ThisisachapteroftheStacksProjectversionfac02ecdcompiledonSep1420211DERIVEDCATEGORIES239Admissiblesubcategories11240Postnikovsystems11441Essentiallyconstantsystems11942Otherchapters121References1221Int ID: 884125
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1 DERIVEDCATEGORIES05QIContents1.Introduct
DERIVEDCATEGORIES05QIContents1.Introduction22.Triangulatedcategories23.Thede nitionofatriangulatedcategory24.Elementaryresultsontriangulatedcategories55.Localizationoftriangulatedcategories136.Quotientsoftriangulatedcategories197.Adjointsforexactfunctors248.Thehomotopycategory269.Conesandtermwisesplitsequences2610.Distinguishedtrianglesinthehomotopycategory3311.Derivedcategories3712.Thecanonicaldelta-functor3913.Filteredderivedcategories4214.Derivedfunctorsingeneral4515.Derivedfunctorsonderivedcategories5216.Higherderivedfunctors5517.Triangulatedsubcategoriesofthederivedcategory5918.Injectiveresolutions6219.Projectiveresolutions6720.Rightderivedfunctorsandinjectiveresolutions6921.Cartan-Eilenbergresolutions7122.Compositionofrightderivedfunctors7223.Resolutionfunctors7324.Functorialinjectiveembeddingsandresolutionfunctors7525.Rightderivedfunctorsviaresolutionfunctors7726.Filteredderivedcategoryandinjectiveresolutions7727.Extgroups8528.K-groups8929.Unboundedcomplexes9030.Derivingadjoints9331.K-injectivecomplexes9432.Boundedcohomologicaldimension9733.Derivedcolimits9934.Derivedlimits10335.Operationsonfullsubcategories10436.Generatorsoftriangulatedcategories10637.Compactobjects10838.Brownrepresentability110 ThisisachapteroftheStacksProject,versionfac02ecd,compiledonSep14,2021.1 DERIVEDCATEGORIES239.Admissiblesubcategories11240.Postnikovsystems11441.Essentiallyconstantsystems11942.Otherchapters121References1221.Introduction05QJWe rstdiscusstriangulatedcategoriesandlocalizationintriangulatedcategories.Next,weprovethatthehomotopycategoryofcomplexesinanadditivecategoryisatriangulatedcategory.Oncethisisdonewede nethederivedcategoryofanabeliancategoryasthelocalizationofthehomotopycategorywithrespecttoquasi-isomorphisms.AgoodreferenceisVerdier'sthesis[Ver96].2.Triangulatedcategories0143Triangulatedcategoriesareaconvenienttooltodescribethetypeofstructureinherentinthederivedcategoryofanabeliancategory.Somereferencesare[Ver96],[KS06],and[Nee01].3.Thede nitionofatriangulatedcategory05QKInthissectionwecollectmostofthede nitionsconcerningtriangul
2 atedandpre-triangulatedcategories.De
atedandpre-triangulatedcategories.De nition3.1.0144LetDbeanadditivecategory.Let[n]:D!D,E7!E[n]beacollectionofadditivefunctorsindexedbyn2Zsuchthat[n][m]=[n+m]and[0]=id(equalityasfunctors).Inthissituationwede neatriangletobeasextuple(X;Y;Z;f;g;h)whereX;Y;Z2Ob(D)andf:X!Y,g:Y!Zandh:Z!X[1]aremorphismsofD.Amorphismoftriangles(X;Y;Z;f;g;h)!(X0;Y0;Z0;f0;g0;h0)isgivenbymorphismsa:X!X0,b:Y!Y0andc:Z!Z0ofDsuchthatbf=f0a,cg=g0banda[1]h=h0c.AmorphismoftrianglesisvisualizedbythefollowingcommutativediagramX// a Y// b Z// c X[1]a[1] X0// Y0// Z0// X0[1]Hereisthede nitionofatriangulatedcategoryasgiveninVerdier'sthesis.De nition3.2.0145Atriangulatedcategoryconsistsofatriple(D;f[n]gn2Z;T)where(1)Disanadditivecategory,(2)[n]:D!D,E7!E[n]isacollectionofadditivefunctorsindexedbyn2Zsuchthat[n][m]=[n+m]and[0]=id(equalityasfunctors),and(3)Tisasetoftrianglescalledthedistinguishedtrianglessubjecttothefollowingconditions DERIVEDCATEGORIES3TR1Anytriangleisomorphictoadistinguishedtriangleisadistinguishedtri-angle.Anytriangleoftheform(X;X;0;id;0;0)isdistinguished.Foranymorphismf:X!YofDthereexistsadistinguishedtriangleoftheform(X;Y;Z;f;g;h).TR2Thetriangle(X;Y;Z;f;g;h)isdistinguishedifandonlyifthetriangle(Y;Z;X[1];g;h;�f[1])is.TR3GivenasoliddiagramXf// a Yg// b Zh// X[1]a[1] X0f0// Y0g0// Z0h0// X0[1]whoserowsaredistinguishedtrianglesandwhichsatis esbf=f0a,thereexistsamorphismc:Z!Z0suchthat(a;b;c)isamorphismoftriangles.TR4GivenobjectsX,Y,ZofD,andmorphismsf:X!Y,g:Y!Z,anddistinguishedtriangles(X;Y;Q1;f;p1;d1),(X;Z;Q2;gf;p2;d2),and(Y;Z;Q3;g;p3;d3),thereexistmorphismsa:Q1!Q2andb:Q2!Q3suchthat(a)(Q1;Q2;Q3;a;b;p1[1]d3)isadistinguishedtriangle,(b)thetriple(idX;g;a)isamorphismoftriangles(X;Y;Q1;f;p1;d1)!(X;Z;Q2;gf;p2;d2),and(c)thetriple(f;idZ;b)isamorphismoftriangles(X;Z;Q2;gf;p2;d2)!(Y;Z;Q3;g;p3;d3).Wewillcall(D;[];T)apre-triangulatedcategoryifTR1,TR2andTR3hold.1TheexplanationofTR4isthatifyouthinkofQ1asY=X,Q2asZ=XandQ3a
3 sZ=Y,thenTR4(a)expressestheisomorphism(Z
sZ=Y,thenTR4(a)expressestheisomorphism(Z=X)=(Y=X)=Z=YandTR4(b)andTR4(c)expressthatwecancomparethetrianglesX!Y!Q1!X[1]etcwithmorphismsoftriangles.ForamoreprecisereformulationofthisideaseetheproofofLemma10.2.ThesigninTR2meansthatif(X;Y;Z;f;g;h)isadistinguishedtriangletheninthelongsequence(3.2.1)05QL:::!Z[�1]�h[�1]����!Xf�!Yg�!Zh�!X[1]�f[1]���!Y[1]�g[1]���!Z[1]!:::eachfourtermsequencegivesadistinguishedtriangle.Asusualweabusenotationandwesimplyspeakofa(pre-)triangulatedcategoryDwithoutexplicitlyintroducingnotationfortheadditionaldata.Thenotionofapre-triangulatedcategoryisusefulin ndingstatementsequivalenttoTR4.Wehavethefollowingde nitionofatriangulatedfunctor.De nition3.3.014VLetD,D0bepre-triangulatedcategories.Anexactfunctor,oratriangulatedfunctorfromDtoD0isafunctorF:D!D0togetherwithgivenfunc-torialisomorphismsX:F(X[1])!F(X)[1]suchthatforeverydistinguishedtri-angle(X;Y;Z;f;g;h)ofDthetriangle(F(X);F(Y);F(Z);F(f);F(g);XF(h))isadistinguishedtriangleofD0. 1Weuse[]asanabbreviationforthefamilyf[n]gn2Z. DERIVEDCATEGORIES4Anexactfunctorisadditive,seeLemma4.17.Whenwesaytwotriangulatedcategoriesareequivalentwemeanthattheyareequivalentinthe2-categoryoftriangulatedcategories.A2-morphisma:(F;)!(F0;0)inthis2-categoryissimplyatransformationoffunctorsa:F!F0whichiscompatiblewithand0,i.e.,F[1]// a?1 [1]F1?a F0[1]0// [1]F0commutes.De nition3.4.05QMLet(D;[];T)beapre-triangulatedcategory.Apre-triangulatedsubcategory2isapair(D0;T0)suchthat(1)D0isanadditivesubcategoryofDwhichispreservedunder[1]and[�1],(2)T0Tisasubsetsuchthatforevery(X;Y;Z;f;g;h)2T0wehaveX;Y;Z2Ob(D0)andf;g;h2Arrows(D0),and(3)(D0;[];T0)isapre-triangulatedcategory.IfDisatriangulatedcategory,thenwesay(D0;T0)isatriangulatedsubcategoryifitisapre-triangulatedsubcategoryand(D0;[];T0)isatriangulatedcategory.InthissituationtheinclusionfunctorD0!DisanexactfunctorwithX:X[1]!X[1]givenbytheidentityonX[1].WewillseeinLemma4.1thatforadistinguishedtriangl
4 e(X;Y;Z;f;g;h)inapre-triangulatedcategor
