; TeX output 1999.03.03:19177 YN cmbx125.ΦAlgebrasXQ cmr12Let& msbm10KbSeacommrutativering.InmostofourapplicationsKwillbea eld.TVensorproSductsofK-moduleswillbesimplywrittenasg cmmi12Mf!", cmsy10 %NB:=^M 2cmmi8K aN@.WEvrerysuchtensorproSductisagainaK-bimodulesinceeacrhK-moduleM!resp.NisaK-bimoSdule(see1.4.14).De nition1.5.1.vaùA}DK*@ cmti12-algebrffa}`isavrectorspaceAtogetherwithamultiplicationrUR:A AUR4!1A꨹thatisassoSciativre:KwX@A A @AQ32fdBw`O line10-kK cmsy8rYxA A AY@A A,{fd*ά- At|{Ycmr8id rHJǠ*Ffe|Ǡ?э;r idH tJǠ*Ffe |Ǡ?`Yr$썹andaunitË:URK4!1A:M ⍑{oK APUR԰n:=APUR԰n:=A K⍒@A A^,{fd)ά-i(id0 HrʟǠ*FfeǠ?`@I{ idH%JǠ*Ffe&|Ǡ?`*rA A BA:Bl32fdXOά-k>rHk}Aϳ,>UH`rׁ̟ ̟ ̟ ̟ 댟>댟> xLet@,AandB2bSeK-algebras.9kA@homomorphismofalgebrffas@,f.߹:A4!MBis@,aK-linearmapsucrhthatthefollowingdiagramscommute:M ABE32fd6^ά- fYCHA AYBE B{fd@ά-if fH4RǠ*FfegǠ?JrX.;cmmi6AHҟǠ*FfeǠ?rX.B̍and@j⍒%(K'X.Aγׁ ɳ ij  d>d> H'tX.Bׁ A A A AD>AD>U AŚB:Rd32fd(Ѝά- Ef/^+o cmr914*7 &eT:5. %ALGEBRAS15YRemark1.5.2.j6EvreryK-algebraAisaringwiththemultiplication]xAA ffUR4!1A Ajr ffUR4!A:2Theunitelemenrtisn9(1),where1istheunitelementofK.ObrviouslyMthecompSositionoftwohomomorphismsofalgebrasisagainahomo-morphismofalgebras.FVurthermoretheidenrtitymapisahomomorphismofalge-bras.iHencetheK-algebrasformacategoryK-Algǹ.ThecategoryofcommrutativeK-algebraswillbSedenotedbryK-cAlgǹ.Problem1.1.cR1.8ShorwthatEndgK!3)(Vp)isaK-algebra.2.%Shorw&that(A;roX:A AoX4!=A;ݑ:oXK4!=A)&isaK-algebraifandonlyifAwiththemrultiplicationA\DA ffW4!<;A Aor ffW4!Aandtheunitn9(1)isaringandË:URK4!1Cenrt->(A)isaringhomomorphisminrtothecenterofA.3.)(1A k 1BN>):A! B A B14!A B4whereH:+B A+4!A B4isthesymmetrymapfromTheorem1.4.15.8Norwthefollowingdiagramscommute捍py%A BE A B A BpyKA A BE B A B4mifd(ά-jE21 r 1-:Aacmr63pypy`(A BE A B4mifd)7ά-jE6ºr r 1-:2Y%A BE A A B BYKA A A BE B B4{fd(ά-MU|ҙ1 X.BI;Aq% cmsy6 A} 1-:3YY`(A A BE B4{fd)7ά-23ir 1 r 1m5H@ `G fe@S4`?, 1-:3* r 1m5HEŸ`G fex`?͍+t1 X.BI B;A 1m5H}0Ÿ`G fe}c`?ht1 r 1鍑X8A A BE A B BH"FB1 r 1-:3P`Z`d`n`x`}Tx}Tx*m5?P5U1-:3* r 14zn@4n@4n@4n@4n@ԟԟ?PH"1-:2* r 1-:2ԟwHԟwHԟwHԟwHԟwH4H4j(A BE A B!A A BE Bl 32fdX ά-m01 r 1xXA BFD32fdXPά-/>FLr rH@ Ǡ*Ffe@S4Ǡ?o1 r rHEŸǠ*FfexǠ?o+t1 r 1 rH}0ŸǠ*Ffe}cǠ?otr r2In>-theleftuppSerrectangleofthediagramthequadranglecommrutesbythepropSertiesofYmthetensorproSductandthetrwoYmtrianglescommrutebyinnerpropSertiesofW..Therighrt=1uppSerandleftlowerrectanglescommutesinceNisanaturaltransformationandtherighrtlowerrectanglecommutesbytheassoSciativityofthealgebrasAandB. 7 ̍16YFVurthermorewreusethehomomorphismË=URA B:K4!1K"u K14!A BCinthefollorwingcommutativediagram~hp WK A BPX԰ @=A BPX԰ @=A BE KpA BE K Kqdmifd'8ά-jܠpp`A BE A B?mifdЍά-iQ>1-:2* I{ m5.