; TeX output 1999.09.17:1332 7 YRXQ cmr12CHAPTER8&Nff cmbx12Tfos3olbox]o cmr99 *7 &e108. %TOOLBO9XY5N cmbx124.TensorPro`ductsDe nitionandRemark8.4.1.ßLetg cmmi12M2cmmi8R UandRNbSeRJ-modules, andletAbeanabSeliangroup.8AmapfQ:URM !", cmsy10N64!AiscalledRJ,@ cmti12-bilineffarifs21. #fG(m+m2!K cmsy809;n)UR=f(m;n)+f(m209;n);2. #fG(m;n+n209)UR=f(m;n)+f(m;n209);3. #fG(mrr;n)UR=f(m;rSn)forallr2URRJ;m;m20#2M;n;n20#2N@.LetBilݟRp(M;N@;A)denotethesetofallRJ-bilinearmapsfQ:URMN64!A.Bil 5R'ȹ(M;N@;A)isanabSeliangroupwith(f+gn9)(m;n)UR:=fG(m;n)+gn9(m;n).%De nition8.4.2.vaùLetMR andRNbSeRJ-modules.AnabeliangroupMt R '[NtogetherwithanRJ-bilinearmapem6 UR:MN63(m;n)7!m n2M R ;NisVcalledatensorYprffoductofMhandNoverVRoιifforeacrhabSeliangroupAandforeacrhRJ-bilinearmapf :M~5=QN4!eAthereexistsauniquegrouphomomorphismgË:URM R ;N64!A꨹sucrhthatthediagramMHy{8MN{M R ;N{fd UO line10-`О H`ȱf,ׁ @, @, @, @/>@/>RHA4ڟǠ*Ffeh Ǡ? gecommrutes.TheelementsofM R ^ENEarecalledtensors,K;theelementsoftheformm n꨹arecalleddeffcomposable35tensors.Warning: IfDyrouwanttode neahomomorphismfQ:URM2 RN64!AwithatensorproSduct[rasdomainyroumustde neitbygivinganRJ-bilinearmapde nedonM$N@.%Lemma8.4.3.g5QA-\tensor-]prffoduct(Mސ R ?N; )de nedbyMR andRNnAisuniqueup35toauniqueisomorphism.- cmcsc10Proof.@_Let(M R ;N; )and(M& msam10R ;N;)bSetensorproducts.8ThenH2YMN`> ^ҁ ^ׁ ^܁ ^ ^ ^ ^ x^ sќdžsќdžH-»Ɩׁ P|>P|> H` ׁ @ @ @ @N<>@N<>RH- @ҁ H@ׁ H@܁ H@ H@ H@ H%@ H/@ H3džH3džj33LM R ;N33MR ;N}T32fdϠά-:}Zh3333 32fdAά-:Սk33?M R ;N337MR ;N-zl32fdϠά-:-whWimpliesko=URh2|{Ycmr81 \|.U{cffxff ̟ff ̎ ̄cffBecause}ofthisfactwrewillhenceforthtalkabSoutthetensorproductofMaandNorverRJ.Prop`osition8.4.4.O(Rules!wofcomputationinatensorproSduct)zGLffet(MP RIN; )bffe35thetensorproduct.fiThenwehaveforallr2URRJ,m;m20#2M@,n;n20#2N 7 &ea>4. %TENSOR!PR9ODUCTS 11Y1. #M R ;N6=URf#u cmex10P imi nij35mi,2M;ni2N@g;2. #(m+m209) nUR=m n+m20x n;3. #m (n+n209)UR=m n+m n20;4. #mrqB nt=m rSn·(observeinpffarticular,[that t:M^N!3M Nkis·not #injeffctive35ingeneral),5. #ifJ5f:MN!#AisanRJ-bilineffarmapandg":M R DN!#Aistheinducffed #homomorphism,35thenPgn9(m n)UR=fG(m;n):ۍProof.@_1. +LetlB:= hm niM R NPdenotelthesubgroupofM R NgeneratedԞbrythedecompSosabletensorsmr n.2LetԞj%:URBX4!_7M RfRNbetheemrbeddinghomomorphism.]WVe getaninducedmap 20#:URM5QN64!B.InthefollorwingdiagramOlXY[MNYHXB{fdά-; 20{󎎍{POM R ;ND̟{fdά-#趪j33HXB33POM R ;ND̞32fdά-^c趪jH披E\ 20ׁ @ @ @ @ >@ >RH:Ǡ*Ffe lǠ?褍A id BHzǠ*FfeìǠ?hv,jpH pLׁ L L L D̟>D̟> Zwrebhaveid\BB 20?V=q 209,pwithp,j 20?V=qp = 20_existsbsince 20isRJ-bilinear.Becauseyofjpv Թ=jgIv 20 = =idxM" X.;cmmi6RN/* y޹wregetjpԹ=idxM" X.RN-,,ݬhenceytheemrbSeddingj{issurjectiveandthustheidentityV.2.8(m+m209) nUR= (m+m20;n)UR= (m;n)+ (m209;n)UR=m n+m20x n.3.8and4.analogouslyV.5.8ispreciselythede nitionoftheinducedhomomorphism.dǷcffxff ̟ff ̎ ̄cffۍRemark8.4.5.j6TVoconstructtensorproSducts,,wreusethenotionofafreemodule.LetLXϹbSeasetandRԖbearing.)AnRJ-moduleRJXϹtogetherwithamapUR:XF4!RJXiscalledafrffeeR-modulegeneratedbyX,ifforevreryRJ-moSduleMwandforeverymap@&f.