gn!1U@(KG)KGisinjectiveandtheimageofGinKGisabasis.src:8284advalg.texThegroupalgebracanbSeconstructedasthefreevrectorspaceKGwithbasisGandthealgebrayCstructureofKGyDisgivrenbyKG KGH 3gy  hH7!gn9h2KGyCandtheunityDF:KH 3 h7!UR e2KG.src:8290advalg.texThegroupalgebraKGisaHopfalgebra.8ThecomrultiplicationisgivenbythediagramKҍj⍒BGj⍒NKG{fd% ά-eH`fׁ @ @ @ @c<>@c<>RH"EKG KGhjǠ*FfeǠ?`N`ʧ7o]96oAdv|rancedTAlgebra{P9areigis[o]src:8294advalg.texwith\ fG(gn9)UR:=g YgBwhicrhde nesagrouphomomorphismfQ:Gn!1U@(KGY XKG). VThecounitisgivrenbyCj⍒ƉnGj⍒KGF{fd% ά-H`Hefd$ׁ @d$ @d$ @d$ @󪤟>@󪤟>RH""8XKҟǠ*FfeǠ?]C "bsrc:8298advalg.texwhereTfG(gn9) =1forTallgw2G.vOneshorwsTeasilybyTusingtheuniversalTpropSertyV,o'thatTiscoassoSciativreandhascounit".8De neanalgebrahomomorphismS):URKGn!1(KG)2op ŹbyI@j⍒Gj⍒KGϴ{fd% ά-6H`fğׁ @ğ @ğ @ğ @4D>@4D>RHP(KG)2op9rǠ*FfelǠ?`$Sf6src:8303advalg.texwith fG(gn9)D:=g21pwhicrh is agrouphomomorphismfC:DG!BU@((KG)2op).kThenone shorwswithPropSosition12.8thatKGisaHopfalgebra.EProp`osition12.11.src:8309advalg.texThe35followingthrffee35monoidalcffategories35aremonoidallyequivalents2(1)%src:8311advalg.texthe35cffategoryM2G RofG-gradedvectorspacesM2G,(2)%src:8312advalg.texthe35cffategoryofG-familiesofvectorspaces(M)2G,(3)%src:8313advalg.texthe35monoidalcffategory35ofKG-cffomodules35M2KG Xz.FPrffoof.#Rsrc:8318advalg.texWVeonlyindicatetheconstructionfortheequvXalencebSetrween(1)and(3).src:8320advalg.texFVor/a.G-gradedvrectorspaceVoneconstructstheKG-comoSduleVwiththestructuremap٧:ftV p!VZ KG,=s2(vn9)fu:=v# gXfor allvԭ2fuVg `andforallgԮ2ftG.GConrversely letV;٦:V!nV< KGh&bSeaKG-comodule. _Thenoneconstructstheh%gradedvrectorspaceVwithgraded(homogenous)DcompSonenrtsVgb:=dfv[2Vpjs2(vn9)e=vU |gg.DItDiseasyCtovrerifyV,ZVthatthisisanequivXalenceofcategories.src:8330advalg.texSinceKGisabialgebra,1thecategoryofKG-comoSdulesisamonoidalcategorybryExercise12.2}(2).Onecrhecks}that}undertheequivXalencebSetrween}M2G }andM2KG`tensorproSductsaremappSedinrtocorrespondingtensorproductssothatwrehaveamonoidalequivXalence.\Exampled12.12.}src:8339advalg.texThefollorwingisabialgebraB=#Khx;yn9i=I,'whereI*isgeneratedbryx22;xyTE+ yn9x. Thediagonalis(y)i=yTE  y,-(x)=x  yTE+1 xandthecounitis(yn9)UR=1;(x)=0.Prop`osition12.13.src:8347advalg.texThemonoidalcffategoryComp-KofchaincffomplexesoverKismonoidallyeffquivalent35tothecategoryofB-comodulesM2B swithB;asintheprecedingexample.FPrffoof.#Rsrc:8353advalg.texWVeFusetheFfollorwingconstruction.5AF}chaincomplexMismappSedFtotheB-comoduleMҹ=i2N֥MiswithCtheDstructuremap:M]!~LM B,*s2(m):=P.USinceyn92i;xyn92iformabasisofBwrehaves2(m)UR=Pidmim byn92i5+P @irm20RAil xyn92i.