ٍ1sठzn@H}ठn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@Hटn@H टn@Hटn@Hटn@H'टn@H1टn@H;टn@HEटn@HOटn@HYटn@Hcटn@Hmटn@HndƑHndƑjm5 zn@Hn@H—䟄n@H̗䟉n@H֗䟎n@H䟓n@H䟘n@H䟝n@H$䟟H$䟟jBA K BE KB`A A BE B? RfdЍά-\퍒;1 I{ 1 m5 br*Ffe?H$1 r 1m5 򟟴*Ffe$?1 r 1 ǠG fe$Ǡ?Gr rm5 9)dzn@HC)dn@HM)dn@HW)dn@Ha)dn@Hk)dn@Hu)dn@H)dn@HdHdjB7K K A BBuJK A K BT Rfdά-͍V:U1 r 1m5 (r*Ffe(?aA BE A BuA A BE BT32fdά-mV:U1 r 1vA B:t32fdt0ά-/;r r (rǠG fe(Ǡ?ևōGI{  1-:2 ǠG fe'$Ǡ?]tdI{ 1  1T %cffxff ̟ff ̎ ̄cffDe nition1.5.4.vaùLetKbSeacommrutativering.?gLetV3beaK-module.?gASK-algebraK1Tƹ(Vp)togetherwithahomomorphismofK-moSdules:V4!AT(Vp)K1iscalledatensor8algebrffaoverV5-ifforeacrhK-algebraAandforeachhomomorphismofK-moSdules^fQ:URV4!`AthereexistsauniquehomomorphismofK-algebrasgË:Tƹ(Vp)4!1AsucrhthatthediagramH;cHQbVH4zTƹ(Vp)Dğ{fd!wЍά- H` }fDğׁ @Dğ @Dğ @Dğ @ߋD>@ߋD>RH\0ArǠ*FfeäǠ? v$gZ!commrutes.Note:"IfW+yrouwanttode neahomomorphismgË:URTƹ(Vp)4!1AW+withatensoralgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVp.Lemma1.5.5.g5QAmtensortalgebrffa(Tƹ(Vp);)de nedbyVisuniqueuptoauniqueisomorphism.Proof.@_Let(Tƹ(Vp);)and(T20o(Vp);209)bSetensoralgebrasorverV.8ThenJbY_V9<ҁ <ׁ <܁ < < < < w< s<džs<džH <20✟ׁ ✟ ✟ ✟ >> H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjURTƹ(Vp)mvTƟ20o(Vp)s32fd@`ά-:{hӌ32fdQά-:ՍkCTƹ(Vp)>6TƟ20o(Vp)#L32fd@`ά-:,hM%impliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cff٠7 &eT:5. %ALGEBRAS17YProp`osition1.5.6.O(Rulesofcomputationinatensoralgebra)ZLffet(Tƹ(Vp);)bethe35tensoralgebrffaoverVp.fiThenwehave$1. #:Vl!Tƹ(Vp)uisinjeffctive(sowemayidentifytheelements(vn9)andvforall #vË2URVp),2. #Tƹ(Vp)UR=f u cmex10P n;'ߍRk!iIviq1 :):::vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;c3. #if%f^[:\V!/AisahomomorphismofK-moffdules,bQAisaK-algebra,bQandg: #Tƹ(Vp)UR!A35istheinducffed35homomorphismofK-algebrffas,then荑i gn9(Xun;'ߍRk!iUVviq1 :):::vin)UR=X/n;'ߍRk!ifG(viq1):::f(vin):#ݍProof.@_1.UseGtheemrbSeddinghomomorphismj`":OVO4!D(Vp),whereD(Vp)isde nedasin1.1.3.=ptoconstructg蠹:zgTƹ(Vp)4![DS(V)sucrhthatgzg=jӹ.=pSincejCisinjectivresois.2.$Let_BX:=URfP n;'ߍRk!iIviq1 /:::)/vinjLNiWy=(i1;:::ʚ;inP)mrultiindexoflengthoBng.Obrviously>Bis(ythesubalgebraofTƹ(Vp)generatedbrytheelementsofVp.TLetjk^:BY4!1Tƹ(V)bSetheemrbSeddinghomomorphism.p*Then/:V34![Tƹ(Vp)factorsthroughalinearmap20#:URV4!`B.8InthefollorwingdiagramO~bYVbYY@Bt{fd*Fά-{G 20HHEjTƹ(Vp)U{fd!wЍά-#=jY@BEjTƹ(Vp)U32fd!wЍά-^c=jH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> wreXhaveid"Bp20ݪ=q209.pwithpj20ݪ=qp=20&:existsXsince20isahomomorphismofK-moSdules.6BecauseofjpUR=jKn20#==id T.:(V)#4K¹wregetjpUR=id T.:(V)!4M,hencetheemrbSeddingj{issurjectiveandthusj{istheidentityV.3.8ispreciselythede nitionoftheinducedhomomorphism.dǷcffxff ̟ff ̎ ̄cff7Prop`osition1.5.7.OGiven-aK-moffduleVp. QThenthereexistsatensoralgebra(Tƹ(Vp);).Proof.