; TeX output 1999.09.17:13347 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox/^o cmr926*7 &e 6. %CO9ALGEBRAS/27YN cmbx126._CoalgebrasDe nition8.6.1.vaùA&* msbm10K@ cmti12-cffoalgebra&isaK-moSduleg cmmi12Ctogetherwithacffomultiplica-tion꨹ordiagonalUR:C1!!", cmsy104!CF CFthatiscoassoSciativre:K_?.CF C9CF C Cܞ32fd)@O line10-a|{Ycmr8id<"K cmsy8 Y6CY1.CF C|{fdAά-ˍJNHǠ*Ffe,Ǡ?`PHǠ*Ffe,Ǡ?э} idϟandacffounitoraugmentationUR:C14!K:YnCY}FCF C{fdY?ά-ˍ&HuǠ*FfeğǠ?`~HǠ*FfeDǠ?эid I 2cmmi8"zCF C"ȎK CP1԰Jع=ܙCP1԰Jع=CF K:t32fd&fpά-aP idHk}idҁ Hׁ H܁ H H H H H H妄džH妄džjAK-coalgebraCFiscffocommutative꨹ifthefollorwingdiagramcommutesG_پCF CCF Cψl32fdά-ÍvIbY-C``ϳ,ׁ ʳ, ų, , >> H`댟ׁ A댟 A댟 A댟 A̟>A̟>UɫLet|CYandDЋbSeK-coalgebras.A|homomorphismofcffoalgebras|fe:NfC+4!DisaK-linearmapsucrhthatthefollowingdiagramscommute:M;?~CF CcD6 Dm,32fdyά- աf fbY&CbYxDr̟{fd5氍ά-iیfH%JǠ*FfeX|Ǡ?XX. ;cmmi6CHʟǠ*FfeǠ?|X.DandEH""%(K-PX.Cdׁ AÐd AȐd A͐d Aγ>Aγ>UH-tX.DDׁ D D D >> bYCbY*DS:Ž{fd(ά-iEfߍRemark8.6.2.j6ObrviouslyhQthecompSositionoftwohomomorphismsofcoalgebrasisagainahomomorphismofcoalgebras.(FVurthermoretheidenrtitymapisahomo-morphism8ofcoalgebras. "{HencetheK-coalgebrasformacategoryK-Coalg!.ThecategoryofcoScommrutativeK-coalgebraswillbedenotedbryK-cCoalg'.7 &e288. %TOOLBO9XYProblem8.6.1.nR1.%]ShorwthatVφ 3Vp2 isacoalgebraforevery nitedimensionalvrectorspaceV>1overa eldKifthecomultiplicationisde nedby(vC ՝vn92.=)@:=$u cmex10P* n U_ i=1v v2n9RAi vi vn92 wherefvidgandfv2n9RAi.=garedualbasesofVresp.Vp2\t.2.ShorwthatthefreeK-moSdulesKX $withthebasisXandthecomrultiplication(x)UR=x x꨹isacoalgebra.8Whatisthecounit?Isthecounitunique?3.&ShorwthatK<,VQwith(1)UR=1<, 1,b(vn9)UR=ve <,1+1 v"ʹde nesacoalgebra.4.Let^CandDbSecoalgebras.ThenC.- QDisacoalgebrawiththecomrultiplicationC DG:=UR(1C R V 1D)(C D)UR:CX DSH C D4!C Dꇹandcounit"UR="C DG:CF D4!K K14!K.8(TheproSofisanalogoustotheproofofLemma8.5.3.)#TVo?describSethecomrultiplicationofaK-coalgebraintermsofelementsweintro-duceanotation rstinrtroSducedbySweedlersimilartothenotationr(a^ b)X=abusedforalgebras.8Insteadof(c)UR=Pci c20RAiOwrewriteƍ(c)UR=Xc(1)$ c(2) \|:ObservrevXthatonlythecompleteexpressionontherighthandsidemakessense,Dnotthe4compSonenrtsc(1)xc(1)$ ::: (c(i) R) ::: c(n)ݭ=URPc(1)$ ::: 1 ::: c(n1)ݭ=URPc(1)$ ::: c(n1)ȆThis4notationanditsapplicationtomrultilinearmapswillalsobSeusedinmoregeneralconrtextslikecomoSdules.]Prop`osition8.6.4.OLffet.MC beacoalgebraandAanalgebra.rThenthecompositionfgË:=URrA(f gn9)C de nes35amultiplicffationeIC{Homa(C5;A) Hom$1(C;A)UR3f gË7!fgË2Hom(C5;A);such.TthatHom(C5;A)bffecomes.Tanalgebrffa.WTheunitelementisgivenbyK&Z3 97!