; TeX output 1994.06.23:1045giIC4;K"V 3 cmbx10ComplementtsandtheKrull-SchmidtTheoreminArbitraryCategories"jKK`y 3 cmr10BoMdofP!areigis Kj cmti9MathematischesN 3 cmmi10fα:ZA '!B,athe Ainduced(anduniquelydetermined)2,morphismfg inthecomm!utativefdiagramO,)kA*jmBHkAHBAnB332fd ά*g{pmAHHUB佛32fdά-*jpmBH0SV'efe'V?PxLٓRcmr7idH0ÕV'efeǟV?P WgidH0 şV'efe>V?H0WHHWHHWHHWHHWHHWHHWHH珞VfH珞VfjH0珞H夗HۤHѤHǤHHHWVfWVf%֟g$|0$-f-2,isfadi erencecok!ernel;,<(IMI)2,if,finthecomm!utativefdiagramE\AA2ClfdH`ά-йϯ_f9͵B˟ج@˟@y۟V @y۟V R9n g V V@@: 2,f"isfadi erencecok!ernelandg haskernel0theng isanisomorphism.*T{K±2L|{Ycmr8Bo!BmbMeanalgebramorphismKandJlet(a;1b) 2AζBbbMeJgiv!en.Byde nitionofg:ݱwehavegdjzA(a) =(a;1f-(a))2AζK´Bױand53gdjzB (b) =(0;1b)2AB."Pic!kbz1|s;1:::l;1bzm αsuchthat!d(bz1|s;1:::l;1f-(a);:::;bzm) =K´b,~thengd(!(jzB (bz1|s);1:::l;1jzA(a);:::;jzB (bzm))) =!d((0;1bz1|s);:::;(a;f-(a));:::;(0;bzm)) =K(!zA(0;1:::l;1a;:::;0);!zB (bz1|s;:::;f-(a);:::;bzm))J =(a;1b):f Th!usgʝissurjectivewhichKinianalgebraiccategorymeansthesameasgisadi erencecok!ernel([2],u3.4Cor.K4).KIf1thereisabinaryopMeration}suc!hthatr(a;1b)G=01ifandonlyifaG=b,cthenK·Calsod]satis es(IMI).Teopro!ved]thisletgԱ:GAB !C,ޱha!ved]kernelzerothengd(b)GA=K´gd(b09) )0=r(g(b);1g(b09))=g(r(b;1b09)))(b;1b09)=0)b=b0,8[henceشgkisinjectiv!e.KHo!wever,finanalgebraiccategoryabijectiv!emorphismisanisomorphism.KInparticularallalgebraiccategorieswheretheobjectsha!veanunderlyinggroupKstructure1andnofurtherdistinguishedelemen!tssatisfyourconditionsforC,Te.g.Krings(withoutunitelemen!ts),ĔassoMciativerings,Lieringsetc.ThebinaryopMeration04=aKb.Similarlythecategoryofloops("non- KassoMciativ!e"fgroups)satisfytheconditionsforC.KWeewillsho!wthatcertaingeneralizationsoftheFittingLemmaandtheKrull-KSc!hmidt:Theoremholdinourcategories.YInsteadofconsideringcongruencerela-Ktions asin[3]oramoMdularlatticeofsubobjectsasin[1]w!eusespeci csubobjectsKtogetherfwithcertainendomorphismstopro!vefthesetheorems.