@_De ne$TƟ2nJ(Vp)/:=Vn 4:::o 4VT=V2 nto$bSethen-foldtensorproductofVp.8De neTƟ20aʹ(V)UR:=KandTƟ21aʹ(Vp):=V.8WVede ne荑8pTƹ(Vp)UR:=M i0wvT i(Vp)=KVG(V Vp)(V V Vp):::uD:ǾThecompSonenrtsTƟ2nJ(Vp)ofTƹ(V)arecalledhomoffgeneous35components.ThecanonicalisomorphismsTƟ2m (Vp)] TƟ2nJ(V)PUR԰n:=TƟ2m+nkR(V)takrenasmultiplicationHJʍ rUR:TƟ2m (Vp) TƟ2nJ(V)UR4!1TƟ2m+nkR(V)sYrUR:Tƹ(Vp) T(Vp)UR4!1T(V)andUtheemrbSeddingy: `K=TƟ20aʹ(Vp)4!0MTƹ(V)UinducethestructureofaK-algebraonTƹ(Vp).8FVurthermorewrehavetheembSeddingUR:V4!`TƟ21aʹ(Vp)Tƹ(V).1̠7 ̍18YWVeharvetoshowthat(Tƹ(Vp);)isatensoralgebra.bLetfhʹ: V;4!AbSeaho-momorphismofK-moSdules.EacrhelementinTƹ(Vp)isasumofdecompSosableten-sors7v1 :::LD vnP. De negz: Tƹ(Vp)4!3A7brygn9(v1 :::LD vnP) :=fG(v1):::ʜf(vn)(and(gC{:BTƟ20aʹ(Vp)4!A)=(:K4!A)). cByinductiononeseesthatgg?isahomomorphismofalgebras. oSince(gJ!:TƟ21aʹ(Vp)4!]A)=(f#:V xX4!mA)wregetg?==fG. S0IfHh:Tƹ(Vp)4!lAisahomomorphismofalgebraswithh==f¹wregeth(v1j ::: vnP)UR=h(v1):::ʜh(vn)=fG(v1):::ʜf(vn)hencehUR=gn9.QWcffxff ̟ff ̎ ̄cff Prop`osition1.5.8.OThekcffonstructionoftensoralgebrasTƹ(Vp)de nesafunctorT4:LK-35Mo`d$3!7,tK-35Alg]thataisleftadjointtotheunderlyingfunctorU0:K-35AlgwH!K-35Mo`do.Proof.@_FVollorwsfromtheuniversalpropSertyand1.9.16.uMcffxff ̟ff ̎ ̄cffProblem1.2.cR1. cLet^XbSeasetandV:=CvKXbSethefreeK-moduleorverX.WShorw&thatX`4!V 4!Tƹ(Vp)de nesafrffeeUalgebra&йoverX,ui.e.WforeveryK-algebra3AandevrerymapfQ:URXF4!AthereisauniquehomomorphismofK-algebrasgË:URTƹ(Vp)4!1A꨹sucrhthatthediagramI׍HXHTƹ(Vp)&<{fd ဍά-H`Xfׁ @ɏ @ӏ @ݏ @l>@l>RHXAۚǠ*Ffe̟Ǡ? Lggcommrutes.WVeG2Tƹ(Vp)$Kl:2fdנά-Í"92Tƹ(Vp)鬟:2fdנά-(7bTƹ(Vp) T(Vp)TTƹ(Vp) T(Vp)Č32fdά-/В-1 SВ-Sr} 1*Ǡ@fe\Ǡ??;*Ǡ@fe\?`6?; zrncommrute.E7 &eT:5. %ALGEBRAS19YDe nition1.5.9.vaùLetKbSeacommrutativering.?gLetV3beaK-module.?gASK-algebraS׹(Vp)togetherwithahomomorphismofK-moSdulesUR:V4!`S(Vp),|sucrhthat(vn9) (v20LK-algebraAandforeachhomomorphismofK-moSdulesf+:V4!}[A,S5suchthatfG(vn9)"zf(vn920@߸\>RHHA񽊟Ǡ*FfeǠ? > H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjUiRS׹(Vp)uSן20(Vp)sY\32fdά-:{hy\32fdRά-:ՍkS׹(Vp)>5Sן20(Vp)#32fdά-:,hvimpliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffProp`osition1.5.11.(Rulespofcomputationinasymmetricalgebra)Lffet(S׹(Vp);)bffe35thesymmetricalgebraoverVp.fiThenwehave{\1. #:V@d!JS׹(Vp)r(isinjeffctive(wewillidentifytheelements(vn9)andvaforall #vË2URVp),2. #S׹(Vp)UR=fP n;'ߍRk!iIviq1 :):::vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;>3. #ifkf׹:sV H!AisahomomorphismofK-moffdulessatisfyingfG(vn9)f(vn920Bӹis5thesubalgebraofS׹(Vp)generatedbrytheelementsofVp.PLetj ::Bp@4!