e(X;Y;Z;f;g;h)inapre-triangulatedcategorythecompositiongf:X!Ziszero.Thusthesequence(3.2.1)isacomplex.Ahomologicalfunctorisonethatturnsthiscomplexintoalongexactsequence.De nition3.5.0147LetDbeapre-triangulatedcategory.LetAbeanabeliancate-gory.AnadditivefunctorH:D!Aiscalledhomologicalifforeverydistinguishedtriangle(X;Y;Z;f;g;h)thesequenceH(X)!H(Y)!H(Z)isexactintheabeliancategoryA.AnadditivefunctorH:Dopp!AiscalledcohomologicalifthecorrespondingfunctorD!Aoppishomological.IfH:D!AisahomologicalfunctorweoftenwriteHn(X)=H(X[n])sothatH(X)=H0(X).OurdiscussionofTR2aboveimpliesthatadistinguishedtriangle(X;Y;Z;f;g;h)determinesalongexactsequence(3.5.1)0148H�1(Z)H(h[�1])// H0(X)H(f)// H0(Y)H(g)// H0(Z)H(h)// H1(X)Thiswillbecalledthelongexactsequenceassociatedtothedistinguishedtriangleandthehomologicalfunctor.Asindicatedwewillnotuseanysignsforthemor-phismsinthelongexactsequence.Thishasthesideeectthatmapsinthelongexactsequenceassociatedtotherotation(TR2)ofadistinguishedtriangledierfromthemapsinthesequenceabovebysomesigns. 2Thisde nitionmaybenonstandard.IfD0isafullsubcategorythenT0istheintersectionofthesetoftrianglesinD0withT,seeLemma4.16.InthiscasewedropT0fromthenotation. DERIVEDCATEGORIES5De nition3.6.0150LetAbeanabeliancategory.LetDbeatriangulatedcategory.A-functorfromAtoDisgivenbyafunctorG:A!Dandarulewhichassignstoeveryshortexactsequence0!Aa�!Bb�!C!0amorphism=A!B!C:G(C)!G(A)[1]suchthat(1)thetriangle(G(A);G(B);G(C);G(a);G(b);A!B!C)isadistinguishedtri-angleofDforanyshortexactsequenceasabove,and(2)foreverymorphism(A!B!C)!(A0!B0!C0)ofshortexactsequencesthediagramG(C) A!B!C// G(A)[1] G(C0)A0!B0!C0// G(A0)[1]iscommutative.Inthissituationwecall(G(A);G(B);G(C);G(a);G(b);A!B!C)theimageoftheshortexactsequenceunderthegiven-functor.Notehowa-functorcomesequippedwithadditionalstructure.StrictlyspeakingitdoesnotmakesensetosaythatagivenfunctorA!Disa-functor,butwewilloftendosoanyway.4.Elementaryresultsontriangulatedcategories05QNMostoftheresultsinthissectionareprovedforpre
5 -triangulatedcategoriesandafortioriholdi
-triangulatedcategoriesandafortioriholdinanytriangulatedcategory.Lemma4.1.0146LetDbeapre-triangulatedcategory.Let(X;Y;Z;f;g;h)beadis-tinguishedtriangle.Thengf=0,hg=0andf[1]h=0.Proof.ByTR1weknow(X;X;0;1;0;0)isadistinguishedtriangle.ApplyTR3toX// 1 X// f 0// X[1]1[1] Xf// Yg// Zh// X[1]Ofcoursethedottedarrowisthezeromap.Hencethecommutativityofthediagramimpliesthatgf=0.Fortheothercasesrotatethetriangle,i.e.,applyTR2.Lemma4.2.0149LetDbeapre-triangulatedcategory.ForanyobjectWofDthefunctorHomD(W;�)ishomological,andthefunctorHomD(�;W)iscohomologi-cal.Proof.Consideradistinguishedtriangle(X;Y;Z;f;g;h).Wehavealreadyseenthatgf=0,seeLemma4.1.Supposea:W!Yisamorphismsuchthat DERIVEDCATEGORIES6ga=0.ThenwegetacommutativediagramW1// b W// a 0// 0 W[1]b[1] X// Y// Z// X[1]Bothrowsaredistinguishedtriangles(useTR1forthetoprow).Hencewecan llthedottedarrowb( rstrotateusingTR2,thenapplyTR3,andthenrotateback).Thisprovesthelemma.Lemma4.3.014ALetDbeapre-triangulatedcategory.Let(a;b;c):(X;Y;Z;f;g;h)!(X0;Y0;Z0;f0;g0;h0)beamorphismofdistinguishedtriangles.Iftwoamonga;b;careisomorphismssoisthethird.Proof.Assumethataandcareisomorphisms.ForanyobjectWofDwriteHW(�)=HomD(W;�).ThenwegetacommutativediagramofabeliangroupsHW(Z[�1])// HW(X)// HW(Y)// HW(Z)// HW(X[1]) HW(Z0[�1])// HW(X0)// HW(Y0)// HW(Z0)// HW(X0[1])Byassumptiontherighttwoandlefttwoverticalarrowsarebijective.AsHWishomologicalbyLemma4.2andthe velemma(Homology,Lemma5.20)itfollowsthatthemiddleverticalarrowisanisomorphism.HencebyYoneda'slemma,seeCategories,Lemma3.5weseethatbisanisomorphism.Thisimpliestheothercasesbyrotating(usingTR2).Remark4.4.09WALetDbeanadditivecategorywithtranslationfunctors[n]asinDe nition3.1.Letuscallatriangle(X;Y;Z;f;g;h)special3ifforeveryobjectWofDthelongsequenceofabeliangroups:::!HomD(W;X)!HomD(W;Y)!HomD(W;Z)!HomD(W;X[1])!:::isexact.TheproofofLemma4.3showsthatif(a;b;c):(X;Y;Z;f;g;h)!(X0;Y0;Z0;f0
6 ;g0;h0)isamorphismofspecialtrianglesandi
;g0;h0)isamorphismofspecialtrianglesandiftwoamonga;b;careisomorphismssoisthethird.Thereisadualstatementforco-specialtriangles,i.e.,triangleswhichturnintolongexactsequencesonapplyingthefunctorHomD(�;W).Thusdistinguishedtrianglesarespecialandco-special,butingeneraltherearemanymore(co-)specialtriangles,thantherearedistinguishedtriangles.Lemma4.5.05QPLetDbeapre-triangulatedcategory.Let(0;b;0);(0;b0;0):(X;Y;Z;f;g;h)!(X;Y;Z;f;g;h)beendomorphismsofadistinguishedtriangle.Thenbb0=0. 3Thisisnonstandardnotation. DERIVEDCATEGORIES7Proof.PictureX// 0 Y// b;b0 �� Z// 0 �� X[1]0 X// Y// Z// X[1]ApplyingLemma4.2we nddottedarrows andsuchthatb0=f andb=g.Thenbb0=gf =0asgf=0byLemma4.1.Lemma4.6.05QQLetDbeapre-triangulatedcategory.Let(X;Y;Z;f;g;h)beadis-tinguishedtriangle.IfZh// c X[1]a[1] Zh// X[1]iscommutativeanda2=a,c2=c,thenthereexistsamorphismb:Y!Ywithb2=bsuchthat(a;b;c)isanendomorphismofthetriangle(X;Y;Z;f;g;h).Proof.ByTR3thereexistsamorphismb0suchthat(a;b0;c)isanendomorphismof(X;Y;Z;f;g;h).Then(0;(b0)2�b0;0)isalsoanendomorphism.ByLemma4.5weseethat(b0)2�b0hassquarezero.Setb=b0�(2b0�1)((b0)2�b0)=3(b0)2�2(b0)3.Acomputationshowsthat(a;b;c)isanendomorphismandthatb2�b=(4(b0)2�4b0�3)((b0)2�b0)2=0.Lemma4.7.014BLetDbeapre-triangulatedcategory.Letf:X!Ybeamor-phismofD.Thereexistsadistinguishedtriangle(X;Y;Z;f;g;h)whichisuniqueupto(nonunique)isomorphismoftriangles.Moreprecisely,givenasecondsuchdistinguishedtriangle(X;Y;Z0;f;g0;h0)thereexistsanisomorphism(1;1;c):(X;Y;Z;f;g;h)�!(X;Y;Z0;f;g0;h0)Proof.ExistencebyTR1.UniquenessuptoisomorphismbyTR3andLemma4.3.Lemma4.8.0FWZLetDbeapre-triangulatedcategory.Let(a;b;c):(X;Y;Z;f;g;h)!(X0;Y0;Z0;f0;g0;h0)beamorphismofdistinguishedtriangles.Ifoneofthefollowingconditionsholds(1)Hom(Y;X0)=0,(2)Hom(Z;Y0)=0,(3)Hom(X;X0)=Hom(Z;X0)=0,(4)Hom(Z;X0)=Hom(Z;Z0)=0,or(5)Hom(X[1];Z0)=Hom(Z;X0)=0thenbistheuniquemorphismfromY!Y0suchthat(a;b;c)isamorphismoftriangles.Pro
7 of.Ifwehaveasecondmorphismoftriangles(a;
of.Ifwehaveasecondmorphismoftriangles(a;b0;c)then(0;b�b0;0)isamorphismoftriangles.Hencewehavetoshow:theonlymorphismb:Y!Y0suchthatX!Y!Y0andY!Y0!Z0arezerois0.WewilluseLemma4.2withoutfurthermention.Inparticular,condition(3)implies(1).Givencondition(1)ifthecompositiong0b:Y!Y0!Z0iszero,thenbliftstoamorphismY!X0whichhastobezero.Thisproves(1). DERIVEDCATEGORIES8Theproofof(2)and(4)aredualtothisargument.Assume(5).ConsiderthediagramXf// 0 Yg// b Zh// 0 X[1]0 X0f0// Y0g0// Z0h0// X0[1]Wemaychoosesuchthatb=g.Theng0g=0whichimpliesthatg0=hforsome2Hom(X[1];Z0).SinceHom(X[1];Z0)=0weconcludethatg0=0.Hence=f0 forsome 2Hom(Z;X0).SinceHom(Z;X0)=0weconcludethat=0andhenceb=0asdesired.Lemma4.9.05QRLetDbeapre-triangulatedcategory.Letf:X!YbeamorphismofD.Thefollowingareequivalent(1)fisanisomorphism,(2)(X;Y;0;f;0;0)isadistinguishedtriangle,and(3)foranydistinguishedtriangle(X;Y;Z;f;g;h)wehaveZ=0.Proof.ByTR1thetriangle(X;X;0;1;0;0)isdistinguished.Let(X;Y;Z;f;g;h)beadistinguishedtriangle.ByTR3thereisamapofdistinguishedtriangles(1;f;0):(X;X;0)!(X;Y;Z).Iffisanisomorphism,then(1;f;0)isanisomor-phismoftrianglesbyLemma4.3andZ=0.Conversely,ifZ=0,then(1;f;0)isanisomorphismoftrianglesaswell,hencefisanisomorphism.Lemma4.10.05QSLetDbeapre-triangulatedcategory.Let(X;Y;Z;f;g;h)and(X0;Y0;Z0;f0;g0;h0)betriangles.Thefollowingareequivalent(1)(XX0;YY0;ZZ0;ff0;gg0;hh0)isadistinguishedtriangle,(2)both(X;Y;Z;f;g;h)and(X0;Y0;Z0;f0;g0;h0)aredistinguishedtriangles.Proof.Assume(2).ByTR1wemaychooseadistinguishedtriangle(XX0;YY0;Q;ff0;g00;h00).ByTR3wecan ndmorphismsofdistinguishedtriangles(X;Y;Z;f;g;h)!(XX0;YY0;Q;ff0;g00;h00)and(X0;Y0;Z0;f0;g0;h0)!(XX0;YY0;Q;ff0;g00;h00).Takingthedirectsumofthesemorphismsweobtainamorphismoftriangles(XX0;YY0;ZZ0;ff0;gg0;hh0)(1;1;c) (XX0;YY0;Q;ff0;g00;h00):IntheterminologyofRemark4.4thisisamapofspecialtriangles(becauseadire
8 ctsumofspecialtrianglesisspecial)andweco