$Apply(3 1)Ȅ=UR(1 )tothisequationandcomparecoSecienrtsthens2(midڹ)UR=miz yn92i+m20RAi1,m xyn92i1; s2(m20RAidڹ)UR=m20RAiO yn92i:޹Henceforeacrhmi,2URMithereisexactlyone@(midڹ)2Mi1AV,sothatM썑cs2(midڹ)UR=mi yn9 i}+@(mi) xyn9 i1; s2(m 0ڍi)UR=m 0ڍi yn9 i:src:8392advalg.texApplyfurthermore(yA 1)s2(m)UR=m͹thenyrougetmUR=PmiM,whereBiscrhosenasinExample12.12.src:8407advalg.tex(4)ShorwthatthebialgebraBfromExample12.12isnotaHopfalgebra.src:8410advalg.tex(5)OFindOabialgebraB20sucrhthatthecategoryofcomplexes:::o@!%ޔM1V .!5M0V .!M@21 V: o!M@22ꍍ!n:::#$'andM2Bd-:0 ꭹaremonoidallyequivXalenrt.8ShowthatB20SisaHopfalgebra.,src:8415advalg.texTheXexampleKGXofaHopfalgebragivresrisetothede nitionofparticularelemenrtsinarbitrary ?Hopf @algebras,R%thatsharecertainpropSertieswithelemenrtsofagroup.WVewillusexandstudytheseelemenrtslateroninthecourseonNonCommutativeGeometryandQuanrtumGroups.,De nition312.14.src:8422advalg.texLet8HbSeaHopf9algebra."AnelemenrtgË2URHF:;g6=09is8calledagrffoup-likeelement꨹ifM퍒u(gn9)UR=g g:src:8427advalg.texObservre[that"(gn9)==1[foreachgroup-likeelementg9inaHopfalgebraHV.Infactwehavegѹ=r(" 1)(gn9)="(g)g6=0 Nhence"(g)=1.Ifthe Mbaseringisnota eldthenoneaddsthispropSertrytothede nitionofagroup-likeelement.,Problem12.6.src:8435advalg.tex(1)LetKbSea eld.JShorwthatanelemenrtx_`2KGsatis es(x)=x xand"(x)UR=1ifandonlyifx=gË2G.src:8439advalg.tex(2)Shorwthatthegroup-likreelementsofaHopfalgebraformagroupundermrultiplicationoftheHopfalgebra.Exampleq12.15.kfsrc:8443advalg.tex(UniversalEnvelopingqAlgebras)ALiealgebrffaconsistsofavrectorspaceg퉹togetherwitha(linear)mrultiplicationgZ Zg 3 x y|/7![x;yn9] 2g툹sucrhthatthefollorwinglawshold:ʍ]Π[x;yn9]UR=[y;x];]Π[a,x;[yn9;z]]+[yn9;[z;x]]+[z;[x;yn9]]UR=0(Jacobiidenrtity).4Rsrc:8453advalg.texA ,homomorphism* of* Liealgebrffas f@|:}g ! h isalinearmapfUsucrhthatfG([x;yn9])}=[fG(x);f(yn9)].8ThrusLiealgebrasformacategoryK-Lie.src:8458advalg.texAnimpSortanrtexampleistheLiealgebraassoSciatedwithanassoSciativrealgebra(withunit).IfAisanalgebrathenthevrectorspaceAwiththeLiemultiplication4Q[x;yn9]UR:=xyyxsrc:8464advalg.texis2aLie2algebradenotedbryA2LGع.ThisconstructionofaLiealgebrade nesacorvXariant2functor-ꨟ2L ҹ:URK-Alg2kK!*JK-Lie.8Thisfunctorleadstothefollorwinguniversalproblem.src:8469advalg.texLetgbSeaLiealgebra.~AnalgebraU@(g)togetherwithaLiealgebrahomomorphismTl:g! U@(g)2L eiscalleda(the)universal_enveloping`algebrffaofg,ifforevreryalgebraAandb7o]98oAdv|rancedTAlgebra{P9areigis[o]forevreryLiealgebrahomomorphismf׹:hghc!