_S׹(V)bSetheIemrbSeddinghomomorphism.Then:VB&4!iS׹(Vp)Ifactorsthroughalinearmap20#:URV4!`B.8InthefollorwingdiagramObYVbYY@Bt{fd*Fά-{G 20HHjS׹(Vp)U{fd!ά-#zjY@BjS׹(Vp)U32fd!ά-^czjH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> Hwrelhaveid77Bs20=2u209,pwithpj20=2up=20:̹existslsince20isahomomorphismofK-moSdulessatisfying209(vn9)-20(vn920@߸\>RHHA񽊟Ǡ*FfeǠ? propSertryjfollowssincethede ningconditionfG(vn9)f(vn920@>RHpAǠ*Ffe;Ǡ? dgscommrutes.ThealgebraK[X]:=S׹(KX)iscalledthepffolynomial=ringoverKinthe(com-muting)35variablesX.2.ELettS׹(Vp)and:VSt4!$S(Vp)tbSeasymmetricalgebra.EShorwthatthereisauniquehomomorphism :S׹(Vp)4!kS(Vp)& S(Vp)with(vn9) =v:_ &1+1 vforallvË2URVp.3.8Shorwthat( 1)UR=(1 )UR:S׹(Vp)4!1S(Vp) S(Vp) S(Vp).4.YShorw5thatthereisauniquehomomorphismofalgebras"W:S׹(Vp)4!.;K5with"(vn9)UR=0forallvË2URVp.5.8Shorwthat(" 1)UR=(1 ")UR=id Sr}(V) .6. Shorw`thatthereisauniquehomomorphismofalgebrasS):URS׹(Vp)4!1S(Vp)`withS׹(vn9)UR=v.7.8Shorwthatthediagrams? J2S׹(Vp)$K<:2fd1Ѝά-ÍÖ"2S׹(Vp)鬟:2fd1Ѝά-(7S׹(Vp) S(Vp)S׹(Vp) S(Vp).<32fdB@ά-/В-1 SВ-Sr} 1*Ǡ@fe\Ǡ??;*Ǡ@fe\?`6?; zrIcommrute.PeDe nition1.5.15.}!ùLetTKbSeacommrutativeTring. wELetV:beaK-module. wEAK-algebraTE(Vp)togetherwithahomomorphismofK-moSdules :V4!=E(Vp),oosucrhthat(vn9)22YF=B0forallv{2Vp,iscalledanexteriorWalgebrffaorGrassmannalgebraoverV]if-foreacrhK-algebraAandforeachhomomorphismofK-moSdulesf u:vV^4!:A,sucrhthatfG(vn9)22V=UR0forallvË2Vp,thereexistsauniquehomomorphismofK-algebrasgË:URE(Vp)4!1A꨹sucrhthatthediagramKr6HBVH`E(Vp){fd ά- eH`]fׁ @ @ @ @O$>@O$>RH ATRǠ*Ffe񇄟Ǡ? :g| 7 ̍22Ycommrutes.ThemrultiplicationinE(Vp)isusuallydenotedbyu^vn9.Note:=If0yrouwanttode neahomomorphismg{: _E(Vp)4!4KA0withanexterioralgebraqasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVsatisfyingfG(vn9)22V=UR0forallv;v202URVp.Problem1.4.cR1.ðLetØf:ƓVc4!C#AbSealinearmapsatisfyingfG(vn9)22 =0forallvË2URVp.8ThenfG(vn9)f(v20> H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjTXE(Vp){E20P(Vp)t+̞32fdOά-:{hK̞32fdQ ά-:ՍkE(Vp)>";E20P(Vp)$Q32fdOά-:,himpliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cffProp`osition1.5.17.(Rules`ofcomputationinanexterioralgebra)sLffet(E(Vp);)bffe35theexterioralgebraoverVp.fiThenwehave 1. #:VT!E(Vp)^isinjeffctive(wewillidentifytheelements(vn9)andvforall #vË2URVp),2. #E(Vp)UR=fP n;'ߍRk!iIviq1 :)^:::^vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;>3. #ifkf׹:sV H!AisahomomorphismofK-moffdulessatisfyingfG(vn9)f(vn920BisthesubalgebraofE(Vp)generatedbrytheelementsofVp.LetjI:B74!5E(V)bSetheemrbSeddinghomomorphism.JThen:V4!)E(Vp)factorsthroughalinearmap7 &eT:5. %ALGEBRAS23Y20#:URV4!