ctsumofspecialtrianglesisspecial)andweconcludethatcisanisomorphism.Thus(1)holds.Assume(1).Wewillshowthat(X;Y;Z;f;g;h)isadistinguishedtriangle.Firstobservethat(X;Y;Z;f;g;h)isaspecialtriangle(terminologyfromRemark4.4)asadirectsummandofthedistinguishedhencespecialtriangle(XX0;YY0;ZZ0;ff0;gg0;hh0).UsingTR1let(X;Y;Q;f;g00;h00)beadistinguishedtriangle.ByTR3thereexistsamorphismofdistinguishedtriangles(XX0;Y DERIVEDCATEGORIES9Y0;ZZ0;ff0;gg0;hh0)!(X;Y;Q;f;g00;h00).Composingthiswiththeinclusionmapwegetamorphismoftriangles(1;1;c):(X;Y;Z;f;g;h)�!(X;Y;Q;f;g00;h00)ByRemark4.4we ndthatcisanisomorphismandweconcludethat(2)holds.Lemma4.11.05QTLetDbeapre-triangulatedcategory.Let(X;Y;Z;f;g;h)beadistinguishedtriangle.(1)Ifh=0,thenthereexistsarightinverses:Z!Ytog.(2)Foranyrightinverses:Z!Yofgthemapfs:XZ!Yisanisomorphism.(3)ForanyobjectsX0;Z0ofDthetriangle(X0;X0Z0;Z0;(1;0);(0;1);0)isdistinguished.Proof.Tosee(1)usethatHomD(Z;Y)!HomD(Z;Z)!HomD(Z;X[1])isexactbyLemma4.2.Bythesametoken,ifsisasin(2),thenh=0andthesequence0!HomD(W;X)!HomD(W;Y)!HomD(W;Z)!0issplitexact(splitbys:Z!Y).HencebyYoneda'slemmaweseethatXZ!Yisanisomorphism.ThelastassertionfollowsfromTR1andLemma4.10.Lemma4.12.05QULetDbeapre-triangulatedcategory.Letf:X!YbeamorphismofD.Thefollowingareequivalent(1)fhasakernel,(2)fhasacokernel,(3)fistheisomorphictoacompositionKZ!Z!ZQofaprojectionandcoprojectionforsomeobjectsK;Z;QofD.Proof.AnymorphismisomorphictoamapoftheformX0Z!ZY0hasbothakernelandacokernel.Hence(3))(1),(2).Nextweprove(1))(3).Suppose rstthatf:X!Yisamonomorphism,i.e.,itskerneliszero.ByTR1thereexistsadistinguishedtriangle(X;Y;Z;f;g;h).ByLemma4.1thecompositionfh[�1]=0.Asfisamonomorphismweseethath[�1]=0andhenceh=0.ThenLemma4.11impliesthatY=XZ,i.e.,weseethat(3)holds.Next,assumefhasakernelK.AsK!XisamonomorphismweconcludeX=KX0andfjX0:X0!Yisamonomorphism.HenceY=X0Y0andwewin.Theimplication(2))(3)isdualtothis.Lemma4.13.0CRGLetDbeapre-triangulatedcategory.LetIbeaset.(1)LetXi,i2IbeafamilyofobjectsofD
9 .(a)IfQXiexists,then(QXi)[1]=QXi[1].(b)I![.(a)IfQXiexists,then(QXi)[1]=QXi[1].(b)I](884125/a-ifqxiexists-then-qxi-1-qxi-1-b-iflxiex.jpg ".(a)IfQXiexists,then(QXi)[1]=QXi[1].(b)I")
.(a)IfQXiexists,then(QXi)[1]=QXi[1].(b)IfLXiexists,then(LXi)[1]=LXi[1].(2)LetXi!Yi!Zi!Xi[1]beafamilyofdistinguishedtrianglesofD.(a)IfQXi,QYi,QZiexist,thenQXi!QYi!QZi!QXi[1]isadistinguishedtriangle.(b)IfLXi,LYi,LZiexist,thenLXi!LYi!LZi!LXi[1]isadistinguishedtriangle.Proof.Part(1)istruebecause[1]isanautoequivalenceofDandbecausedirectsumsandproductsarede nedintermsofthecategorystructure.Letusprove(2)(a).ChooseadistinguishedtriangleQXi!QYi!Z!QXi[1].ForeachjwecanuseTR3tochooseamorphismpj:Z!Zj ttingintoamorphismof DERIVEDCATEGORIES10distinguishedtriangleswiththeprojectionmapsQXi!XjandQYi!Yj.Usingthede nitionofproductsweobtainamapQpi:Z!QZi ttingintoamorphismoftrianglesfromthedistinguishedtriangletothetrianglemadeoutoftheproducts.ObservethattheproducttriangleQXi!QYi!QZi!QXi[1]isspecialintheterminologyofRemark4.4becauseproductsofexactsequencesofabeliangroupsareexact.HenceRemark4.4showsthatthemorphismoftrianglesisanisomorphismandweconcludebyTR1.Theproofof(2)(b)isdual.Lemma4.14.05QWLetDbeapre-triangulatedcategory.IfDhascountableproducts,thenDisKaroubian.IfDhascountablecoproducts,thenDisKaroubian.Proof.AssumeDhascountableproducts.ByHomology,Lemma4.3itsu cestocheckthatmorphismswhichhavearightinversehavekernels.Anymorphismwhichhasarightinverseisanepimorphism,hencehasakernelbyLemma4.12.Thesecondstatementisdualtothe rst.Thefollowinglemmamakesitslightlyeasiertoprovethatapre-triangulatedcate-goryistriangulated.Lemma4.15.014CLetDbeapre-triangulatedcategory.InordertoproveTR4itsu cestoshowthatgivenanypairofcomposablemorphismsf:X!Yandg:Y!Zthereexist(1)isomorphismsi:X0!X,j:Y0!Yandk:Z0!Z,andthensettingf0=j�1fi:X0!Y0andg0=k�1gj:Y0!Z0thereexist(2)distinguishedtriangles(X0;Y0;Q1;f0;p1;d1),(X0;Z0;Q2;g0f0;p2;d2)and(Y0;Z0;Q3;g0;p3;d3),suchthattheassertionofTR4holds.Proof.ThereplacementofX;Y;ZbyX0;Y0;Z0isharmlessbyourde nitionofdistinguishedtrianglesandtheirisomorphisms.Thelemmafollowsfromthefactthatthedistinguishedtriangles(X0;Y0;Q1;f0;p1;d1),(X0;Z0;Q2;g0f0;p2;d2)and(Y0;Z0;Q3;g0;p3;d3)areuniqueuptoisomorphismbyLem
10 ma4.7.Lemma4.16.05QXLetDbeapre-trian
ma4.7.Lemma4.16.05QXLetDbeapre-triangulatedcategory.AssumethatD0isanadditivefullsubcategoryofD.Thefollowingareequivalent(1)thereexistsasetoftrianglesT0suchthat(D0;T0)isapre-triangulatedsubcategoryofD,(2)D0ispreservedunder[1];[�1]andgivenanymorphismf:X!YinD0thereexistsadistinguishedtriangle(X;Y;Z;f;g;h)inDsuchthatZisisomorphictoanobjectofD0.InthiscaseT0asin(1)isthesetofdistinguishedtriangles(X;Y;Z;f;g;h)ofDsuchthatX;Y;Z2Ob(D0).Finally,ifDisatriangulatedcategory,then(1)and(2)arealsoequivalentto(3)D0isatriangulatedsubcategory.Proof.Omitted.Lemma4.17.05QYAnexactfunctorofpre-triangulatedcategoriesisadditive.Proof.LetF:D!D0beanexactfunctorofpre-triangulatedcategories.Since(0;0;0;10;10;0)isadistinguishedtriangleofDthetriangle(F(0);F(0);F(0);1F(0);1F(0);F(0)) DERIVEDCATEGORIES11isdistinguishedinD0.Thisimpliesthat1F(0)1F(0)iszero,seeLemma4.1.HenceF(0)isthezeroobjectofD0.ThisalsoimpliesthatFappliedtoanyzeromorphismiszero(sinceamorphisminanadditivecategoryiszeroifandonlyifitfactorsthroughthezeroobject).Next,usingthat(X;XY;Y;(1;0);(0;1);0)isadistinguishedtriangle,weseethat(F(X);F(XY);F(Y);F(1;0);F(0;1);0)isonetoo.ThisimpliesthatthemapF(1;0)F(0;1):F(X)F(Y)!F(XY)isanisomorphism,seeLemma4.11.Weomittherestoftheargument.Lemma4.18.05SQLetF:D!D0beafullyfaithfulexactfunctorofpre-triangulatedcategories.Thenatriangle(X;Y;Z;f;g;h)ofDisdistinguishedifandonlyif(F(X);F(Y);F(Z);F(f);F(g);F(h))isdistinguishedinD0.Proof.Theonlyifpartisclear.Assume(F(X);F(Y);F(Z))isdistinguishedinD0.Pickadistinguishedtriangle(X;Y;Z0;f;g0;h0)inD.ByLemma4.7thereexistsanisomorphismoftriangles(1;1;c0):(F(X);F(Y);F(Z))�!(F(X);F(Y);F(Z0)):SinceFisfullyfaithful,thereexistsamorphismc:Z!Z0suchthatF(c)=c0.Then(1;1;c)isanisomorphismbetween(X;Y;Z)and(X;Y;Z0).Hence(X;Y;Z)isdistinguishedbyTR1.Lemma4.19.014YLetD;D0;D00bepre-triangulatedcategories.LetF:D!D0andF0:D0!D00beexactfunctors.ThenF0Fisanexactfunctor.Proof.Omitted.Lemma4.20.05QZLetDbeapre-triangulatedcategory.LetAbeanabeliancategory.LetH:D!Abeahomologicalfunctor.(1)LetD0beapre-triangulated
11 category.LetF:D0!Dbeanexactfunctor.Thent
category.LetF:D0!Dbeanexactfunctor.ThenthecompositionHFisahomologicalfunctoraswell.(2)LetA0beanabeliancategory.LetG:A!A0beanexactfunctor.ThenGHisahomologicalfunctoraswell.Proof.Omitted.Lemma4.21.0151LetDbeatriangulatedcategory.LetAbeanabeliancategory.LetG:A!Dbea-functor.(1)LetD0beatriangulatedcategory.LetF:D!D0beanexactfunctor.ThenthecompositionFGisa-functoraswell.(2)LetA0beanabeliancategory.LetH:A0!Abeanexactfunctor.ThenGHisa-functoraswell.Proof.Omitted.Lemma4.22.05SRLetDbeatriangulatedcategory.LetAandBbeabeliancategories.LetG:A!Dbea-functor.LetH:D!Bbeahomologicalfunctor.AssumethatH�1(G(A))=0forallAinA.ThenthecollectionfHnG;Hn(A!B!C)gn0isa-functorfromA!B,seeHomology,De nition12.1. DERIVEDCATEGORIES12Proof.Thenotationsigni esthefollowing.If0!Aa�!Bb�!C!0isashortexactsequenceinA,then=A!B!C:G(C)!G(A)[1]isamorphisminDsuchthat(G(A);G(B);G(C);a;b;)isadistinguishedtriangle,seeDe nition3.6.ThenHn():Hn(G(C))!Hn(G(A)[1])=Hn+1(G(A))isclearlyfunctorialintheshortexactsequence.Finally,thelongexactcohomologysequence(3.5.1)combinedwiththevanishingofH�1(G(C))givesalongexactsequence0!H0(G(A))!H0(G(B))!H0(G(C))H0()����!H1(G(A))!:::inBasdesired.TheproofofthefollowingresultusesTR4.Proposition4.23.05R0LetDbeatriangulatedcategory.AnycommutativediagramX// Y X0// Y0canbeextendedtoadiagramX// Y// Z// X[1] X0// Y0// Z0// X0[1] X00// Y00// Z00// X00[1] X[1]// Y[1]// Z[1]// X[2]whereallthesquaresarecommutative,exceptforthelowerrightsquarewhichisanticommutative.Moreover,eachoftherowsandcolumnsaredistinguishedtrian-gles.Finally,themorphismsonthebottomrow(resp.rightcolumn)areobtainedfromthemorphismsofthetoprow(resp.leftcolumn)byapplying[1].Proof.Duringthisproofweavoidwritingthearrowsinordertomaketheproofleg-ible.Choosedistinguishedtriangles(X;Y;Z),(X0;Y0;Z0),(X;X0;X00),(Y;Y0;Y00),and(X;Y0;A).Notethatthe