(cUR7!n9( (c)))2Hom(C5;A).- cmcsc10Proof.@_TheamrultiplicationofHom(C5;A)obviouslyisabilinearmap. Themul-tiplicationPisassoSciativresince(fgn9)hbZ=rA((rA(f gn9)C) h)C =bZrA(rA 1)((f lgn9) h)(CC 1)C t=URrA(1 rA)(f (g h))(1 C)C t=URrA(f (rA(g h)C))C <=fab(g<h).NFVurthermore"itisunitarywithunit1HomD(C;A)+_l=AC >sinceAC fQ=URrA(AC fG)C t=URrA(A { 1A)(1K fG)(C 1C)C t=URf,{and|similarlyfAC t=URfG.d cffxff ̟ff ̎ ̄cffDe nition8.6.5.vaùThe*$mrultiplicationUR:Hom(C5;A)!o Hom(C;A)UR4!1Hom-=(C;A)iscalledcffonvolution.Corollary8.6.6.sWLffet~C[zbeaK-coalgebra. I_ThenCܞ2 X=zHom5K&pŹ(C5;K)isanK-algebrffa.Proof.@_UsethatKitselfisaK-algebra.ìecffxff ̟ff ̎ ̄cffRemark8.6.7.j6IfwrewritetheevXaluationasCܞ2bj Cu3#a c7!ha;ci2Kthenan-elemenrta2Cܞ2 Ϲis-completelydeterminedbythevXaluesofha;ciforallc2Cܞ.oSotheproSductofaandbinCܞ2 JisuniquelydeterminedbrytheformulaWiHhab;ciUR=ha b;(c)iUR=Xa(c(1) \|)b(c(2)):XTheunitelemenrtofCܞ2 JisUR2Cܞ2.Lemma8.6.8.g5QLffetKbea eldandAbea nitedimensionalK-algebra.ThenA2V=URHom۟K$ (A;K)35isaK-cffoalgebra.Proof.@_De nethecomrultiplicationonCܞ2 Jby~;UR:A 2 r-:#q% cmsy6pVj!(A A) 2Vcanb-:Aacmr61p Î!"tA j A :eTheBcanonicalmapcan:URA2 A2V4!5(A A)2fFisBinrvertible,#sinceAis nitedimensional.By&[adiagrammaticproSoforbrycalculationwithelementsitiseasytoshowthatA2bSecomesaK-coalgebra./ׄcffxff ̟ff ̎ ̄cffRemark8.6.9.j6IfYKisanarbitrarycommrutativeYring,tthenA2X=THomݟK$Ɵ(A;K)isaK-coalgebraifAisa nitelygeneratedprojectivreK-moSdule.V7 &e308. %TOOLBO9XYProblem8.6.2.nRFindisucienrtconditionsforanalgebraAresp."acoalgebraCsucrhthatHomd1(A;Cܞ)bSecomesacoalgebrawithco-convolutionascomultiplication.De nition8.6.10.}!ùLetCbSeaK-coalgebra.AleftYCܞ-cffomoduleisaK-moduleM+togetherwithahomomorphismM B:URM64!CF M@,sucrhthatthediagramsK𺍍CF MeCF C M32fd&ά-a>id˾ YMYCF M\{fd>y@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?э} id)and@bYMŸǠ*FfeHǠ?k}"CF M"VK MP6԰=@M:32fd:]ά-aF idHk}؄4idҁ Hׁ H܁ HĹ Hι Hع H⹴ H카 HFdžHFdžjcommrute.Let~џ2C iMand~џ2CNbSe~Cܞ-comodulesandletf:QMf4!uNbeaK-linearmap.\Themapf2iscalledahomomorphism35ofcffomodules꨹ifthediagramMYzNCF N>,32fd@Z@ά-EX.NY MY^CF M.