`B.8InthefollorwingdiagramOebYVbYY@Bt{fd*Fά-{G 20HHpE(Vp)U{fd ά-#jY@BpE(Vp)U32fd ά-^cjH 20tׁ @t @t @t @>@>RH"Ǡ*FfeTǠ?褍QidBH bǠ*Ffe ԔǠ?hjpH p4ׁ 4 4 ݜ4 U>U> +wrelhaveid77Bs20=2u209,pwithpj20=2up=20:̹existslsince20isahomomorphismofA8K-moSdulessatisfying209(vn9)20(vn920 cmmi10n oiG^"*.2. bShorw)thatthesymmetricgroupSn gyopSerates(fromtheleft)onTƟ2nJ(Vp)byn9(v1j ::: vnP)UR=vI{1 ;;(1) ::: vI{1 ;;(n)F/withË2Sn andvi,2Vp.؍3./Atensorsymmetrictensorifn9(a)r=aforall2SnP.]܍Letx^Sן2n(Vp)bSethesubspaceofsymmetrictensorsinTƟ2nJ(V).a)ShorwthatS;:URTƟ2nJ(Vp)3a7!PI{2Sn'Hn9(a)2TƟ2n(Vp)isalinearmap.ፑb)ShorwthatS hasitsimageinx^Sן2n(Vp).c)ShorwthatImq(Sb)UR=xVb^Sן2nj(Vp)ifn!isinvertibleinK.rjd)oShorwthatxp.^Sן2n6(Vp)6,!TƟ2nJ(V)2p!mSן2n['(V)oisanisomorphismifn!isinrvertibleoinK andqK:TƟ2nJ(Vp)4!xSן2n['(V) istherestrictionofqK:Tƹ(V)4!xS׹(V),the symmetricalgebra.4.Actensorda#2TƟ2nJ(Vp)iscalledanantisymmetrictensorifn9(a)#="()adforallk22Sn Ϲwhere_"(n9)isthesignofthepSermrutation.fLetx^E2nn(Vp)bSethesubspaceofanrtisymmetrictensorsinTƟ2nJ(Vp).a)ShorwthatEh:URTƟ2nJ(Vp)3a7!PI{2Sn'H"(n9)(a)UR2TƟ2n(Vp)isalinearmap.ፑb)ShorwthatEkhasitsimageinx^E2n(Vp).c)ShorwthatImq(Eù)UR=x\^E2nd^(Vp)ifn!isinvertibleinK.rjd)UShorwthatx^E2nd(Vp)UR,!TƟ2nJ(V)2p!E2n\g(V)Uisanisomorphismifn!isinrvertibleUinKand:URTƟ2nJ(Vp)4!1E2n\g(V)istherestrictionof:URTƹ(V)4!1E(V),theexterioralgebra.K7 ̍24YDe nition1.5.19.}!ùLetAbSeaK-algebra.رAleftO{A-moffduleisaK-moduleMtogetherwithahomomorphismM B:URA M64!M@,sucrhthatthediagramsS?A MXM232fd>`ά-WYwA A MY?A M,{fd'8ά-ipid HJǠ*Ffe|Ǡ?э;r idH tJǠ*Ffe |Ǡ?'Y0(andKύ⍒nMP6԰=@K M⍒oA M{fd@ά-i?~I{ idH7M zǠ*Ffe Ǡ?',Hk}lid"ҁ H"ׁ H"܁ H" H" H" H" H" HdžHdžjcommrute.LetkA OMOandkANbSekA-modulesandletfQ:URM64!NObeaK-linearmap./Themapf2iscalledahomomorphism35ofmoffdules꨹ifthediagramPh+A NqN,32fd@`ά-W <X.NY0A MYMĺ{fd>`ά-iwX.MH Ǡ*Ffe=<Ǡ?`!1 fH Ǡ*Ffe /<Ǡ?`f<[commrutes.ThepleftA-moSdulesandtheirhomomorphismsformthecffategoryA MofA-moffdules.OProblem1.6.cRShorwņthatanabSeliangroupMjisaleftmoduleorverņtheringAifandonlyifM+isaK-moSduleandanA-moduleinthesenseofDe nition1.5.19.*g6._CoalgebrasDe nition1.6.1.vaùA&K-cffoalgebra&isaK-moSduleCtogetherwithacffomultiplica-tion꨹ordiagonalUR:C14!CF CFthatiscoassoSciativre:Q_?.CF C9CF C Cܞ32fd)@ά-aid< Y6CY1.CF C|{fdAά-ˍJNHǠ*Ffe,Ǡ?`PHǠ*Ffe,Ǡ?э} idk7 &e 6. %CO9ALGEBRAS/25YandacffounitoraugmentationUR:C14!K:KYnCY}FCF C{fdY?ά-ˍ&HuǠ*FfeğǠ?`~HǠ*FfeDǠ?эid I "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džj4AK-coalgebraCFiscffocommutative꨹ifthefollorwingdiagramcommutesH9پCF CCF Cψl32fdά-ÍvIbY-C``ϳ,ׁ ʳ, ų, , >> H`댟ׁ A댟 A댟 A댟 A̟>A̟>U/ Let|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.CHʟǠ*FfeǠ?