12 morphismX!Y0isbothequaltothecompositionX
morphismX!Y0isbothequaltothecompositionX!Y!Y0andequaltothecompositionX!X0!Y0.Hence,wecan ndmorphisms(1)a:Z!Aandb:A!Y00,and(2)a0:X00!Aandb0:A!Z0asinTR4.Denotec:Y00!Z[1]thecompositionY00!Y[1]!Z[1]anddenotec0:Z0!X00[1]thecompositionZ0!X0[1]!X00[1].TheconclusionofourapplicationTR4arethat(1)(Z;A;Y00;a;b;c),(X00;A;Z0;a0;b0;c0)aredistinguishedtriangles, DERIVEDCATEGORIES13(2)(X;Y;Z)!(X;Y0;A),(X;Y0;A)!(Y;Y0;Y00),(X;X0;X00)!(X;Y0;A),(X;Y0;A)!(X0;Y0;Z0)aremorphismsoftriangles.Firstusingthat(X;X0;X00)!(X;Y0;A)and(X;Y0;A)!(Y;Y0;Y00).aremor-phismsoftrianglesweseethe rstofthediagramsX0// Y0 X00ba0// Y00 X[1]// Y[1]andY// Zb0a // X[1] Y0// Z0// X0[1]iscommutative.Thesecondiscommutativetoousingthat(X;Y;Z)!(X;Y0;A)and(X;Y0;A)!(X0;Y0;Z0)aremorphismsoftriangles.Atthispointwechooseadistinguishedtriangle(X00;Y00;Z00)startingwiththemapba0:X00!Y00.NextweapplyTR4onemoretimetothemorphismsX00!A!Y00andthetri-angles(X00;A;Z0;a0;b0;c0),(X00;Y00;Z00),and(A;Y00;Z[1];b;c;�a[1])togetmor-phismsa00:Z0!Z00andb00:Z00!Z[1].Then(Z0;Z00;Z[1];a00;b00;�b0[1]a[1])isadistinguishedtriangle,hencealso(Z;Z0;Z00;�b0a;a00;�b00)andhencealso(Z;Z0;Z00;b0a;a00;b00).Moreover,(X00;A;Z0)!(X00;Y00;Z00)and(X00;Y00;Z00)!(A;Y00;Z[1];b;c;�a[1])aremorphismsoftriangles.Atthispointwehavede nedallthedistinguishedtrianglesandallthemorphisms,andallthat'sleftistoverifysomecommutativityrelations.Toseethatthemiddlesquareinthediagramcommutes,notethatthearrowY0!Z0factorsasY0!A!Z0because(X;Y0;A)!(X0;Y0;Z0)isamor-phismoftriangles.Similarly,themorphismY0!Y00factorsasY0!A!Y00because(X;Y0;A)!(Y;Y0;Y00)isamorphismoftriangles.Hencethemid-dlesquarecommutesbecausethesquarewithsides(A;Z0;Z00;Y00)commutesas(X00;A;Z0)!(X00;Y00;Z00)isamorphismoftriangles(byTR4).Thesquarewithsides(Y00;Z00;Y[1];Z[1])commutesbecause(X00;Y00;Z00)!(A;Y00;Z[1];b;c;�a[1])isamorphismoftrianglesandc:Y00!Z[1]isthecompositionY00!Y[1]!Z[1].Thesquarewithsides(Z0;X0[1];X00[1];Z00)iscommutativebecause(X00;A;Z0)!(X00;Y00;Z00)isamorphis
13 moftrianglesandc0:Z0!X00[1]isthecompo-si![moftrianglesandc0:Z0!X00[1]isthecompo-si](884125/moftrianglesandc0-z0-x00-1-isthecompo-si.jpg "moftrianglesandc0:Z0!X00[1]isthecompo-si")
moftrianglesandc0:Z0!X00[1]isthecompo-sitionZ0!X0[1]!X00[1].Finally,wehavetoshowthatthesquarewithsides(Z00;X00[1];Z[1];X[2])anticommutes.Thisholdsbecause(X00;Y00;Z00)!(A;Y00;Z[1];b;c;�a[1])isamorphismoftrianglesandwe'redone.5.Localizationoftriangulatedcategories05R1Inordertoconstructthederivedcategorystartingfromthehomotopycategoryofcomplexes,wewillusealocalizationprocess.De nition5.1.05R2LetDbeapre-triangulatedcategory.WesayamultiplicativesystemSiscompatiblewiththetriangulatedstructureifthefollowingtwoconditionshold:MS5Fors2Swehaves[n]2Sforalln2Z. DERIVEDCATEGORIES14MS6GivenasolidcommutativesquareX// s Y// s0 Z// X[1]s[1] X0// Y0// Z0// X0[1]whoserowsaredistinguishedtriangleswiths;s02Sthereexistsamor-phisms00:Z!Z0inSsuchthat(s;s0;s00)isamorphismoftriangles.Itturnsoutthattheseaxiomsarenotindependentoftheaxiomsde ningmulti-plicativesystems.Lemma5.2.05R3LetDbeapre-triangulatedcategory.LetSbeasetofmorphismsofDandassumethataxiomsMS1,MS5,MS6hold(seeCategories,De nition27.1andDe nition5.1).ThenMS2holds.Proof.Supposethatf:X!YisamorphismofDandt:X!X0anelementofS.Chooseadistinguishedtriangle(X;Y;Z;f;g;h).Next,chooseadistinguishedtriangle(X0;Y0;Z;f0;g0;t[1]h)(hereweuseTR1andTR2).ByMS5,MS6(andTR2torotate)wecan ndthedottedarrowinthecommutativediagramX// t Y// s0 Z// 1 X[1]t[1] X0// Y0// Z// X0[1]withmoreovers02S.ThisprovesLMS2.TheproofofRMS2isdual.Lemma5.3.05R4LetF:D!D0beanexactfunctorofpre-triangulatedcategories.LetS=ff2Arrows(D)jF(f)isanisomorphismgThenSisasaturated(seeCategories,De nition27.20)multiplicativesystemcom-patiblewiththetriangulatedstructureonD.Proof.WehavetoproveaxiomsMS1MS6,seeCategories,De nitions27.1and27.20andDe nition5.1.MS1,MS4,andMS5aredirectfromthede nitions.MS6followsfromTR3andLemma4.3.ByLemma5.2weconcludethatMS2holds.To nishtheproofwehavetoshowthatMS3holds.Todothisletf;g:X!YbemorphismsofD,andlett:Z!XbeanelementofSsuchthatft=gt.AsDisadditivethissimplymeansthatat=0witha=f�g.Choosead
14 istinguishedtriangle(Z;X;Q;t;d;h)usingTR
istinguishedtriangle(Z;X;Q;t;d;h)usingTR1.Sinceat=0weseebyLemma4.2thereexistsamorphismi:Q!Ysuchthatid=a.Finally,usingTR1againwecanchooseatriangle(Q;Y;W;i;j;k).HereisapictureZt// Xd// 1 Q// i Z[1]Xa// Yj W DERIVEDCATEGORIES15OK,andnowweapplythefunctorFtothisdiagram.Sincet2SweseethatF(Q)=0,seeLemma4.9.HenceF(j)isanisomorphismbythesamelemma,i.e.,j2S.Finally,ja=jid=0asji=0.Thusjf=jgandweseethatLMS3holds.TheproofofRMS3isdual.Lemma5.4.05R5LetH:D!Abeahomologicalfunctorbetweenapre-triangulatedcategoryandanabeliancategory.LetS=ff2Arrows(D)jHi(f)isanisomorphismforalli2ZgThenSisasaturated(seeCategories,De nition27.20)multiplicativesystemcom-patiblewiththetriangulatedstructureonD.Proof.WehavetoproveaxiomsMS1MS6,seeCategories,De nitions27.1and27.20andDe nition5.1.MS1,MS4,andMS5aredirectfromthede nitions.MS6followsfromTR3andthelongexactcohomologysequence(3.5.1).ByLemma5.2weconcludethatMS2holds.To nishtheproofwehavetoshowthatMS3holds.Todothisletf;g:X!YbemorphismsofD,andlett:Z!XbeanelementofSsuchthatft=gt.AsDisadditivethissimplymeansthatat=0witha=f�g.Chooseadistinguishedtriangle(Z;X;Q;t;g;h)usingTR1andTR2.Sinceat=0weseebyLemma4.2thereexistsamorphismi:Q!Ysuchthatig=a.Finally,usingTR1againwecanchooseatriangle(Q;Y;W;i;j;k).HereisapictureZt// Xg// 1 Q// i Z[1]Xa// Yj WOK,andnowweapplythefunctorsHitothisdiagram.Sincet2SweseethatHi(Q)=0bythelongexactcohomologysequence(3.5.1).HenceHi(j)isanisomorphismforallibythesameargument,i.e.,j2S.Finally,ja=jig=0asji=0.Thusjf=jgandweseethatLMS3holds.TheproofofRMS3isdual.Proposition5.5.05R6LetDbeapre-triangulatedcategory.LetSbeamultiplica-tivesystemcompatiblewiththetriangulatedstructure.Thenthereexistsauniquestructureofapre-triangulatedcategoryonS�1DsuchthatthelocalizationfunctorQ:D!S�1Disexact.Moreover,ifDisatriangulatedcategory,soisS�1D.Proof.WehaveseenthatS�1DisanadditivecategoryandthatthelocalizationfunctorQisadditiveinHomology,Lemm
15 a8.2.Itisclearthatwemayde neQ(X)[n]=![a8.2.Itisclearthatwemayde neQ(X)[n]=](884125/a8-2-itisclearthatwemayde-28-neq-x-n-q-x.jpg "a8.2.Itisclearthatwemayde neQ(X)[n]=")