{fd>y@ά-zX.MHDleftCܞ-comoSdulesandtheirhomomorphismsformthecffategory2C XMofcffomod-ules.LetiNMbSeanarbitraryK-moduleandMMbeaCܞ-comodule. #ThenthereisabijectionbSetrweenallmultilinearmaps$LfQ:URCF:::M64!Nandalllinearmaps=`fG 0k:URCF ::: M64!N:ӍThesemapsareassoSciatedtoeacrhotherbytheformulaq,fG(c1;:::ʚ;cnP;m)UR=f 08(c1j ::: cnR m):FVormUR2M+wrede ne5n邟X>fG(m(1) \|;:::ʚ;m(n) D;m(M") h)UR:=f 08(s2 n(m));wheres22n *denotesthen-foldapplicationofs2,i.e.82n pԹ=UR(1 ::: 1 )(1 )..7 &e 6. %CO9ALGEBRAS/31YInparticularwreobtainforthebilinearmap UR:CFM64!C MX Jm(1)$ m(M")u=URs2(m);IIandforthemrultilinearmap 22V:URCFCM64!C C MPUȟXcm(1)$ m(2) m(M")u=UR(1 s2)(c)UR=( 1)s2(m):+Problem8.6.3.nRShorw'!thata nitedimensionalvectorspaceVÑisacomoSduleorverthecoalgebraV ^Vp2 OJasde nedinproblem8.11.1withthecoactions2(vn9):=Pv v2n9RAi vi,2UR(VG Vp2\t) VwhereꨟPSv2n9RAi viOisthedualbasisofVinVp2  Vp.8Theorem8.6.11.wX(Fundamental~ThefforemforComodules)LetKbea eld.ILetMKbffe aleftCܞ-comoduleandletmUR2MKbe given.Thenthereexistsa nitedimensionalsubffcoalgebrahCܞ20طCanda nitedimensionalCܞ20-cffomodulehM@20 withm2M@20Mwherffe35M@20doURMtisaK-submodule,suchthatthediagramNWMϮCF M32fd>y@ά-͍/ n:M@20/ uCܞ20U M@20L{fd:?ά- L̟-:0HǠ*Ffe$̟Ǡ?H㚟Ǡ*Ffe̟Ǡ?Ncffommutes.8Corollary8.6.12.z1.~Each;?elementcd42Cof;?acffoalgebra;?iscffontained;?ina nitedimensional35subffcoalgebraofCܞ.2. wEachpelementmu2M$Tofpacffomodulepiscffontainedpina nitedimensionalsubffcomodule35ofM@.Corollary8.6.13.z1. ^Eachl nitedimensionalsubspffacelVofacffoalgebralCI%iscffontained35ina nitedimensionalsubffcoalgebra35Cܞ20 ofCܞ.2.ËEach nitedimensionalsubspffaceVDofacffomoduleMziscffontainedina nitedimensional35subffcomoduleM@20BRofM@.Corollary8.6.14.z1.GEachQcffoalgebraisaunionof nitedimensionalsubcoalge-brffas.2.fiEach35cffomoduleisaunionof nitedimensionalsubcomodules.Proof.@_(oftheThefforem)@WVecanassumethatmUR6=0@forelsewrecanuseM@20do=UR0andCܞ20)=UR0.Undertherepresenrtationsofs2(m)UR2CF M8as nitesumsofdecompSosabletensorspicrkone@Os2(m)UR= sX ㇍Si=1ci mi :Ԡ7 &e328. %TOOLBO9XYofshortestlengths.WThenthefamilies(cidjix=1;:::ʚ;s)and(mijix=1;:::ʚ;s)arelinearlyindepSendenrt.8Choosecoecienrtscij 62URCFsuchthat͍{N(cjf )UR= wtX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s;bryrsuitablyextendingthelinearlyindepSendentfamily(cidji<̹=1;:::ʚ;s)rtoalinearlyindepSendenrtfamily(cidjiUR=1;:::ʚ;t)andtURs.WVeɨ rstshorwthatwecanchoSoset=s.ByɨcoassociativitrywehaveP*tUs U_tUi=1!ciV As2(midڹ)t=P*]s U_]jv=1"a(cjf ) mjں=tP*]s U_]jv=1P*/ t U_/ i=1?Mcit cij h~ mjf .3]Since|theciVandthemjareclinearlyindepSendenrtwecancomparecoSecientsandget~4zs2(midڹ)UR= sX ㇍jv=1cij mjf ; 8i=1;:::ʚ;s(1)!oand0UR=P*s U_jv=1!Bcij mjPfori>s.8Thelaststatemenrtimpliesu{Ucij 6=UR0; 8i>s;j%=1;:::ʚ;s:⍹HencewregettUR=s꨹and{N(cjf )UR= sX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s:De neu_ nitedimensionalsubspacesCܞ20 =,hcijJji;j=1;:::ʚ;siCQandu_M@20 I=hmidjin=1;:::ʚ;siM@.Then&^bry(1)weget:nM@20 }4!Cܞ20,r M@20.WVeshowthatmL2M@20,and|thattherestrictionoftoCܞ20&givresalinearmapL:Cܞ204!^Cܞ20 4  Cܞ20so6zthattherequiredpropSertiesofthetheoremaresatis ed.VFirstobservrethatm`=P"(cidڹ)mi,2URM@20Źandcj\=P"(ci)cij 62Cܞ20׹.8UsingcoassoSciativitrywegetQ$P*]Ϭn U_]Ϭi;jv=1tS~ci (cijJ) mj=URP*s U_k6;jv=1(B(ck#) ck6j D mj=URP*s U_i;j$;k6=1- ci cik  ck6j D mj>hence( (cijJ)UR= sX'؍?k6=1cik  ck6j :(2)P %cffxff ̟ff ̎ ̄cffGRemark8.6.15.qN6WVegivreasketchofasecondproSofwhichissomewhatmoretecrhnical.ͩSinceCisaK-coalgebra,VthedualCܞ2Eisanalgebra.ThecomoSdulestructureѹ:Mʃ4!mC %M୹leadstoamoSdulestructurebry=(ev / 1)(1% s2):Cܞ2œ Mʃ4!Cܞ2E ]C M64!M@.Consider[thesubmoSduleN6:=URCܞ2m.ThenN0?is nitedimensional,since|c2mUR=P*n U_i=1AUhc2;cidimi>Vfor|allc2V2URCܞ2 vwhereP*)n U_)i=1 }cin mi,=s2(m).3'ObservrethatCܞ2m jisasubspaceofthespacegeneratedbrythemidڹ.'ButitdoSesnotdependonthedcrhoiceofthemidڹ. FVurthermoreifwetakes2(m)ؖ=PAci minwithdashortest!G7 &e 6. %CO9ALGEBRAS/33YrepresenrtationthenthemiqareinCܞ2msincec2m%4=Phc2;cidimi =miqforc2 Qanelemenrtofadualbasisofthecidڹ.NisaCܞ-comoSdulesinces2(c2m)#=PShc2;cidis2(mi)#=PShc2;ci(1)AVici(2)@ bmi 2CF Cܞ2m.Norw9weconstructasubScoalgebraDXǹofC׹suchthatNFisaDS-comodulewiththeinducedcoaction.:LetDQ:=xN` ;N@2.By8.13NŹisacomoSduleorverthecoalgebraN jN@2.Constructalinearmap5[:D4!CLbryn n2 _7!Pn(1) \|hn2;n(N")i.Byde nitionofthedualbasiswrehavenUR=Pnidhn2RAi;ni.8Thusweget.jʍ)#( )D(n n2)گ=UR( )(Pn n2RAij ni n2)گ=URPn(1) \|hn2RAi;n(N")i ni(1)AVhn2;ni(N")iگ=URPn(1)$ ni(1)AVhn2;ni(N")ihn2RAi;n(N")iگ=URPn(1)$ n(2) \|hn2;n(N")iUR=PC(n(1) \|)hn2;n(N")iگ=URC(n n2):FVurthermore~"C(n n2) ="(Pn(1) \|hn2;n(N")i=hn2;P"(n(1) \|)n(N")i=hn2;ni="(n n2).Hence UR:D4!Cԩisahomomorphismofcoalgebras,(DKis nitedimensionalandtheimageCܞ20T:=(DS)isa nitedimensionalsubcoalgebraofCܞ.#ClearlyNisalsoaCܞ20׹-comoSdule,sinceitisaD-comodule.FinallyǦwreshowthattheDS-comoduleǦstructureonNifliftedtotheCܞ-comoSdulestructurecoincideswiththeonede nedonM@.8WVeharve$eʍ8C(c2m)