|X.D^andFd卍H""%(K-PX.Cdׁ AÐd AȐd A͐d Aγ>Aγ>UH-tX.DDׁ D D D >> bYCbY*DS:Ž{fd(ά-iEf̍Remark1.6.2.j6ObrviouslyhQthecompSositionoftwohomomorphismsofcoalgebrasisagainahomomorphismofcoalgebras.(FVurthermoretheidenrtitymapisahomo-morphism8ofcoalgebras. "{HencetheK-coalgebrasformacategoryK-Coalg!.ThecategoryofcoScommrutativeK-coalgebraswillbedenotedbryK-cCoalg'.Problem1.7.cR1.Shorwy thatV  Vp2 ~isacoalgebraforevery nitedimensionalvrectorspaceV>1overa eldKifthecomultiplicationisde nedby(vC ՝vn92.=)@:=P* n U_ i=1v v2n9RAi vi vn92 wherefvidgandfv2n9RAi.=garedualbasesofVresp.Vp2\t.2.ShorwthatthefreeK-moSdulesKX $withthebasisXandthecomrultiplication(x)UR=x x꨹isacoalgebra.8Whatisthecounit?Isthecounitunique?3.&ShorwthatK<,VQwith(1)UR=1<, 1,b(vn9)UR=ve <,1+1 v"ʹde nesacoalgebra.4.Let^CandDbSecoalgebras.ThenC.- QDisacoalgebrawiththecomrultiplicationC DG:=UR(1C R V 1D)(C D)UR:CX DSH C D4!C Dꇹandcounit"UR="C DG:CF D4!K K14!K.8(TheproSofisanalogoustotheproofofLemma1.5.3.)7 ̍26YTVo?describSethecomrultiplicationofaK-coalgebraintermsofelementsweintro-duceanotation rstinrtroSducedbySweedlersimilartothenotationr(a^ b)X=abusedforalgebras.8Insteadof(c)UR=Pci c20RAiOwrewrite[(c)UR=Xc(1)$ c(2) \|:ObservrevXthatonlythecompleteexpressionontherighthandsidemakessense,Dnotthe4compSonenrtsc(1)andforthemrultilinearmap 22V:URCFCC14!C C CVXjITc(1)$ c(2) c(3)ι=UR( 1)(c)=(1 )(c):Withthisnotationonevreri eseasilyDeXX c(1)$ ::: (c(i) R) ::: c(n)=URXc(1)$ ::: c(n+1)andxc(1)$ ::: (c(i) R) ::: c(n)ݭ=URPc(1)$ ::: 1 ::: c(n1)ݭ=URPc(1)$ ::: c(n1)ٍThis4notationanditsapplicationtomrultilinearmapswillalsobSeusedinmoregeneralconrtextslikecomoSdules.Prop`osition1.6.4.OLffet.MC beacoalgebraandAanalgebra.rThenthecompositionfgË:=URrA(f gn9)C de nes35amultiplicffation"IC{Homa(C5;A) Hom$1(C;A)UR3f gË7!fgË2Hom(C5;A);~7 &e 6. %CO9ALGEBRAS/27Ysuch.TthatHom(C5;A)bffecomes.Tanalgebrffa.WTheunitelementisgivenbyK&Z3 97!(cUR7!n9( (c)))2Hom(C5;A).Proof.@_TheamrultiplicationofHom(C5;A)obviouslyisabilinearmap. Themul-tiplicationPisassoSciativresince(fgn9)hbZ=rA((rA(f gn9)C) h)C =bZrA(rA 1)((f lgn9) h)(CC 1)C t=URrA(1 rA)(f (g h))(1 C)C t=URrA(f (rA(g h)C))C <=fab(g<h).NFVurthermore"itisunitarywithunit1HomD(C;A)+_l=AC >sinceAC fQ=URrA(AC fG)C t=URrA(A { 1A)(1K fG)(C 1C)C t=URf,{and|similarlyfAC t=URfG.d cffxff ̟ff ̎ ̄cffDe nition1.6.5.vaùThe*$mrultiplicationUR:Hom(C5;A)!o Hom(C;A)UR4!1Hom-=(C;A)iscalledcffonvolution.Corollary1.6.6.sWLffet~C[zbeaK-coalgebra. I_ThenCܞ2 X=zHom5K&pŹ(C5;K)isanK-algebrffa.Proof.@_UsethatKitselfisaK-algebra.ìecffxff ̟ff ̎ ̄cffRemark1.6.7.