a8.2.Itisclearthatwemayde neQ(X)[n]=Q(X[n])sinceSispreservedundertheshiftfunctors[n]byMS5.Finally,wesayatriangleofS�1DisdistinguishedifitisisomorphictotheimageofadistinguishedtriangleunderthelocalizationfunctorQ.ProofofTR1.Theonlythingtoprovehereisthatifa:Q(X)!Q(Y)isamorphismofS�1D,thena tsintoadistinguishedtriangle.Writea=Q(s)�1Q(f)forsomes:Y!Y0inSandf:X!Y0.Chooseadistinguishedtriangle(X;Y0;Z;f;g;h)inD.Thenweseethat(Q(X);Q(Y);Q(Z);a;Q(g)Q(s);Q(h))isadistinguishedtriangleofS�1D. DERIVEDCATEGORIES16ProofofTR2.Thisisimmediatefromthede nitions.ProofofTR3.Notethattheexistenceofthedottedarrowwhichisrequiredtoexistmaybeprovenafterreplacingthetwotrianglesbyisomorphictriangles.Hencewemayassumegivendistinguishedtriangles(X;Y;Z;f;g;h)and(X0;Y0;Z0;f0;g0;h0)ofDandacommutativediagramQ(X)Q(f)// a Q(Y)b Q(X0)Q(f0)// Q(Y0)inS�1D.NowweapplyCategories,Lemma27.10to ndamorphismf00:X00!Y00inDandacommutativediagramXf k// X00f00 X0f0 soo Yl// Y00Y0too inDwiths;t2Sanda=s�1k,b=t�1l.AtthispointwecanuseTR3forDandMS6to ndacommutativediagramX// k Y// l Z// m X[1]g[1] X00// Y00// Z00// X00[1]X0// sOO Y0// tOO Z0// rOO X0[1]s[1]OO withr2S.Itfollowsthatsettingc=Q(r)�1Q(m)weobtainthedesiredmorphismoftriangles(Q(X);Q(Y);Q(Z);Q(f);Q(g);Q(h))(a;b;c) (Q(X0);Q(Y0);Q(Z0);Q(f0);Q(g0);Q(h0))Thisprovesthe rststatementofthelemma.IfDisalsoatriangulatedcategory,thenwestillhavetoproveTR4inordertoshowthatS�1Distriangulatedaswell.TodothiswereducebyLemma4.15tothefollowingstatement:Givencomposablemorphismsa:Q(X)!Q(Y)andb:Q(Y)!Q(Z)wehavetoproduceanoctahedronafterpossiblyreplacingQ(X);Q(Y);Q(Z)byisomorphicobjects.Todothiswemay rstreplaceYbyanobjectsuchthata=Q(f)forsomemorphismf:X!YinD.(Moreprecisely,writea=s�1fwiths:Y!Y0inSandf:X!Y0.ThenreplaceYbyY0.)AfterthiswesimilarlyreplaceZbyanobjectsuchthatb=Q(g)forsomemorphismg:Y!Z.Nowwecan nddistinguishedtriangles(X;Y;Q1;f;p1;d1),(X;Z;Q2;gf;p2;d2),and(Y;Z;Q3;g;p3;d3)inD(byTR1),andmorph
16 ismsa:Q1!Q2andb:Q2!Q3asinTR4.Thenitisimm
ismsa:Q1!Q2andb:Q2!Q3asinTR4.Thenitisimmediatelyveri edthatapplyingthefunctorQtoallthesedatagivesacorrespondingstructureinS�1D DERIVEDCATEGORIES17Theuniversalpropertyofthelocalizationofatriangulatedcategoryisasfollows(weformulatethisforpre-triangulatedcategories,henceitholdsafortiorifortri-angulatedcategories).Lemma5.6.05R7LetDbeapre-triangulatedcategory.LetSbeamultiplicativesystemcompatiblewiththetriangulatedcategory.LetQ:D!S�1Dbethelocalizationfunctor,seeProposition5.5.(1)IfH:D!AisahomologicalfunctorintoanabeliancategoryAsuchthatH(s)isanisomorphismforalls2S,thentheuniquefactorizationH0:S�1D!AsuchthatH=H0Q(seeCategories,Lemma27.8)isahomologicalfunctortoo.(2)IfF:D!D0isanexactfunctorintoapre-triangulatedcategoryD0suchthatF(s)isanisomorphismforalls2S,thentheuniquefactorizationF0:S�1D!D0suchthatF=F0Q(seeCategories,Lemma27.8)isanexactfunctortoo.Proof.Thislemmaprovesitself.Detailsomitted.Thefollowinglemmadescribesthekernel(seeDe nition6.5)ofthelocalizationfunctor.Lemma5.7.05R8LetDbeapre-triangulatedcategory.LetSbeamultiplicativesystemcompatiblewiththetriangulatedstructure.LetZbeanobjectofD.Thefollowingareequivalent(1)Q(Z)=0inS�1D,(2)thereexistsZ02Ob(D)suchthat0:Z!Z0isanelementofS,(3)thereexistsZ02Ob(D)suchthat0:Z0!ZisanelementofS,and(4)thereexistsanobjectZ0andadistinguishedtriangle(X;Y;ZZ0;f;g;h)suchthatf2S.IfSissaturated,thenthesearealsoequivalentto(5)themorphism0!ZisanelementofS,(6)themorphismZ!0isanelementofS,(7)thereexistsadistinguishedtriangle(X;Y;Z;f;g;h)suchthatf2S.Proof.Theequivalenceof(1),(2),and(3)isHomology,Lemma8.3.If(2)holds,then(Z0[�1];Z0[�1]Z;Z;(1;0);(0;1);0)isadistinguishedtriangle(seeLemma4.11)with02S.Byrotatingweconcludethat(4)holds.If(X;Y;ZZ0;f;g;h)isadistinguishedtrianglewithf2SthenQ(f)isanisomorphismhenceQ(ZZ0)=0henceQ(Z)=0.Thus(1)(4)areallequivalent.Next,assumethatSissaturated.Notethateachof(5),(6),(7)impliesoneoftheequivalentconditions(1)(4).SupposethatQ(Z)=0.Then0!ZisamorphismofDwhichbecomesanisomorphisminS�1D.AccordingtoCategories,Lemma27.21thef
17 actthatSissaturatedimpliesthat0!ZisinS.H
actthatSissaturatedimpliesthat0!ZisinS.Hence(1))(5).Dually(1))(6).Finally,if0!ZisinS,thenthetriangle(0;Z;Z;0;idZ;0)isdistinguishedbyTR1andTR2andisatriangleasin(4).Lemma5.8.05R9LetDbeatriangulatedcategory.LetSbeasaturatedmultiplicativesysteminDthatiscompatiblewiththetriangulatedstructure.Let(X;Y;Z;f;g;h)beadistinguishedtriangleinD.ConsiderthecategoryofmorphismsoftrianglesI=f(s;s0;s00):(X;Y;Z;f;g;h)!(X0;Y0;Z0;f0;g0;h0)js;s0;s002Sg DERIVEDCATEGORIES18ThenIisa lteredcategoryandthefunctorsI!X=S,I!Y=S,andI!Z=Sareco nal.Proof.Westronglysuggestthereaderskiptheproofofthislemmaandinsteadworkitoutonanapkin.The rstremarkisthatusingrotationofdistinguishedtriangles(TR2)givesanequivalenceofcategoriesbetweenIandthecorrespondingcategoryforthedistin-guishedtriangle(Y;Z;X[1];g;h;�f[1]).UsingthisweseeforexamplethatifweprovethefunctorI!X=Sisco nal,thenthesamethingistrueforthefunctorsI!Y=SandI!Z=S.Notethatifs:X!X0isamorphismofS,thenusingMS2wecan nds0:Y!Y0andf0:X0!Y0suchthatf0s=s0f,whereuponwecanuseMS6tocompletethisintoanobjectofI.HencethefunctorI!X=Sissurjectiveonobjects.UsingrotationasabovethisimpliesthesamethingistrueforthefunctorsI!Y=SandI!Z=S.Supposegivenobjectss1:X!X1ands2:X!X2inX=Sandamorphisma:X1!X2inX=S.SinceSissaturated,weseethata2S,seeCategories,Lemma27.21.Bytheargumentofthepreviousparagraphwecancompletes1:X!X1toanobject(s1;s01;s001):(X;Y;Z;f;g;h)!(X1;Y1;Z1;f1;g1;h1)inI.Thenwecanrepeatand nd(a;b;c):(X1;Y1;Z1;f1;g1;h1)!(X2;Y2;Z2;f2;g2;h2)witha;b;c2Scompletingthegivena:X1!X2.Butthen(a;b;c)isamorphisminI.InthiswayweconcludethatthefunctorI!X=Sisalsosurjectiveonarrows.Usingrotationasabove,thisimpliesthesamethingistrueforthefunctorsI!Y=SandI!Z=S.ThecategoryIisnonemptyastheidentityprovidesanobject.Thisprovesthecondition(1)ofthede nitionofa lteredcategory,seeCategories,De nition19.1.Wecheckcondition(2)ofCategories,De nition19.1forthecategoryI.Supposegivenobjects(s1;s01;s001):(X;Y;Z;f;g;h)!(X1;Y1;Z1;f1;g1;h1)and(s2;s02;s002):(X;Y;Z;f;g;h)!(X2;Y2;Z2;f2;g2;h2)inI.Wewantto ndanobjectofIwhichisthetargetofanarrowfr
18 omboth(X1;Y1;Z1;f1;g1;h1)and(X2;Y2;Z2;f2
omboth(X1;Y1;Z1;f1;g1;h1)and(X2;Y2;Z2;f2;g2;h2).ByCategories,Remark27.7thecategoriesX=S,Y=S,Z=Sare ltered.Thuswecan ndX!X3inX=Sandmorphismss:X2!X3anda:X1!X3.Bytheabovewecan ndamorphism(s;s0;s00):(X2;Y2;Z2;f2;g2;h2)!(X3;Y3;Z3;f3;g3;h3)withs0;s002S.Afterreplacing(X2;Y2;Z2)by(X3;Y3;Z3)wemayassumethatthereexistsamorphisma:X1!X2inX=S.RepeatingtheargumentforYandZ(byrotatingasabove)wemayassumethereisamorphisma:X1!X2inX=S,b:Y1!Y2inY=S,andc:Z1!Z2inZ=S.However,thesemorphismsdonotnecessarilygiverisetoamorphismofdistinguishedtriangles.Ontheotherhand,thenecessarydiagramsdocommuteinS�1D.Hencewesee(forexample)thatthereexistsamorphisms02:Y2!Y3inSsuchthats02f2a=s02bf1.Anotherreplacementof(X2;Y2;Z2)asabovethengetsustothesituationwheref2a=bf1.Rotatingandapplyingthesameargumenttwomoretimesweseethatwemayassume(a;b;c)isamorphismoftriangles.Thisprovescondition(2).Nextwecheckcondition(3)ofCategories,De nition19.1.Suppose(s1;s01;s001):(X;Y;Z)!(X1;Y1;Z1)and(s2;s02;s002):(X;Y;Z)!(X2;Y2;Z2)areobjectsofI,andsuppose(a;b;c);(a0;b0;c0)aretwomorphismsbetweenthem.Sinceas1=a0s1thereexistsamorphisms3:X2!X3suchthats3a=s3a0.Usingthe DERIVEDCATEGORIES19surjectivitystatementwecancompletethistoamorphismoftriangles(s3;s03;s003):(X2;Y2;Z2)!(X3;Y3;Z3)withs3;s03;s0032S.Thus(s3s2;s03s02;s003s002):(X;Y;Z)!(X3;Y3;Z3)isalsoanobjectofIandaftercomposingthemaps(a;b;c);(a0;b0;c0)with(s3;s03;s003)weobtaina=a0.Byrotatingwemaydothesametogetb=b0andc=c0.Finally,wecheckthatI!X=Sisco nal,seeCategories,De nition17.1.The rstconditionistrueasthefunctorissurjective.Supposethatwehaveanobjects:X!X0inX=Sandtwoobjects(s1;s01;s001):(X;Y;Z;f;g;h)!(X1;Y1;Z1;f1;g1;h1)and(s2;s02;s002):(X;Y;Z;f;g;h)!(X2;Y2;Z2;f2;g2;h2)inIaswellasmorphismst1:X0!X1andt2:X0!X2inX=S.Byproperty(2)ofIprovedabovewecan ndmorphisms(s3;s03;s003):(X1;Y1;Z1;f1;g1;h1)!(X3;Y3;Z3;f3;g3;h3)and(s4;s04;s004):(X2;Y2;Z2;f2;g2;h2)!(X3;Y3;Z3;f3;g3;h3)inI.WewouldbedoneifthecompositionsX0!X1!X3andX0!X1!X3whereequal(seedisplayedequationinCategories,De nition17.