@ٜ>RH"A2LG:ʟǠ*FfeǠ? |gysrc:8479advalg.texTheunivrersalenvelopingalgebraU@(g)is(ifitexists)uniqueuptoisomorphism. Itisgeneratedasanalgebrabrytheimageofg.src:8483advalg.texTheAunivrersal@envelopingalgebra@canbSeconstructedasU@(g)UR=Tƹ(g)=(x% yJ%y %x[x;yn9])whereTƹ(g)UR=Kgg g:::Fisthetensoralgebra.8ThemapUR:gn!1U@(g)2L 2isinjectivre.src:8490advalg.texThedwunivrersalenvelopingalgebradxU@(g)isaHopfalgebra.MThecomrultiplicationisgivenbythediagramFGHgHDU@(g)ŗ̟{fd$ά-h[H`>fZׁ @Z @Z @Z @<>@<>RHU@(g) U(g)jǠ*FfeٜǠ?`N*src:8494advalg.texwithLfG(x)UR:=x_ `1+1 xLwhicrhMde nesaLiealgebrahomomorphismfQ:URgn!1(U@(g) _U(g))2LGع.ThecounitisgivrenbyFHHgHr|U@(g)f{fd$ά-6H` 5f(ׁ @( @( @( @ot>@ot>RH""(KtǠ*FfeԟǠ?]CZT"src:8499advalg.texwith+fG(x)=0for*allx2g.gOneshorws*easilyby*usingtheunivrersalpropSertyV, that*iscoassoSciativreandhascounit".8De neanalgebrahomomorphismS):URU@(g)n!1(U(g))2op ŹbryKXHe}gHU@(g){fd$ά-۱+H`Շfׁ̣ @֣ @࣌ @꣌ @ >@ >RH¹(U@(g))2op:Ǡ*Ffe"lǠ?`S^src:8506advalg.texwithfG(x)UR:=xwhicrhisaLie algebrahomomorphismfQ:gn!1(U@(g)2op)2LGع.0]ThenoneshorwswithPropSosition12.8thatU@(g)isaHopfalgebra.src:8511advalg.tex(Observre,thatڳtheڴmeaningofUinthisexampleandthepreviousexample(groupofunits,univrersalenvelopingalgebra)istotallydi erenrt,;pinthe rstcaseU8canbSeappliedtoanalgebraandgivresagroup,inthesecondcaseU˹canbSeappliedtoaLiealgebraandgivresanalgebra.)+%src:8518advalg.texTheB@precedingBAexampleofaHopfalgebragivresrisetothede nitionofparticularelemenrtsinarbitraryHopfalgebras,thatsharecertainpropSertieswithelemenrtsofaLiealgebra.src:8523advalg.texWVe3will4useandstudytheseelemenrtslateroninthecourseonNonCommrutative3GeometryandQuanrtumGroups.+&De nition12.16._Nsrc:8527advalg.texLetH3bSeaHopfalgebra.AnelemenrtxUR2H2iscalledaprimitiver_elementif (x)UR=x 1+1 x:c j7o][BialgebrasTandHopfAlgebras99[o]src:8530advalg.texLetgʀ2\GHbSeagroup-likreelement.ˊAnelementx\H2\GHiscalledaskewKsprimitiveKtorgn9-primitive35element꨹if^(x)UR=x 1+g x:eBProblem 12.7.Usrc:8536advalg.texShorwthatthesetofprimitivreelementsPƹ(HV)UR=fx2Hj(x)=x 1+1 xgofaHopfalgebraHisaLiesubalgebraofHV2L5..eCProp`ositionД12.17.src:8542advalg.texLffetH5beaHopfalgebrawithantipodeS.7Thefollowingareequivalent:src:8545advalg.tex(1)35Sן22-=id.src:8547advalg.tex(2)35PS׹(a(2) \|)a(1)ι=URn9"(a)35foralla2HV.src:8550advalg.tex(3)35Pa(2) \|S׹(a(1))UR=n9"(a)35forallaUR2HV.Prffoof.#Rsrc:8555advalg.texLetSן22-=id.8Then;ʍX@aPf S׹(a(2) \|)a(1)=URSן22r۹(PS׹(a(2) \|)a(1))=S׹(PS(a(1) \|)S22r۹(a(2)))=URS׹(PS(a(1) \|)a(2))=S׹(n9"(a))="(a)src:8562advalg.texbryusingProblem12.4.src:8564advalg.texConrverselyassumethat(2)holds.8TheneʍqS]Sן22r۹(a)=URPS׹(a(1) \|S22r۹(a(2))UR=S׹(PS(a(2))a(1)=URS׹(n9"(a))="(a):isrc:8570advalg.texThrusSן22 ]andidareinversesofSintheconvolutionalgebraHomd1(HF:;HV),henceSן22-=URid .src:8573advalg.texAnalogouslyoneshorwsthat(1)and(3)areequivXalent.eCCorollaryT12.18.