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.Lemma1.6.8.g5QLffetKbea eldandAbea nitedimensionalK-algebra.ThenA2V=URHom۟K$ (A;K)35isaK-cffoalgebra.Proof.@_De nethecomrultiplicationonCܞ2 Jby~;UR:A 2 r-:pVj!(A A) 2Vcanb-:1p Î!"tA j A :eTheBcanonicalmapcan:URA2 A2V4!5(A A)2fFisBinrvertible,#sinceAis nitedimensional.By&[adiagrammaticproSoforbrycalculationwithelementsitiseasytoshowthatA2bSecomesaK-coalgebra./ׄcffxff ̟ff ̎ ̄cffRemark1.6.9.j6IfYKisanarbitrarycommrutativeYring,tthenA2X=THomݟK$Ɵ(A;K)isaK-coalgebraifAisa nitelygeneratedprojectivreK-moSdule.Problem1.8.cRFindisucienrtconditionsforanalgebraAresp.,"acoalgebraCsucrhthatHomd1(A;Cܞ)bSecomesacoalgebrawithco-convolutionascomultiplication.De nition1.6.10.}!ùLetCbSeaK-coalgebra.AleftYCܞ-cffomoduleisaK-moduleM+togetherwithahomomorphismM B:URM64!CF M@,sucrhthatthediagramsJCF MeCF C M32fd&ά-a>id˾ YMYCF M\{fd>y@ά--(HǠ*Ffe,Ǡ?k}HǠ*Ffe,Ǡ?э} id7 ̍28YandAjbYMŸǠ*FfeHǠ?k}"CF M"VK MP6԰=@M:32fd:]ά-aF idHk}؄4idҁ Hׁ H܁ HĹ Hι Hع H⹴ H카 HFdžHFdžjccommrute.Let~џ2C iMand~џ2CNbSe~Cܞ-comodulesandletf:QMf4!uNbeaK-linearmap.\Themapf2iscalledahomomorphism35ofcffomodules꨹ifthediagramN;㍍NCF N>,32fd@Z@ά-EX.NY MY^CF M.{fd>y@ά-zX.MHDleftCܞ-comoSdulesandtheirhomomorphismsformthecffategory2C XMofcffomod-ules.LetiNMbSeanarbitraryK-moduleandMMbeaCܞ-comodule. #ThenthereisabijectionbSetrweenallmultilinearmapsoLfQ:URCF:::M64!Nandalllinearmaps `fG 0k:URCF ::: M64!N:fG(m(1) \|;:::ʚ;m(n) D;m(M") h)UR:=f 08(s2 n(m));wheres22n *denotesthen-foldapplicationofs2,i.e.82n pԹ=UR(1 ::: 1 )(1 ).Inparticularwreobtainforthebilinearmap UR:CFM64!C MaX Jm(1)$ m(M")u=URs2(m);"andforthemrultilinearmap 22V:URCFCM64!C C MPUȟXcm(1)$ m(2) m(M")u=UR(1 s2)(c)UR=( 1)s2(m):%ӍProblem1.9.cRShorwthata nitedimensionalvectorspaceVRGisacomoSduleoverthe+coalgebraV MVp2 韹asde nedinproblem1.7.1withthecoactions2(vn9)i:=Pv v2n9RAi vi,2UR(VG Vp2\t) VwhereꨟPSv2n9RAi viOisthedualbasisofVinVp2  Vp.D7 &e 6. %CO9ALGEBRAS/29YTheorem1.6.11.wX(Fundamental~ThefforemforComodules)LetKbea eld.ILetMKbffe aleftCܞ-comoduleandletmUR2MKbe given.Thenthereexistsa nitedimensionalsubffcoalgebrahCܞ20طCanda nitedimensionalCܞ20-cffomodulehM@20 withm2M@20Mwherffe35M@20doURMtisaK-submodule,suchthatthediagramO0̍WMϮCF M32fd>y@ά-͍/ n:M@20/ uCܞ20U M@20L{fd:?ά- L̟-:0HǠ*Ffe$̟Ǡ?H㚟Ǡ*Ffe̟Ǡ?_cffommutes.)卍Corollary1.6.12.z1.~Each;?elementcd42Cof;?acffoalgebra;?iscffontained;?ina nitedimensional35subffcoalgebraofCܞ.2. wEachpelementmu2M$Tofpacffomodulepiscffontainedpina nitedimensionalsubffcomodule35ofM@.Corollary1.6.13.z1. ^Eachl nitedimensionalsubspffacelVofacffoalgebralCI%iscffontained35ina nitedimensionalsubffcoalgebra35Cܞ20 ofCܞ.2.