19 1).Ifnot,then,becauseX=Sis ltered,we
1).Ifnot,then,becauseX=Sis ltered,wecanchooseamorphismX3!X4inSsuchthatthecompositionsX0!X1!X3!X4andX0!X1!X3!X4areequal.Thenwe nallycompleteX3!X4toamorphism(X3;Y3;Z3)!(X4;Y4;Z4)inIandcomposewiththatmorphismtoseethattheresultistrue.6.Quotientsoftriangulatedcategories05RAGivenatriangulatedcategoryandatriangulatedsubcategorywecanconstructanothertriangulatedcategorybytakingthequotient.Theconstructionusesalocalization.ThisissimilartothequotientofanabeliancategorybyaSerresubcategory,seeHomology,Section10.Beforewedotheactualconstructionwebrie ydiscusskernelsofexactfunctors.De nition6.1.05RBLetDbeapre-triangulatedcategory.Wesayafullpre-triangulatedsubcategoryD0ofDissaturatedifwheneverXYisisomorphictoanobjectofD0thenbothXandYareisomorphictoobjectsofD0.Asaturatedtriangulatedsubcategoryissometimescalledathicktriangulatedsub-category.Insomereferences,thisisonlyusedforstrictlyfulltriangulatedsub-categories(andsometimesthede nitioniswrittensuchthatitimpliesstrictness).Thereisanothernotion,thatofanépaissetriangulatedsubcategory.Thede nitionisthatgivenacommutativediagramS X ?? // Y// T// X[1]wherethesecondlineisadistinguishedtriangleandSandTisomorphictoobjectsofD0,thenalsoXandYareisomorphictoobjectsofD0.Itturnsoutthatthisisequivalenttobeingsaturated(thisiselementaryandcanbefoundin[Ric89])andthenotionofasaturatedcategoryiseasiertoworkwith.Lemma6.2.05RCLetF:D!D0beanexactfunctorofpre-triangulatedcategories.LetD00bethefullsubcategoryofDwithobjectsOb(D00)=fX2Ob(D)jF(X)=0g DERIVEDCATEGORIES20ThenD00isastrictlyfullsaturatedpre-triangulatedsubcategoryofD.IfDisatriangulatedcategory,thenD00isatriangulatedsubcategory.Proof.ItisclearthatD00ispreservedunder[1]and[�1].If(X;Y;Z;f;g;h)isadistinguishedtriangleofDandF(X)=F(Y)=0,thenalsoF(Z)=0as(F(X);F(Y);F(Z);F(f);F(g);F(h))isdistinguished.HencewemayapplyLemma4.16toseethatD00isapre-triangulatedsubcategory(respectivelyatrian-gulatedsubcategoryifDisatriangulatedcategory).The nalassertionofbeingsaturatedfollowsfromF(X)F(Y)=0)F(X)=F(Y)=0.Lemma6.3.05RDLetH:D!Abeahomologi
20 calfunctorofapre-triangulatedcategoryint
calfunctorofapre-triangulatedcategoryintoanabeliancategory.LetD0bethefullsubcategoryofDwithobjectsOb(D0)=fX2Ob(D)jH(X[n])=0foralln2ZgThenD0isastrictlyfullsaturatedpre-triangulatedsubcategoryofD.IfDisatriangulatedcategory,thenD0isatriangulatedsubcategory.Proof.ItisclearthatD0ispreservedunder[1]and[�1].If(X;Y;Z;f;g;h)isadistinguishedtriangleofDandH(X[n])=H(Y[n])=0foralln,thenalsoH(Z[n])=0forallnbythelongexactsequence(3.5.1).HencewemayapplyLemma4.16toseethatD0isapre-triangulatedsubcategory(respectivelyatri-angulatedsubcategoryifDisatriangulatedcategory).TheassertionofbeingsaturatedfollowsfromH((XY)[n])=0)H(X[n]Y[n])=0)H(X[n])H(Y[n])=0)H(X[n])=H(Y[n])=0foralln2Z.Lemma6.4.05RELetH:D!Abeahomologicalfunctorofapre-triangulatedcategoryintoanabeliancategory.LetD+H;D�H;DbHbethefullsubcategoryofDwithobjectsOb(D+H)=fX2Ob(D)jH(X[n])=0foralln 0gOb(D�H)=fX2Ob(D)jH(X[n])=0foralln 0gOb(DbH)=fX2Ob(D)jH(X[n])=0foralljnj 0gEachoftheseisastrictlyfullsaturatedpre-triangulatedsubcategoryofD.IfDisatriangulatedcategory,theneachisatriangulatedsubcategory.Proof.LetusprovethisforD+H.Itisclearthatitispreservedunder[1]and[�1].If(X;Y;Z;f;g;h)isadistinguishedtriangleofDandH(X[n])=H(Y[n])=0foralln 0,thenalsoH(Z[n])=0foralln 0bythelongexactsequence(3.5.1).HencewemayapplyLemma4.16toseethatD+Hisapre-triangulatedsubcategory(respectivelyatriangulatedsubcategoryifDisatriangulatedcategory).TheassertionofbeingsaturatedfollowsfromH((XY)[n])=0)H(X[n]Y[n])=0)H(X[n])H(Y[n])=0)H(X[n])=H(Y[n])=0foralln2Z. DERIVEDCATEGORIES21De nition6.5.05RFLetDbea(pre-)triangulatedcategory.(1)LetF:D!D0beanexactfunctor.ThekernelofFisthestrictlyfullsaturated(pre-)triangulatedsubcategorydescribedinLemma6.2.(2)LetH:D!Abeahomologicalfunctor.ThekernelofHisthestrictlyfullsaturated(pre-)triangulatedsubcategorydescribedinLemma6.3.ThesearesometimesdenotedKer(F)orKer(H).TheproofofthefollowinglemmausesTR4.Lemma6.6.05RGLetDbeatriangulatedcategory.LetD0Dbeafulltriangulatedsubcategory.Set(6.6.1)05RHS=f2Arrows(D)suchthatthereexistsadistinguishedtriang
21 le(X;Y;Z;f;g;h)ofDwithZisomorphictoanobj
le(X;Y;Z;f;g;h)ofDwithZisomorphictoanobjectofD0ThenSisamultiplicativesystemcompatiblewiththetriangulatedstructureonD.Inthissituationthefollowingareequivalent(1)Sisasaturatedmultiplicativesystem,(2)D0isasaturatedtriangulatedsubcategory.Proof.Toprovethe rstassertionwehavetoprovethatMS1,MS2,MS3andMS5,MS6hold.ProofofMS1.ItisclearthatidentitiesareinSbecause(X;X;0;1;0;0)isdistin-guishedforeveryobjectXofDandbecause0isanobjectofD0.Letf:X!Yandg:Y!ZbecomposablemorphismscontainedinS.Choosedistinguishedtriangles(X;Y;Q1;f;p1;d1),(X;Z;Q2;gf;p2;d2),and(Y;Z;Q3;g;p3;d3).ByassumptionweknowthatQ1andQ3areisomorphictoobjectsofD0.ByTR4weknowthereexistsadistinguishedtriangle(Q1;Q2;Q3;a;b;c).SinceD0isatrian-gulatedsubcategoryweconcludethatQ2isisomorphictoanobjectofD0.Hencegf2S.ProofofMS3.Leta:X!Ybeamorphismandlett:Z!XbeanelementofSsuchthatat=0.ToproveLMS3itsu cesto ndans2Ssuchthatsa=0,comparewiththeproofofLemma5.3.Chooseadistinguishedtriangle(Z;X;Q;t;g;h)usingTR1andTR2.Sinceat=0weseebyLemma4.2thereexistsamorphismi:Q!Ysuchthatig=a.Finally,usingTR1againwecanchooseatriangle(Q;Y;W;i;s;k).HereisapictureZt// Xg// 1 Q// i Z[1]Xa// Ys WSincet2SweseethatQisisomorphictoanobjectofD0.Hences2S.Finally,sa=sig=0assi=0byLemma4.1.WeconcludethatLMS3holds.TheproofofRMS3isdual.ProofofMS5.FollowsasdistinguishedtrianglesandD0arestableundertransla-tions DERIVEDCATEGORIES22ProofofMS6.SupposegivenacommutativediagramX// s Ys0 X0// Y0withs;s02S.ByProposition4.23wecanextendthistoaninesquarediagram.Ass;s0areelementsofSweseethatX00;Y00areisomorphictoobjectsofD0.SinceD0isafulltriangulatedsubcategoryweseethatZ00isalsoisomorphictoanobjectofD0.WhencethemorphismZ!Z0isanelementofS.ThisprovesMS6.MS2isaformalconsequenceofMS1,MS5,andMS6,seeLemma5.2.This nishestheproofofthe rstassertionofthelemma.Let'sassumethatSissaturated.(Inthefollowingwewilluserotationofdistin-guishedtriangleswithoutfurthermention.)LetXYbeanobjectisomorphictoanobjectofD0.Considerthemorphismf:0!X.Thecomposition0!X!X
22 YisanelementofSas(0;XY;XY;0;
