; TeX output 1999.09.17:13337 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox/^o cmr915*7 &e168. %TOOLBO9XYN cmbx125.ΦAlgebrasLet( msbm10KbSeacommrutativering.InmostofourapplicationsKwillbea eld.TVensorproSductsofK-moduleswillbesimplywrittenasg cmmi12Mf!", cmsy10 %NB:=^M 2cmmi8K aN@.WEvrerysuchtensorproSductisagainaK-bimodulesinceeacrhK-moduleM!resp.NisaK-bimoSdule(see8.4.14).De nition8.5.1.vaùA}DK,@ cmti12-algebrffa}`isavrectorspaceAtogetherwithamultiplicationrUR:A AUR4!1A꨹thatisassoSciativre:KwX@A A @AQ32fdBw`O line10-k K 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- Ef7 &eT:5. %ALGEBRAS17YRemark8.5.2.j6EvreryK-algebraAisaringwiththemultiplication]xAA ffUR4!1A Ajr ffUR4!A:2Theunitelemenrtisn9(1),where1istheunitelementofK.ObrviouslyMthecompSositionoftwohomomorphismsofalgebrasisagainahomo-morphismofalgebras.FVurthermoretheidenrtitymapisahomomorphismofalge-bras.iHencetheK-algebrasformacategoryK-Algǹ.ThecategoryofcommrutativeK-algebraswillbSedenotedbryK-cAlgǹ.Problem8.5.1.nR1.8ShorwthatEndgK!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 B4isthesymmetrymapfromTheorem8.4.15.8Norwthefollowingdiagramscommute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;A!q% 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 &e188. %TOOLBO9XYFVurthermorewreusethehomomorphismË=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 nition8.5.4.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$gZ!commrutes.Note:"IfW+yrouwanttode neahomomorphismgË:URTƹ(Vp)4!1AW+withatensoralgebraasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVp.Lemma8.5.5.g5QAmtensortalgebrffa(Tƹ(Vp);)de nedbyVisuniqueuptoauniqueisomorphism.Proof.@_Let(Tƹ(Vp);)and(T20o(Vp);209)bSetensoralgebrasorverV.8ThenJbY_V9<ҁ <ׁ <܁ < < < < w< s<džs<džH <20✟ׁ ✟ ✟ ✟ >> H9S\ׁ @S\ @S\ @S\ @ܟ>@ܟ>RH V<20苼ҁ H򋼟ׁ H܁ H H H H$ H. H3džH3džjURTƹ(Vp)mvTƟ20o(Vp)s32fd@`ά-:{hӌ32fdQά-:ՍkCTƹ(Vp)>6TƟ20o(Vp)#L32fd@`ά-:,hM%impliesko=URh21 \|.U{cffxff ̟ff ̎ ̄cff 7 &eT:5. %ALGEBRAS19YProp`osition8.5.6.O(Rulesofcomputationinatensoralgebra)ZLffet(Tƹ(Vp);)bethe35tensoralgebrffaoverVp.fiThenwehave$1. #:Vl!Tƹ(Vp)uisinjeffctive(sowemayidentifytheelements(vn9)andvforall #vË2URVp),2. #Tƹ(Vp)UR=f"u cmex10P n;'ߍRk!iIviq1 :):::vinjLNiWy=(i1;:::ʚ;inP)35multiindexoflengthnng;c3. #if%f^[:\V!/AisahomomorphismofK-moffdules,bQAisaK-algebra,bQandg: #Tƹ(Vp)UR!A35istheinducffed35homomorphismofK-algebrffas,then荑i gn9(Xun;'ߍRk!iUVviq1 :):::vin)UR=X/n;'ߍRk!ifG(viq1):::f(vin):#ݍProof.@_1.UseGtheemrbSeddinghomomorphismj`":OVO4!D(Vp),whereD(Vp)isde nedasin8.5.3.=ptoconstructg蠹:zgTƹ(Vp)4![DS(V)sucrhthatgzg=jӹ.=pSincejCisinjectivresois.2.$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`osition8.5.7.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).27 &e208. %TOOLBO9XYWVeharvetoshowthat(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`osition8.5.8.OThekcffonstructionoftensoralgebrasTƹ(Vp)de nesafunctorT4:LK-35Mo`d$3!7,tK-35Alg]thataisleftadjointtotheunderlyingfunctorU0:K-35AlgwH!K-35Mo`do.Proof.@_FVollorwsfromtheuniversalpropSertyand8.9.16.uMcffxff ̟ff ̎ ̄cffProblem8.5.2.nR1.LetXobSeasetandV:=@l>RHXAۚǠ*Ffe̟Ǡ? Lggcommrutes.WVeG2Tƹ(Vp)$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.F\7 &eT:5. %ALGEBRAS21YDe nition8.5.9.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`osition8.5.11.(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 nition8.5.15.}!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 &e248. %TOOLBO9XYcommrutes.ThemrultiplicationinE(Vp)isusuallydenotedbyu^vn9.Note:=If0yrouwanttode neahomomorphismg{: _E(Vp)4!4KA0withanexterioralgebraqasdomainyroushouldde neitbygivingahomomorphismofK-moSdulesde nedonVsatisfyingfG(vn9)22V=UR0forallv;v202URVp.Problem8.5.4.nR1.XLetJf@:V 4!/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`osition8.5.17.(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. %ALGEBRAS25Y20#: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.^7 &e268. %TOOLBO9XYDe nition8.5.19.}!ùLetAbSeaK-algebra.رAleftO{A-moffduleisaK-moduleMtogetherwithahomomorphismM B:URA M64!M@,sucrhthatthediagramsLW?A MXM232fd>`ά-WYwA A MY?A M,{fd'8ά-ipid HJǠ*Ffe|Ǡ?э;r idH tJǠ*Ffe |Ǡ?'YandE@⍒nMP6԰=@K M⍒oA M{fd@ά-i?~I{ idH7M zǠ*Ffe Ǡ?',Hk}lid"ҁ H"ׁ H"܁ H" H" H" H" H" HdžHdžjs2commrute.LetkA OMOandkANbSekA-modulesandletfQ:URM64!NObeaK-linearmap./Themapf2iscalledahomomorphism35ofmoffdules꨹ifthediagramJ:ٍ+A NqN,32fd@`ά-W <X.NY0A MYMĺ{fd>`ά-iwX.MH Ǡ*Ffe=<Ǡ?`!1 fH Ǡ*Ffe /<Ǡ?`f̍commrutes.ThepleftA-moSdulesandtheirhomomorphismsformthecffategoryA MofA-moffdules.Problem8.5.6.nRShorwthatanabSeliangroupMisaleftmoduleorvertheringAifandonlyifM+isaK-moSduleandanA-moduleinthesenseofDe nition8.5.19.;7  ,@ cmti12+- cmcsc10( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12 b> cmmi10O line10u cmex10n