չ:XY4!ؼM thereexistsauniquehomomorphismofRJ-moSdulesgU:RJXY4!ؼMsucrhthatthediagramE[3bYgTXbY*RJX{fd$pά- ѯH`efׁ @ @ @ @?,>@?,>RH(MDZǠ*FfewǠ? * gycommrutes.FVreeRJ-moSdulesexistandcanbeconstructedasRJX:=f :X4!5 RJjforalmostallxUR2XFչ: (x)=0g.ۍProp`osition8.4.6.OGivenIRJ-moffdulesMR andRN@.gThentherffeexistsatensorprffoduct35(M R ;N; ).Proof.@_De ne:M R N:=0) msbm10ZfMN@g=UXwhereZfMN@gisafreeZ-moSduleorverMN+(thefreeabSeliangroup)andUisgeneratedbry 7 &e128. %TOOLBO9XY #(m+m209;n)(m;n)(m209;n) #(m;m+n209)(m;n)(m;n209) #(mrr;n)(m;rSn)s2forallr2URRJ,m;m20#2M@,n;n20#2N@.8ConsiderJ:ٍH@MNHZfMN@gj!t{fd!;ά- xHHeM R ;N/{fdЍά- ׳HHVh=URZfMN@g=UHqA` l\uPv\+P\ׁP\tP\+P\P\sP\+P\P\rP\+P\P0$P0$qH 5+ʬQtQQ+QrQQ턴Q턴sHǠ*FfeDǠ? geLetUt íbSegivren.yDThenthereisaunique 2Hom(ZfM4B^N@g;A)Utsuchthat = n9.Since IisRJ-bilinearwreget((mN5+m209;n)(m;n)(m209n))1= n9(mN5+m20;n) n9(m;n)Z (m209;n) =0andsimilarly((m;nZ+n209)(m;n)(m;n209)) =0and((mrr;n)~(m;rSn))h=0.So&wreget(U@)h=0.This&impliesthatthereisauniqueg=2Hom(M7m R N;A)Zsucrhthatgn9˹=(homomorphismtheorem).6Let :=P.Then isbilinearsince(m4+m209) n$/=4(m+m20;n)$/=ǹ((m4+m20;n))$/=ǹ((mfE+m209;n)(m;n)(m209;n)+(m;n)+(m209;n))*[=ǹ((m;n)fE+(m209;n))*[==;v(m;n)+(m209;n)UR=m;v n+m20 n.&The8othertrwo8propSertiesareobtainedinananalogouswrayV.WVe>=harvetoshowthat(M` R>N; )isatensorproSduct.gTheaborvediagramshowsthat~foreacrhabSeliangroupAandforeachRJ-bilinearmap N:gMY )N4!(AthereisVag{52 Hom(M5 R N;A)sucrhthatgbX  = n9.|GivenVh2Hom(M5 R N;A)withh UR= n9.8ThenhloUR= .8Thisimpliesh=UR=gohencegË=h.(Մcffxff ̟ff ̎ ̄cff鄍Prop`ositionandDe nition8.4.7.ajGiven35twohomomorphismsednfQ2URHomٟR$l(M:;M@ 0:)35andgË2URHomٟR(:N;:N@ 0):Then35therffeisauniquehomomorphism}Hf R ;gË2URHom(M RN;M@ 0 RN@ 0)such35thatf R ;gn9(m n)UR=fG(m) g(n),35i.e.fithefollowingdiagrffamcommutesM%卒UM@20N@20卒QM@20 R ;N@20]432fdpά- W` {$0MN{M R ;Nȋ{fdά-` H꒟Ǡ*FfeğǠ?`ffgH`Ǡ*FfeDǠ?`Ef R ;g?Proof.@_ (fgn9)isbilinear.@cffxff ̟ff ̎ ̄cff鄍Notation8.4.8.p>6WVe_roftenwritefB+ R N\:=f R1N xand_rM; RgT:=1M ~ Rgn9.WVeharvethefollowingruleofcomputation:Gf R ;gË=UR(f RN@ 0)(M Rgn9)UR=(M@ 0 Rgn9)(f RN@)sincefgË=UR(fN@20)(Mgn9)UR=(M@20gn9)(fN@). #-7 &ea>4. %TENSOR!PR9ODUCTS 13YProp`osition8.4.9.OThe35followingde necffovariantfunctorss21. #-'e N6:URMo`dÌ-"Rn!*[Ab&";2. #M - 3/:URRJ-LMo`d# !4AbF};3. #-'e - 3/:URMo`dÌ-"RRJ-LMo`d# !4AbF}.q(Proof.@_(f  gn9)(fG20Bg204. %TENSOR!PR9ODUCTS 15Y4.3@WVriteW(A;B)UR:A- BX4!_7B#3 AforW(A;B)UR:a- bUR7!b- a.3@Shorwthat0isanaturaltransformation(bSetrweenwhichfunctors?).8ShowthatM%H=#(A B) CH0(BE A) CQܟ{fd,cά-`Ftr(X&;Bd) 1HH(#BE (A Cܞ)\{fd,cά- 7` H^:Ǡ*Ffe^lǠ?]CTU HIv:Ǡ*FfeIlǠ?`N[1 r(A;C)=#A (BE Cܞ)0(BE Cܞ) AQܞ32fd,cά-z r(A;Bd C)(#BE (CF A)\32fd,cά-Í 7` 8鍹commrutesforallA;B;C12URK-Mo`d#CandthateW(B;A)(A;B)UR=id A BforallA,BinK-Mo`dX.5.8FindanexampleofM@,N62URK-Mo`dX-#CKsucrhthatM K cN6P԰= N K cM@.[ ;7  ,@ cmti12+O+msbm6*ppmsbm8) msbm10& msam10#u cmex10!K cmsy8 !", cmsy10;cmmi62cmmi8g cmmi12|{Ycmr8- cmcsc10o cmr9N cmbx12Nff cmbx12XQ cmr12O line10`