|H2[h>Complementts2andinternaldirectproYducts>GK±Let;C$±satisfytheaxiomsdiscussedinsection1.yWeearein!terestedinthequestion,KwhetherqinsideanobjectAGBinqC5therecanbMeother"subobjects"XniofAGBK±suc!h>NthatX)1Bw=ӴAB.Feor>NthiswehavetomakeprecisewhattheequalityKmeansandho!wAandBVaresubMobjectsofAaB. AUsubobjectwillbeusedKin.thesenseof[2],A_thatisasarepresen!tative.oftheusualequivdDalenceclassofKmonomorphisms.KIn thefollo!wingde nitionpzA H:AwYB\ ]q!A (resp.ȴpzB ı:AB\ ]q!B) denotesKthefcanonicalprojection.K½De nition22.1\~1.ixAqosubMobjectq|zX ˱: X,,Ϸ!AB iscalleda!': 3 cmti10wepakekPmBHH ⦴A>32fd򐍑ά-ȾpmA0M|rӱKe(f-)9*.rZgJHJHJHJHJHj̏Hj̏js9VT2@T2@B̏@B̏Rs֧2hAAAAAAAΟTAΟTU9>(VtfeqZV?9hVtfe7V?nmX9㲵fಟج@ಟ@TŸV @TŸV R}K´BݵemB"KK H`!FpzAzX )Kisanisomorphismifandonlyifthereisanepimorphism K´g withf(pzAzX$)go:= id *.Th!usf"canbMeconstructeduniquelysuchthatf8c= pzB zX$gd.K(1)f)(3):IfpzAzX @isanisomorphismthen,ob!viouslye,hisanisomorphism.K(3)f)(1):LethbMeanisomorphism.Inthecomm!utativefdiagramgi@5=vA5=UAnBFzfd ά-(SeemȂ%X̑%3]XJnBf{bfd?䍑ά"UpmXHvAHUAnBF32fd άqpmAHH0^ fe:?uapmXrhrZcmr51se QϵmAH0 WAnB^ fe:?4GmX^V fe:V?%c pmAHH0؟ feI ?.𩪵hr1H0؟V'efeI V?P𩪵hODK±w!e\lhavepzAJHCehzA!L=:id Z.Hence(A: !:X !A)=idqintheabMo!vediagram.ThusKitsucestosho!wthatpzAzX @isamonomorphism.SuppMosethatpzAzX$f=մpzAzXgd.KThenfw!ehavethecommutativediagram{}xʟiȬʟiʟi*͟*͟ `;R (;R (;R (ղlղl x&g`&P02fN"#yY9`H0%$'efeXV?!qǍ#e9Hf`H0υğ'efeϸ?ځӝeg`H0-צּ019x@19x@19x@^O@^OȒ%٪X̑%rXJnB{bfd?䍑ά`ZNpmX̑%̑%Bf{bfdά-2⍒)pr0BHAHNAnB%32fd ά*pmAHHBJ32fdά-<pmBH0V'efe֟V?w:SpmA>mXH0UtV'efe͈V?PFhH0 DV'efe@vV?Pid2K±De neګeLrůf andJ>ehůg (b!yůtheuniversalpropMertyoftheproMductXxB.ThenpzAhQ+eLrf ˱= pzAhJUehgK±andHpzB hQ+eLrf =u0=pzBhJUehg andHhencehQ+eLrf =uhJUehgandKseLrf=Jehug .Ysincehisanisomorphism.KSoff8c= g andpzAzX @isamonomorphism.`Offǟffffffffǎ%K½Remark22.5Hb1)Insomesense2.4,(3),mepansthatA!BnisgeneratedbytheKsubpobjects|XXandB.Observe,|however,that|therpeisalsoacanonicalmorphismK´X)C0` #BK H`! AзBwhich,Lin%generpal,does%notfactorthroughXȷдBinthecanonicalKway(e.g. v#F C 3 cmbxti10GrUX)norisitanepimorphism(e.g. vcpommutativemonoids).K2)In2.4, /(2),onecpanconsiderXasthegraphofthemorphismfg':9kA !B.KSothisppartofProposition2.4mayberephrasedas:KtherpeisabijectionbetweenthecomplementsofB>inABandthemorphismsK´f8c: A !B.KT)oshowthatepachfdeterminesasubobjectXnofAaB,mleteLrδf ': A !AaBrbeKthe5%morphismwithpzAԻeLrfԱ= idFandpzB 37eLr fP= f-.'Then(A;$\eLr1f]U)isasubpobject5%ofAB,Knamelythegrpaphoff-.04krs.tex-Date: TJune23,1994Time:10:45C͠T{K±6LBo;VtfeqmV?9{Vtfe7V?P(NUjlfdά-Ȳ;gp-B'X@<:32fd+ά-@f1B+A9WޟmAiHiHiHiH iHQ̏HQ̏js9Vw@w@4̏@4̏Rs{,Q0럺AAAAAAA TA TU9mVtfeݟV?9,pVtfe,ߟV?99 pK±TheyC'inducethecanonicalmorphismA1C0` +BK H`! P A!U1,Wwhic!hC'isadi erenceKcok!ernel.SincefKe(f-) =0fthediagramnA:P3Uܾҟlfdά-ȲZgLBX֞32fdά-@iBf0M{Ke (gd)9VH½H̽HֽH཮HΟ̏HΟ̏js9V@@È̏@È̏Rs8LVH:Pfek~V?9Vtfe1V?9̟VtfeV?͍\0 =Kel(f-)͎ϰ͎͎32fdά-ϰ[K±comm!utesandthecanonicalmorphismKe$(gd) 8!PVisthezeromorphism.ByKaxiom (IMI)gisanisomorphism.Th!uswecanreplaceP1`byUݩandgbyidxsothatKthereuisafactorizationU= ! B Lf}O I!IXmofU=! X,'xwhic!histhecokernelofK´A ! U1.KSince(A߷ !ߴU~ !X)=0andKe(f-)=0w!eget(A !U~ !B)=0Kandhenceafactorization(U !X h!!B)=(U !B).!SinceU !XisKthe5%cok!ernelofA !Ufıweget(Xԇ !B ȵf}O3 6H!" X)=id .Now5%fbhaskernelzeroKandXisadi erencecok!ernelof(B sƵf}O1 F! XH !lB;1B t9idk1 F!B). 'HenceXb!yaxiomK(IMI){w!e{obtainthatfE:nB- B!XWis{anisomorphism.^wThereforewemayreplaceK´Xb!yB>3andconsiderU !Bascok!ernelofA x!U1.XFeurthermorewehaveK(B2 JG! U>- B!B)=idb!ydiagram().IAnalogouslywegetamorphismU>- B! A,Kwhic!hfisacokernelofBK H`! U1,suchthat(A ! UHAN"YÒfd'FTά`N"N"[B˺dÒfd'eά-`>HAU{bfd'հ*Pά򎎍[B˞`{bfd'ά-H>HAHחAnB32fd άϰHH[B֏32fdά-ϰ`H0&J'efeY|?Dj=`H0'efeL?`H0'efe?Dj)=H0&JV'efeY|V?:=H0V'efeLV?PqhH0V'efeV?:)=`H09x@9x@9x@h4O@h4OR`H0 9x 9x 9x˺dO˺dO  K±Teo pro!vetheexistenceofYy *! AobservethatA ! UǩisthekernelofUHAU{bfd'հά-򎎍[B˞`{bfd'~\άH>HAHחAnB32fd άϰHH[B֏32fdά-ϰM,A1C0` +BH0Hh4;Ah4;A`H0H @ @ @˺d;A@˺d;AIH0H@H@H@h4U@h4URH0 H H H˺dU˺dU H0&JV'efeY|V?:=H0V'efeLV?PqhH0V'efeV?:)=`H0'efeL?K±whic!h1impliesbyaxiom(I)(thatA1C0` +Bϝ !BUt %!AsBTձis1adi erencecokernel.KByaxiom(IMI)lhisanisomorphismandtheabo!vediagrampro!vesthatthediagramKinfthetheoremcomm!utes.4ffǟffffffffǎ%3"Summable2morphismsK±In-thisparagraphw!eshallintroMduceanadditionofcertainmorphisms.OneoftheKaimsi istheproMofofaform!ulaid=JZeh zA n_pzAw+JehazB pzB K,forAaB.qLeti C beasinsectionsK1fand2.04krs.tex-Date: TJune23,1994Time:10:45 zT{Krull-ScÎhmidtXTheorem@19;K½De nition23.1SLet#:A1C0` +A p-!AAbMethecanonicalmorphismde nedb!y Kthefcomm!utativediagramJmxAEA1C0` +A{bfdp4ά-j1^A>R{bfdp4̞άMJj2HxAHdLAnA32fd8,Ԟά*sٵp1HH^AvZ32fd8,ά-*5p2H0`V'efeV?PidH0`V'efeV?Pd2idH0V'efeKŸV?H0"HH"HH"HH"HH"HH"HH"HHbVfHbVfjH0bHڱbHбbHƱbHbHbHbH"Vf"VfӍˬb$|[0$|0K±Byuaxiom(I)uJisadi erencecok!ernel.K(Letf;1gb2cC(A;B).f;andugarecalledKsummablefifthecanonicalmorphismhof+`䍒xA`䍒EA1C0` +AJfdp4ά-sȎ`䎎`䍒^A>RJfdp4̞άsȎPB ?f"M0H"M0HV%HV%jV feKŸV?Dˬbh%騪gbM0ڱbM0֝V%֝V%%dK±factorsޒ(necessarilyuniquely)throughd.AThefactorizationmorphismAFA ! BK±willfbMewrittenashf;1gdi.KAysimilaryde nitioncanbMegiv!enfornmorphisms.Thefamily(fziTLji =1;1:::l;1n)Kis9summableifforalli;1jوwith1÷i=UgS=h(fz1oб+ K´:1::+nfznq~):fThesecondpartfollo!wsfromthecommutativediagramsl41BJD1B;DMޟnjfd/*ά-StfeRxi1B1BDA.fd/*DάT rnAϷAnOVlfd0B䍑ά-NtfeRxi1ִAеlfd0B䍑Ͻά鍒 rns9̐tfeJ̐?؍|.ks9px̐tfeƣ̐?uJk+Brns9؟̐tfeM ̐?؍ k9fiw⍂Qw8,QwQwQ&VQ&Vs9UCpxVtfeƣV?9 KJhf1 ;::: ;fnli9]Vf1 +::: +fnf⍂f8,ff!NV!NV+TK±b!yfhfz1|s;1:::l;1fznq~ikX?nd= hfz1kX?;:::l;fznq~kifand(fz1_+n:::;+nfznq~)kb= fz1|sk++:::;+fznq~kX?."L8ffǟffffffffǎ%K½Lemma23.3EjSumsofsummablemorphismssatisfytheassopciativelaw. 'Inpartic-Kularhif(fz1|s;1fz2;fz3)pl2C(A;B)3 ,ishsummablethen(fz1z{+fz2|s)+fz3߱=plfz1+fz2+fz3߱=K´fz1_+n(fz2+fz3|s):⍑KProMof: WeeYpro!veonlythesecondstatement.The rstfollowsbystandardKreasoning.Considerfthefollo!wingcommutativediagram]U AUAnA*y^Òfd_ά-cڍζeȵr21UUgWA.*ÒfdSάcڍ5˲e6ĵr22̑%dAnȂ%OAnAAr{bfd#ά-/)weUr31;1_R^A``H'efeH?yd2`8du"6"ͬP@""PJ"9wPT"P^"!Ph"9vPr"P|" P"9uP"P"Pwn\Pwn\q`H0`'efe?Ս"id`H0%e&ݟr33R!9xH!9x>!9x4!9x*!9x !9x!9x UŸ UŸH0d޲hf1 ;f2im~כQm~FQm~FQm~Qm~FQUQUs`0sԪf1 +f22v9xHf8c= f"b!yfLemma3.2.Butthefactorizationh0;1id 2iispz2:AnA !AfsincePB̑%xȂ%dLAnA{bfd8,ά-'*e1̑%̑%^AvZ{bfd8,Ԟά'e 2H00A儍*o0= H@= H@= H@U@URH0V'efeKŸV?wˬbp2H0PpidZzHZzHZzHڟUڟU K±comm!utesfbyde nitionofJkehz1gandJkehz2 .Hence0n+id= h0;1id 2i=pz2|s=id *."3bffǟffffffffǎ%K½PropYosition23.5\LpetUpmBK±comm!utes, whencepzA<pzB 쳱= hJijehzA cpzA;Jeh1zB epzB iandid zUjY=(pzA<pzB )=JZehzA n_pzA+Jeh<zB ;pzB .KTeo|indicatetheinductionstepassumeV(=p9ABC8=U/Cȁ.aThen|id zV=JuehK´zUާpzU 0^+JBTeh}zCpzC iF=JtehlzUid5zU#qpzU+JBTeh}zCpzC iF=JtehlzUBQ(JijehUAA pUAA+JBTeh}UAB?