^7src:8577advalg.texIfqH^gisqacffommutativeHopfalgebraoracocommutativeHopfalgebrawithantipffode35S,thenS22-=id.eBRemark12.19.src:8586advalg.texKaplansky:TenconjecturesonHopfalgebrassrc:8589advalg.texIn~aset~oflecturenotesonbialgebrasbasedonacoursegivrenatChicagounivrersity~in1973,madeYpublicZin1975,I.KaplanskyformrulatedtenconjecturesonHopfalgebrasthatharvebSeentheaimofinrtensiveresearch.s2(1)%src:8596advalg.texIfCFisaHopfsubalgebraoftheHopfalgebraBthenBisafreeleftCܞ-moSdule.1src:8599advalg.tex(YVes,8if)2His)3 nitedimensional[Nicrhols-ZoSeller];HxNoforin nitedimensionalHopf%algebras[ObSerst-Scrhneider];BX:URCFisnotnecessarilyfaithfully at[Schauenburg])(2)%src:8604advalg.texCallGacoalgebraC$,admissibleifitadmitsanalgebrastructuremakingitaHopf%algebra.}TheconjecturestatesthatC(isadmissibleifandonlyifevrery nitesubset%ofCFliesina nite-dimensionaladmissiblesubScoalgebra.1src:8610advalg.tex(Remarks.)((a)=Ѭsrc:8612advalg.texBothimplicationsseemhard.)' (b)=Ѭsrc:8613advalg.texThereisacorrespSondingconjecturewhere\Hopfalgebra"isreplacedbry\bial-=Ѭgebra".*uD(c)=Ѭsrc:8615advalg.texThereisadualconjectureforloScally nitealgebras.)1src:8619advalg.tex(Noresultsknorwn.)(3)%src:8621advalg.texAHopfalgebraofcrharacteristic0hasnonon-zerocentralnilpSotentelements.1src:8624advalg.tex(FirstIcounrterJexamplegivenbyJ[Schmidt-Samoa].IfH isIunimoSdularandnot%semisimple,e.g.+aDrinfel'ddoubleofanotsemisimple nitedimensionalHopfalge-%bra,_thenNtheMinrtegralsatis esUR6=0,`22V="()=0MsinceDS(HV)isnotsemisimple,%andaUR="(a)="(a)=asinceDS(HV)isunimodular[Sommerh auser].)(4)%src:8633advalg.tex(Nicrhols).Let+_x+`bSeanelementina+_HopfalgebraHwithantipSode+`S׹.Assumethat%foranryainHwehave򍍍FXʜbidxS׹(ci)UR="(a)x ֍%src:8636advalg.texwhereaUR=Pbi cidڹ.8Conjecture:xisinthecenrterofHV.dK7o]100oAdv|rancedTAlgebra{P9areigis[o]1src:8639advalg.tex(YVes,sinceaxUR=Pa(1) \|x"(a(2))UR=Pa(1) \|xS׹(a(2))a(3))UR=P"(a(1) \|)xa(2)ι=xa:)1src:8643advalg.texIntheremainingsixconjecturesHVisa nite-dimensionalHopfalgebraorveran%algebraicallyclosed eld.(5)%src:8646advalg.texIfs!H`vissemisimpleoneithers side(i.e.JeitherH`worthedualHV2 {issemisimpleasan%algebra)thesquareoftheanrtipSodeistheidentityV.1src:8650advalg.tex(YVesifcrhar(K)=0[Larson-Radford],yesifchar(K)islarge[Sommerh auser])(6)%src:8653advalg.texTheݠsizeݟofthematricesoSccurringinanryfullmatrixconstituenrtofHdividesthe%dimensionofHV.1src:8656advalg.tex(YVes'=P2?H(c)[bsrc:8910advalg.texP1jQPUR԰n:=P2Q꨹=)P1PV԰.>=P2?#,(10)=Ѭsrc:8913advalg.texZ=(2)Z=(6)Z=(6):::Pʚ԰ス= uCZ=(6)Z=(6)Z=(6):::uH.#,(11)=Ѭsrc:8916advalg.texZ=(2)Z=(4)Z=(4):::ʚ6P԰= Z=(4)Z=(4)Z=(4):::uH.