ËEach nitedimensionalsubspffaceVDofacffomoduleMziscffontainedina nitedimensional35subffcomoduleM@20BRofM@.Corollary1.6.14.z1.GEachQcffoalgebraisaunionof nitedimensionalsubcoalge-brffas.2.fiEach35cffomoduleisaunionof nitedimensionalsubcomodules.Proof.@_(oftheThefforem)@WVecanassumethatmUR6=0@forelsewrecanuseM@20do=UR0andCܞ20)=UR0.Undertherepresenrtationsofs2(m)UR2CF M8as nitesumsofdecompSosabletensorspicrkoneåOs2(m)UR= sX ㇍Si=1ci miS?ofshortestlengths.WThenthefamilies(cidjix=1;:::ʚ;s)and(mijix=1;:::ʚ;s)arelinearlyindepSendenrt.8Choosecoecienrtscij 62URCFsuchthat"){N(cjf )UR= wtX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s;!rbryrsuitablyextendingthelinearlyindepSendentfamily(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|theciVandthemjarey7 ̍30YlinearlyindepSendenrtwecancomparecoSecientsandget!zs2(midڹ)UR= sX ㇍jv=1cij mjf ; 8i=1;:::ʚ;s(1)$and0UR=P*s U_jv=1!Bcij mjPfori>s.8Thelaststatemenrtimpliesm;{Ucij 6=UR0; 8i>s;j%=1;:::ʚ;s: HencewregettUR=s꨹and{N(cjf )UR= sX ㇍Si=1ci cijJ; 8j%=1;:::ʚ;s:# bDe neu_ nitedimensionalsubspacesCܞ20 =,hcijJji;j=1;:::ʚ;siCQandu_M@20 I=hmidjin=1;:::ʚ;siM@.Then&^bry(1)weget:nM@20 }4!Cܞ20,r M@20.WVeshowthatmL2M@20,and|thattherestrictionoftoCܞ20&givresalinearmapL:Cܞ204!^Cܞ20 4  Cܞ20so6zthattherequiredpropSertiesofthetheoremaresatis ed.VFirstobservrethatm`=P"(cidڹ)mi,2URM@20Źandcj\=P"(ci)cij 62Cܞ20׹.8UsingcoassoSciativitryweget ?Q$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$;k6=1- ci cik  ck6j D mj:hence (cijJ)UR= sX'؍?k6=1cik  ck6j :(2)"S %cffxff ̟ff ̎ ̄cffRemark1.6.15.qN6WVegivreasketchofasecondproSofwhichissomewhatmoretecrhnical.ͩSinceCisaK-coalgebra,VthedualCܞ2Eisanalgebra.ThecomoSdulestructureѹ:Mʃ4!mC %M୹leadstoamoSdulestructurebry=(ev / 1)(1% s2):Cܞ2œ Mʃ4!Cܞ2E ]C M64!M@.Consider[thesubmoSduleN6:=URCܞ2m.ThenN0?is nitedimensional,since|c2mUR=P*n U_i=1AUhc2;cidimi>Vfor|allc2V2URCܞ2 vwhereP*)n U_)i=1 }cin mi,=s2(m).3'ObservrethatCܞ2m jisasubspaceofthespacegeneratedbrythemidڹ.'ButitdoSesnotdependonthedcrhoiceofthemidڹ. FVurthermoreifwetakes2(m)ؖ=PAci minwithdashortestrepresenrtationthenthemiqareinCܞ2msincec2m%4=Phc2;cidimi =miqforc2 Qanelemenrtofadualbasisofthecidڹ.NisaCܞ-comoSdulesinces2(c2m)#=PShc2;cidis2(mi)#=PShc2;ci(1)AVici(2)@ bmi 2CF Cܞ2m.Norw9weconstructasubScoalgebraDXǹofC׹suchthatNFisaDS-comodulewiththeinducedNcoaction.LetDG:= No PN@2.By1.9N2isacomoSduleorverNthecoalgebraN jN@2.Constructalinearmap5[:D4!CLbryn n2 _7!Pn(1) \|hn2;n(N")i.By7 &e 6. %CO9ALGEBRAS/31Yde nitionofthedualbasiswrehavenUR=Pnidhn2RAi;ni.8Thusweget.jʍ)#( )D(n n2)گ=UR( )(Pn n2RAij ni n2)گ=URPn(1) \|hn2RAi;n(N")i ni(1)AVhn2;ni(N")iگ=URPn(1)$ 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) cmmi10O line10u cmex10