մpUAB )pzU+JBTeh}zCpzC iF=JtehlzUJehBQUAA6pUAApzU+JuehK´zUJZehާUAB۴pUAB pzU !+J3ehnzC pzC ā=JZeh zA n_pzA|+J3ehnzB 1 pzB P+J3ehnzCpzCڱ.YVffǟffffffffǎ%K½Example23.6L[WeUwanttoprpoveinthecaseC.= GrV, thecategoryofgroups, thatKtwozmorphismsf;1gt::A !Barpezsummableifandonlyiff-(x)gd(y)=gd(y)f(x)KforalFlx;1y;k2شA. SnGivenhf;gdiwehavehf;gdi(x;y)=hf;gdi((x;e)ͷ(e;yd))=K·hf;1gdi(x;e);hf;1gdi(e;y)A=f-(x);g(y). \Sincpe](x;1e)and(e;yd)cpommutewegetK´f-(x)gd(y)Tf=gd(y)f(x). {hConverselyitisawelFl-knownexerpcisethatthiscondi-Ktionimpliesthathf;1gdiisahomomorphism.>Thesumfb+5=ga+isthende nepdbyK±(f+ngd)(x) =f-(x)g(x).Xi;Y(4NAJfdHhά^ffrn1,CoimFe(f-n:)Coim~(f-n:)]̲lfd#άj?mA>AJ32fdHhάyލffrn11B1BF>Aj~fdn荒_άJbfggfCoim(f-n1ۮ)Y2lfdDά$r~fr0ngg4CoimM~(f-n:) nlfdDάnŽF>Aj~32fdn荒_ά@Jbfs9CU̐tfeC̐?9CUVtfeCV?s9@̐tfer̐?9@VtferV?s9I@̐tfeJ.r̐?9I@VtfeJ.rV?s9VgHgHgHgHgHOŸ̏HOŸ̏j9"H "H"H"H)"H0BV H0BV jC̍K±whereBtheleftpartisan(n1)-foldBrepMetitionoftherigh!tpartandtherightpartKcomm!utesfbyLemma4.1andLemma4.2withgo:= f-n1ۮ.KFeorsuitablylargenw!ehavealsoCoimtt(f-n:) ! Aakernel.GDe ne'0:= (A !7K±Coimؑ(f-n:)#bݲ(fr0nln)rn ݍ /"7!,CoimE](f-n)). CSince(f-0Anq~zn)nisanisomorphism,hw!ehavethatƍK´'0: A !Coimv(f-n:)isacoimageoff-n Nasw!ellasof' =(AS'r0}O!Coim1Fg(f-n:)!A).KTh!us7*Coim(') !A7*isakernelandKeN(')=Ke^(A !Coim('))=Ke(A !04krs.tex-Date: TJune23,1994Time:10:45T{Krull-ScÎhmidtXTheoremƗ15;KCoimؑ(f-n:))=Ke(f-n).Sinceaf"isbMounded,'satis es(Gz')andLemma4.4holds Kforf'.Teranslatedbac!kintotermsoff"givestherequiredresult.N|ffǟffffffffǎ%K½Example24.7L[Consider;thecpategory;ofgrpoupsGrT.aiAmorphismf=B:G !GKisIcpalFlednormaliff-(aba1 t)l=af(b)a1jforIalFla;1bl2G.SIfIfisnormalandGKhas!a.c.c.andd.c.c.thenforalFlnwehavef-n:(aba1 t)<=af-n(b)a1݊and!thusK±Coimؑ(f-n:)Z=Imo(f-n)znormalinG.6&ThechainsofsubpgroupszKe(f-n)ZзKe(f-n+1=)KandۆIm(f-n:)MIm(f-n+1=)ۆbpecomestationary. KFinalFlyaM7!a(f-0Anq~zn)1 tf-n:(a1)ȷK±(f-0Anq~zn)1 tf-n:(a)isasepctionforthecanonicalmapKe4(f-n:)1C0` +Im,p(f-n) ! G.