YisanelementofSas(0;XY;XY;0;1;0)isadistinguishedtriangle.Thecom-positionY[�1]!0!XisanelementofSas(X;XY;Y;(1;0);(0;1);0)isadistinguishedtriangle,seeLemma4.11.Hence0!XisanelementofS(asSissaturated).ThusXisisomorphictoanobjectofD0asdesired.Finally,assumeD0isasaturatedtriangulatedsubcategory.LetWh�!Xg�!Yf�!ZbecomposablemorphismsofDsuchthatfg;gh2S.Wewillbuildupapictureofobjectsasinthediagrambelow.Q12 !! Q23 !! Q1 +1~~ == Q2 +1}} +1oo == Q3 +1}} +1oo W// X aa // Y aa // Z `` Firstchoosedistinguishedtriangles(W;X;Q1),(X;Y;Q2),(Y;Z;Q3)(W;Y;Q12),and(X;Z;Q23).Denotes:Q2!Q1[1]thecompositionQ2!X[1]!Q1[1].Denotet:Q3!Q2[1]thecompositionQ3!Y[1]!Q2[1].ByTR4appliedtothecompositionW!X!YandthecompositionX!Y!Zthereexistdistinguishedtriangles(Q1;Q12;Q2)and(Q2;Q23;Q3)whichusethemorphismssandt.TheobjectsQ12andQ23areisomorphictoobjectsofD0asW!YandX!ZareassumedinS.Hencealsos[1]tisanelementofSasSisclosedundercompositionsandshifts.Notethats[1]t=0asY[1]!Q2[1]!X[2]iszero,seeLemma4.1.HenceQ3[1]Q1[2]isisomorphictoanobjectofD0,seeLemma4.11.ByassumptiononD0weconcludethatQ3andQ1areisomorphictoobjectsofD0.Lookingatthedistinguishedtriangle(Q1;Q12;Q2)weconcludethatQ2isalsoisomorphictoanobjectofD0.Lookingatthedistinguishedtriangle(X;Y;Q2)we nallyconcludethatg2S.(Itisalsofollowsthath;f2S,butwedon'tneedthis.) DERIVEDCATEGORIES23De nition6.7.05RILetDbeatriangulatedcategory.LetBbeafulltriangulatedsubcategory.Wede nethequotientcategoryD=BbytheformulaD=B=S�1D,whereSisthemultiplicativesystemofDassociatedtoBviaLemma6.6.ThelocalizationfunctorQ:D!D=Biscalledthequotientfunctorinthiscase.NotethatthequotientfunctorQ:D!D=Bisanexactfunctoroftriangulatedcategories,seeProposition5.5.Theuniversalpropertyofthisconstructionisthefollowing.Lemma6.8.05RJLetDbeatriangulatedcategory.LetBbeafulltriangulatedsub-categoryofD.LetQ:D!D=Bbethequotientfunctor.(1)IfH:D!AisahomologicalfunctorintoanabeliancategoryAsuchthatBKer(H)thenthereexistsauniquefactorizationH0:D=B!AsuchthatH=H0QandH0isahomologicalfunctortoo.(2)IfF:D!D0isanexactfunctorintoapre-triangulatedc
23 ategoryD0suchthatBKer(F)thenthereex
ategoryD0suchthatBKer(F)thenthereexistsauniquefactorizationF0:D=B!D0suchthatF=F0QandF0isanexactfunctortoo.Proof.ThislemmafollowsfromLemma5.6.Namely,iff:X!YisamorphismofDsuchthatforsomedistinguishedtriangle(X;Y;Z;f;g;h)theobjectZisisomorphictoanobjectofB,thenH(f),resp.F(f)isanisomorphismundertheassumptionsof(1),resp.(2).Detailsomitted.Thekernelofthequotientfunctorcanbedescribedasfollows.Lemma6.9.05RKLetDbeatriangulatedcategory.LetBbeafulltriangulatedsub-category.ThekernelofthequotientfunctorQ:D!D=BisthestrictlyfullsubcategoryofDwhoseobjectsareOb(Ker(Q))=Z2Ob(D)suchthatthereexistsaZ02Ob(D)suchthatZZ0isisomorphictoanobjectofBInotherwordsitisthesmalleststrictlyfullsaturatedtriangulatedsubcategoryofDcontainingB.Proof.Firstnotethatthekernelisautomaticallyastrictlyfulltriangulatedsub-categorycontainingsummandsofanyofitsobjects,seeLemma6.2.Thedescriptionofitsobjectsfollowsfromthede nitionsandLemma5.7part(4).LetDbeatriangulatedcategory.Atthispointwehaveconstructionswhichinduceorderpreservingmapsbetween(1)thepartiallyorderedsetofmultiplicativesystemsSinDcompatiblewiththetriangulatedstructure,and(2)thepartiallyorderedsetoffulltriangulatedsubcategoriesBD.Namely,theconstructionsaregivenbyS7!B(S)=Ker(Q:D!S�1D)andB7!S(B)whereS(B)isthemultiplicativesetof(6.6.1),i.e.,S(B)=f2Arrows(D)suchthatthereexistsadistinguishedtriangle(X;Y;Z;f;g;h)ofDwithZisomorphictoanobjectofBNotethatitisnotthecasethattheseoperationsaremutuallyinverse.Lemma6.10.05RLLetDbeatriangulatedcategory.Theoperationsdescribedabovehavethefollowingproperties DERIVEDCATEGORIES24(1)S(B(S))isthesaturationofS,i.e.,itisthesmallestsaturatedmulti-plicativesysteminDcontainingS,and(2)B(S(B))isthesaturationofB,i.e.,itisthesmalleststrictlyfullsaturatedtriangulatedsubcategoryofDcontainingB.Inparticular,theconstructionsde nemutuallyinversemapsbetweenthe(partiallyordered)setofsaturatedmultiplicativesystemsinDcompatiblewiththetriangulatedstructureonDandthe(partiallyordered)setofstrictlyfullsaturatedtriangulatedsubcategoriesofD.Proof.First,let'
24 sstartwithafulltriangulatedsubcategoryB.
sstartwithafulltriangulatedsubcategoryB.ThenB(S(B))=Ker(Q:D!D=B)andhence(2)isthecontentofLemma6.9.Next,supposethatSismultiplicativesysteminDcompatiblewiththetriangula-tiononD.ThenB(S)=Ker(Q:D!S�1D).Hence(usingLemma4.9inthelocalizedcategory)S(B(S))=f2Arrows(D)suchthatthereexistsadistinguishedtriangle(X;Y;Z;f;g;h)ofDwithQ(Z)=0=ff2Arrows(D)jQ(f)isanisomorphismg=^S=S0inthenotationofCategories,Lemma27.21.The nalstatementofthatlemma nishestheproof.Lemma6.11.05RMLetH:D!AbeahomologicalfunctorfromatriangulatedcategoryDtoanabeliancategoryA,seeDe nition3.5.ThesubcategoryKer(H)ofDisastrictlyfullsaturatedtriangulatedsubcategoryofDwhosecorrespondingsaturatedmultiplicativesystem(seeLemma6.10)isthesetS=ff2Arrows(D)jHi(f)isanisomorphismforalli2Zg:ThefunctorHfactorsthroughthequotientfunctorQ:D!D=Ker(H).Proof.ThecategoryKer(H)isastrictlyfullsaturatedtriangulatedsubcategoryofDbyLemma6.3.ThesetSisasaturatedmultiplicativesystemcompatiblewiththetriangulatedstructurebyLemma5.4.RecallthatthemultiplicativesystemcorrespondingtoKer(H)isthesetf2Arrows(D)suchthatthereexistsadistinguishedtriangle(X;Y;Z;f;g;h)withHi(Z)=0foralliBythelongexactcohomologysequence,see(3.5.1),itisclearthatfisanelementofthissetifandonlyiffisanelementofS.Finally,thefactorizationofHthroughQisaconsequenceofLemma6.8.7.Adjointsforexactfunctors0A8CResultsonadjointfunctorsbetweentriangulatedcategories.Lemma7.1.0A8DLetF:D!D0beanexactfunctorbetweentriangulatedcategories.IfFadmitsarightadjointG:D0!D,thenGisalsoanexactfunctor. DERIVEDCATEGORIES25Proof.LetXbeanobjectofDandAanobjectofD0.SinceFisanexactfunctorweseethatMorD(X;G(A[1])=MorD0(F(X);A[1])=MorD0(F(X)[�1];A)=MorD0(F(X[�1]);A)=MorD(X[�1];G(A))=MorD(X;G(A)[1])ByYoneda'slemma(Categories,Lemma3.5)weobtainacanonicalisomorphismG(A)[1]=G(A[1]).LetA!B!C!A[1]beadistinguishedtriangleinD0.ChooseadistinguishedtriangleG(A)!G(B)!X!G(A)[1]inD.ThenF(G(A))!F(G(B))!F(X)!F(G(A))[1]isadistinguishedtriangleinD0.ByTR3wecanchooseamorphismofdistinguishedtrianglesF(G(A))// F(G(B))// F(X)// F(G(A))[1]
25 A// B// C// A[1]SinceGisthead![A//
B//
C//
A[1]SinceGisthead](884125/15-15-a-b-c-a-1-sincegistheadjointthenew.jpg "A//
B//
C//
A[1]SinceGisthead")
A// B// C// A[1]SinceGistheadjointthenewmorphismdeterminesamorphismX!G(C)suchthatthediagramG(A)// G(B)// X// G(A)[1] G(A)// G(B)// G(C)// G(A)[1]commutes.ApplyingthehomologicalfunctorHomD0(W;�)foranobjectWofD0wededucefromthe5lemmathatHomD0(W;X)!HomD0(W;G(C))isabijectionandusingtheYonedalemmaoncemoreweconcludethatX!G(C)isanisomorphism.HenceweconcludethatG(A)!G(B)!G(C)!G(A)[1]isadistinguishedtrianglewhichiswhatwewantedtoshow.Lemma7.2.09J1LetD,D0betriangulatedcategories.LetF:D!D0andG:D0!Dbefunctors.Assumethat(1)FandGareexactfunctors,(2)Fisfullyfaithful,(3)GisarightadjointtoF,and(4)thekernelofGiszero.ThenFisanequivalenceofcategories.Proof.SinceFisfullyfaithfultheadjunctionmapid!GFisanisomorphism(Categories,Lemma24.4).LetXbeanobjectofD0.ChooseadistinguishedtriangleF(G(X))!X!Y!F(G(X))[1] DERIVEDCATEGORIES26inD0.ApplyingGandusingthatG(F(G(X)))=G(X)we ndadistinguishedtriangleG(X)!G(X)!G(Y)!G(X)[1]HenceG(Y)=0.ThusY=0.ThusF(G(X))!Xisanisomorphism.8.Thehomotopycategory05RNLetAbeanadditivecategory.ThehomotopycategoryK(A)ofAisthecategoryofcomplexesofAwithmorphismsgivenbymorphismsofcomplexesuptohomotopy.Hereistheformalde nition.De nition8.1.013HLetAbeanadditivecategory.(1)WesetComp(A)=CoCh(A)bethecategoryof(cochain)complexes.(2)AcomplexKissaidtobeboundedbelowifKn=0foralln 0.(3)AcomplexKissaidtobeboundedaboveifKn=0foralln 0.(4)AcomplexKissaidtobeboundedifKn=0foralljnj 0.(5)WeletComp+(A),Comp�(A),resp.Compb(A)bethefullsubcategoryofComp(A)whoseobjectsarethecomplexeswhichareboundedbelow,boundedabove,resp.bounded.(6)WeletK(A)bethecategorywiththesameobjectsasComp(A)butasmorphismshomotopyclassesofmapsofcomplexes(seeHomology,Lemma13.7).(7)WeletK+(A),K�(A),resp.Kb(A)bethefullsubcategoryofK(A)whoseobjectsareboundedbelow,boundedabove,resp.boundedcomplexesofA.ItwillturnoutthatthecategoriesK(A),K+(A),K�(A),andKb(A)aretrian-gulatedcategories.Toprovethiswe rstdevelopsomemachineryrelatedtoconesandsplitexactsequences.9.Conesandtermwisesplitsequences014DLetAbea