#,(12)=Ѭsrc:8919advalg.texMan3 nde4zwreiabSelscheGruppSen4PzundQ,ܱsodaPzisomorphzueinerUnrter-=ѬgruppSeBvronQBistundQisomorphzueinerUnrtergruppSevonPOistundBP6P԰= @=Q=Ѭgilt.*ISI.%src:8927advalg.texTVensorproSdukteT)((1)=Ѭsrc:8932advalg.texInC CCgilt1 ii 1UR=0.=Ѭsrc:8935advalg.texInC RCgilt1 ii 1UR6=0.)((2)=Ѭsrc:8938advalg.texFSvurjedenRJ-ModulgiltR R ;MP6԰=@M@.)((3)=Ѭsrc:8940advalg.texSeiderQ-VVektorraumV¹=URQ2n gegebSen.G(a)[bsrc:8943advalg.texBestimmedimRG(R Q cVp).GH(b)[bsrc:8945advalg.texGibexpliziteinenIsomorphismrusR Q cVP԰ =kR2n an.)((4)=Ѭsrc:8949advalg.texSeiVeinQ-VVektorraumundWneinR-Vektorraum.G(a)[bsrc:8952advalg.texHomyR(:RQ;:Wƹ)PUR԰n:=WninQ-MoSdi.GH(b)[bsrc:8954advalg.texHomyQ2k(:V;:Wƹ)PUR԰n:=Hom(yR.(:R Q cV;:Wƹ).H(c)[bsrc:8956advalg.texSeisIdimQVQ<=1unddimRGWߧ<1.WiekXannmanvrerstehen,qdain4b[blinksunendlicrheMatrizenundrechtsendlicheMatrizenstehen?f9Ơ7o]102oAdv|rancedTAlgebra{P9areigis[o]GH¹(d)[bsrc:8960advalg.texHomyQ2k(:V;HomyR (:R;:Wƹ)PUR԰n:=Hom(yR.(:R Q cV;:Wƹ).U)((5)=Ѭsrc:8964advalg.texZ=(18) ZZ=(30)UR6=0.)((6)=Ѭsrc:8966advalg.texmUR:Z=(18) ZLZ=(30)3dzRKx dz(ߟKy p)7!dz 1Kxy2Z=(6)=ist6vuberKܞ,Z,K[x].)((2)=Ѭsrc:9115advalg.texFindealleeinfacrhenMoSduln>6vuberC[x],M2(Kܞ),Q[x]=(x22).)((3)=Ѭsrc:9118advalg.texFindealleeinfacrhenMoSduln>6vuber4ʍџqʍc)K>K0>Kq:)((4)=Ѭsrc:9126advalg.texStelleEndgK[x]*(Kܞ[x]=(x)K[x]=(x1))alsRingvronMatrizendar.* 3VISI.%src:9132advalg.texRadikXalundSoScrkel⍍)((1)=Ѭsrc:9137advalg.texRadikXalundSoScrkelendlicherzeugterabSelscherGruppSen.8BestimmeG(a)[bsrc:9140advalg.texRad (ZTZ=(p)),SoSc*(ZZ=(p)).UGH(b)[bsrc:9141advalg.texRad (Z=(p2nP)),SoSc*(Z=(p2n)).H(c)[bsrc:9142advalg.texRad (Z=(p2nP)Z=(p2mĹ)),SoSc*(Z=(p2n)Z=(p2mĹ)).GH(d)[bsrc:9143advalg.texFSvurwrelchenUR2NistRadg(ZTZ=(n))=0?)((2)=Ѭsrc:9147advalg.texBestimmeRadikXalundSoScrkelderabelscrhenGruppenG(a)[bsrc:9149advalg.texZ,GH(b)[bsrc:9150advalg.texQ,H(c)[bsrc:9151advalg.texQ=Z.*rVISII.%src:9156advalg.texLokXaleRingeፍ)((1)=Ѭsrc:9161advalg.texSeiReinlokXalerRing.8DannistRJ=meinScrhiefk orpSer.hbO7o]104oAdv|rancedTAlgebra{P9areigis[o])((2)=Ѭsrc:9164advalg.texDerRingderformalenProtenzreihenKܞ[[x]]isteinlokXalerRing.U)((3)=Ѭsrc:9167advalg.texDerProlynomringKܞ[x]istkeinlokXalerRing.*IX.%src:9172advalg.texLokXalizations2)((1)=Ѭsrc:9177advalg.texS):=UR2Znf0g꨹istmrultiplikXativabgeschlossen.8Sן21 SZUR$Q.)((2)G(a)[bsrc:9182advalg.texWVennxS)URTD>mrultiplikXativyabgeschlosseneMengensind,dannwirddadurch[beinHomomorphismrus Ë:URSן21 SM6!TƟ21 BM+induziert.GH(b)[bsrc:9185advalg.texFindeeinehinreicrhendeBedingungdafSvur,da Xinjektivist.H(c)[bsrc:9187advalg.texFSvur=S):=URZn͹(p)und cmmi10O line10u cmex10