$ThusfKisbpoundedandtheFittingLemmaholds.C^o52The2Krull-Scthmidt-Theorem!KDe nition25.1SLetsA 6=0bMeinC.WeecallAindepcomposableifA =XYFbimpliesK´X柱= 0forYy=0.K½Lemma25.2EjLpet7Abeindecomposableandf8c: A !A7aboundedendomorphism.KThenfiseithernilppotentoranautomorphism.KProMof: By1theFittingLemmathereisann2Nбsuc!h1thatA=Ke(f-n:)K±Coimؑ(f-n:). c>If(1Coim(f-n)=0(1thenf-n ,۱=0andfUisnilpMoten!t.IfKeU(f-n:)=0Kthen}CoimL(f-n:)# }8!#A}m!ustbMetheidentitye.:#Hencef-n k]:#A }8!A}haskernelzeroKandbisadi erencecok!ernel.Byaxiom(IMI)b~forC5wegetthatf-n andalsofjareKautomorphisms.4ffǟffffffffǎ%K½De nition25.3SGiv!en )A[=B輷\Cȁ. 'Weede nethesubsetX7End(B)ofA-Kprpoductivefendomorphismsasfollo!ws:e L1.2,IffA =B0ɷnCȁ0= then(BK H`!A !B0e !A!B)2X. L2.2,Ifff;1go:2 X,thenf-g2 X. L3.2,Ifff8c2 XJ\nAut(B),thenf-1׷2X. L4.2,Ifff;1go:2 X^aresummablethenf+ng2 X. L5.2,X柱=fthesmallestsetsatisfying(1);1:::l;1(4).K½De nition25.4SAisbpoundedifforallA =Bȷ$CallA-proMductiv!eendomorphisms K´f"arefbMounded.K½Lemma25.5EjLpetʴBnnbeindecomposableandA=Bu!ѴCKbpebounded.Letf;1gF]beK´A-prpoductiveandsummable. vIffandgO}arpenilpotent,thensoisf+ngd.04krs.tex-Date: TJune23,1994Time:10:45T{K±16LBo(B)bpesummableandA-productive.TIffz1r+r:1::a+rfznT=id%`thenoneofthefziKisanautomorphism.KProMof: ,If[allfziarenilpoten!tthenasimpleinductionproofsho!wsthatfz1dG+K´:1::+fzn rѱisSnilpMoten!t.SooneofthefziUcannotbenilpoten!t,andhenceitmustbMeKanfautomorphism.)9ffǟffffffffǎK½Theorem25.7N=(Krull-Sc!hmidt)Lpetq5A =Az1_n:1::;nAzm B=Bz1n:1::;nBznKbpextwodecompositionsofAintointernaldirectproductsofindecomposablesubob- Kjepcts|9AziЅresp.dBzj6.LpetAbebounded.dThenm=nandAzif0lf0=ZBziЅforalFliandaKsuitablerpeorderingoftheBziTLs.KProMof: Weefpro!vethefollowingstatementbyinductionfort minJ(m;1n).K´PV(t):gBzt+1!:1::Bznq~).{Let->X柱= Az1:1::Azt1>gBzt+1!:1::BznK±as|subMobjectofA.asThenAztCisacomplemen!tforXXinAp2=BztFXsince|p0AtVzt:K´Azt9 !WA el!Bztis 7anisomorphism.OByPropMosition2.6w!egetAzt8"\̴XO=W0andK´Aztq[XT=xA.oFeurthermoreAztt!AandXT +!Aarek!ernels.oHenceAisanKin!ternalfdirectproMductA =AztBnX^offAzt+andX. ;ffǟffffffffǎ%References1.]P9aulTM.Cohn:pUniversalN 3 cmmi10Zcmr5ٓRcmr7K`y 3 cmr10