26 nadditivecategory,andletK(A)denotethecat
nadditivecategory,andletK(A)denotethecategoryofcomplexesofAwithmorphismsgivenbymorphismsofcomplexesuptohomotopy.Notethattheshiftfunctors[n]oncomplexes,seeHomology,De nition14.7,giverisetofunctors[n]:K(A)!K(A)suchthat[n][m]=[n+m]and[0]=id.De nition9.1.014ELetAbeanadditivecategory.Letf:K!LbeamorphismofcomplexesofA.TheconeoffisthecomplexC(f)givenbyC(f)n=LnKn+1anddierentialdnC(f)=dnLfn+10�dn+1KItcomesequippedwithcanonicalmorphismsofcomplexesi:L!C(f)andp:C(f)!K[1]inducedbytheobviousmapsLn!C(f)n!Kn+1.Inotherwords(K;L;C(f);f;i;p)formsatriangle:K!L!C(f)!K[1]Theformationofthistriangleisfunctorialinthefollowingsense. DERIVEDCATEGORIES27Lemma9.2.014FSupposethatK1f1// a L1b K2f2// L2isadiagramofmorphismsofcomplexeswhichiscommutativeuptohomotopy.Thenthereexistsamorphismc:C(f1)!C(f2)whichgivesrisetoamor-phismoftriangles(a;b;c):(K1;L1;C(f1);f1;i1;p1)!(K2;L2;C(f2);f2;i2;p2)ofK(A).Proof.Lethn:Kn1!Ln�12beafamilyofmorphismssuchthatbf1�f2a=dh+hd.De necnbythematrixcn=bnhn+10an+1:Ln1Kn+11!Ln2Kn+12Amatrixcomputationshowthatcisamorphismofcomplexes.Itistrivialthatci1=i2b,anditistrivialalsotocheckthatp2c=ap1.Notethatthemorphismc:C(f1)!C(f2)constructedintheproofofLemma9.2ingeneraldependsonthechosenhomotopyhbetweenf2aandbf1.Lemma9.3.08RISupposethatf:K!Landg:L!Maremorphismsofcomplexessuchthatgfishomotopictozero.Then(1)gfactorsthroughamorphismC(f)!M,and(2)ffactorsthroughamorphismK!C(g)[�1].Proof.TheassumptionssaythatthediagramKf// Lg 0// Mcommutesuptohomotopy.Sincetheconeon0!MisMthemapC(f)!C(0!M)=MofLemma9.2isthemapin(1).TheconeonK!0isK[1]andapplyingLemma9.2givesamapK[1]!C(g).Applying[�1]weobtainthemapin(2).NotethatthemorphismsC(f)!MandK!C(g)[�1]constructedintheproo
27 fofLemma9.3ingeneraldependonthechosenhom
fofLemma9.3ingeneraldependonthechosenhomotopy.De nition9.4.014GLetAbeanadditivecategory.Atermwisesplitinjection :A!BisamorphismofcomplexessuchthateachAn!Bnisisomorphictotheinclusionofadirectsummand.Atermwisesplitsurjection:B!CisamorphismofcomplexessuchthateachBn!Cnisisomorphictotheprojectionontoadirectsummand.Lemma9.5.014HLetAbeanadditivecategory.LetAf// a Bb Cg// D DERIVEDCATEGORIES28beadiagramofmorphismsofcomplexescommutinguptohomotopy.Iffisatermwisesplitinjection,thenbishomotopictoamorphismwhichmakesthedia-gramcommute.Ifgisasplitsurjection,thenaishomotopictoamorphismwhichmakesthediagramcommute.Proof.Lethn:An!Dn�1beacollectionofmorphismssuchthatbf�ga=dh+hd.Supposethatn:Bn!Anaremorphismssplittingthemorphismsfn.Takeb0=b�dh�hd.Supposesn:Dn!Cnaremorphismssplittingthemorphismsgn:Cn!Dn.Takea0=a+dsh+shd.Computationsomitted.Thefollowinglemmacanbeusedtoreplaceamorphismofcomplexesbyamor-phismwhereineachdegreethemapistheinjectionofadirectsummand.Lemma9.6.013NLetAbeanadditivecategory.Let :K!LbeamorphismofcomplexesofA.ThereexistsafactorizationK~ // 66 ~L// Lsuchthat(1)~ isatermwisesplitinjection(seeDe nition9.4),(2)thereisamapofcomplexess:L!~Lsuchthats=idLandsuchthatsishomotopictoid~L.Moreover,ifbothKandLareinK+(A),K�(A),orKb(A),thensois~L.Proof.Weset~Ln=LnKnKn+1andwede nedn~L=0@dnL000dnKidKn+100�dn+1K1AInotherwords,~L=LC(1K).Moreover,weset~ =0@ idKn01Awhichisclearlyasplitinjection.Itisalsoclearthatitde nesamorphismofcomplexes.Wede ne=�idLn00sothatclearly~ = .Wesets=0@idLn001Asothats=idL.Finally,lethn:~Ln!~Ln�1bethemapwhichmapsthesummandKnof~LnviatheidentitymorphismtothesummandKnof~Ln�1.Thenitisatrivialmatter(seecomputationsinremarkbelow)toprovethatid~L�s=dh+hdwhich nishestheproofofthelemma. DERIVEDCATEGORIES29Remark
28 9.7.013OToseethelastdisplayedequalityint
9.7.013OToseethelastdisplayedequalityintheproofabovewecanarguewithelementsasfollows.Wehaves(l;k;k+)=(l;0;0).Hencethemorphismofthelefthandsidemaps(l;k;k+)to(0;k;k+).Ontheotherhandh(l;k;k+)=(0;0;k)andd(l;k;k+)=(dl;dk+k+;�dk+).Hence(dh+hd)(l;k;k+)=d(0;0;k)+h(dl;dk+k+;�dk+)=(0;k;�dk)+(0;0;dk+k+)=(0;k;k+)asdesired.Lemma9.8.0642LetAbeanadditivecategory.Let :K!LbeamorphismofcomplexesofA.ThereexistsafactorizationKi// 66 ~K~ // Lsuchthat(1)~ isatermwisesplitsurjection(seeDe nition9.4),(2)thereisamapofcomplexess:~K!Ksuchthatsi=idKandsuchthatisishomotopictoid~K.Moreover,ifbothKandLareinK+(A),K�(A),orKb(A),thensois~K.Proof.DualtoLemma9.6.Take~Kn=KnLn�1Lnandwede nedn~K=0@dnK000�dn�1LidLn00dnL1Ainotherwords~K=KC(1L[�1]).Moreover,weset~ =� 0idLnwhichisclearlyasplitsurjection.Itisalsoclearthatitde nesamorphismofcomplexes.Wede nei=0@idKn001Asothatclearly~ i= .Wesets=�idKn00sothatsi=idK.Finally,lethn:~Kn!~Kn�1bethemapwhichmapsthesummandLn�1of~KnviatheidentitymorphismtothesummandLn�1of~Kn�1.Thenitisatrivialmattertoprovethatid~K�is=dh+hdwhich nishestheproofofthelemma.De nition9.9.014ILetAbeanadditivecategory.AtermwisesplitexactsequenceofcomplexesofAisacomplexofcomplexes0!A �!B�!C!0togetherwithgivendirectsumdecompositionsBn=AnCncompatiblewith nandn.Weoftenwritesn:Cn!Bnandn:Bn!Anforthemapsinducedby DERIVEDCATEGORIES30thedirectsumdecompositions.AccordingtoHomology,Lemma14.10wegetanassociatedmorphismofcomplexes:C�!A[1]whichindegreenisthemapn+1dnBsn.Inotherwords(A;B;C; ;;)formsatriangleA!B!C!A[1]Thiswillbethetriangleassociatedtothetermwisesplitsequenceofcomplexes.Lemma9.10.05SSLetAbeanadditivecategory.Let0!A!B!C!0betermwisesplitexactsequencesasinDe nition9.9.Let(0)n,(s0)nbease
29 condcollectionofsplittings.Denote0:
condcollectionofsplittings.Denote0:C�!A[1]themorphismassociatedtothissecondsetofsplittings.Then(1;1;1):(A;B;C; ;;)�!(A;B;C; ;;0)isanisomorphismoftrianglesinK(A).Proof.Thestatementsimplymeansthatand0arehomotopicmapsofcom-plexes.ThisisHomology,Lemma14.12.Remark9.11.014JLetAbeanadditivecategory.Let0!Ai!Bi!Ci!0,i=1;2betermwisesplitexactsequences.Supposethata:A1!A2,b:B1!B2,andc:C1!C2aremorphismsofcomplexessuchthatA1a // B1// b C1c A2// B2// C2commutesinK(A).Ingeneral,theredoesnotexistamorphismb0:B1!B2whichishomotopictobsuchthatthediagramabovecommutesinthecategoryofcomplexes.Namely,considerExamples,Equation(62.0.1).Ifwecouldreplacethemiddlemaptherebyahomotopiconesuchthatthediagramcommutes,thenwewouldhaveadditivityoftraceswhichwedonot.Lemma9.12.086LLetAbeanadditivecategory.Let0!Ai!Bi!Ci!0,i=1;2;3betermwisesplitexactsequencesofcomplexes.Letb:B1!B2andb0:B2!B3bemorphismsofcomplexessuchthatA10 // B1// b C10 A2// B2// C2andA20 // B2// b0 C20 A3// B3// C3commuteinK(A).Thenb0b=0inK(A).Proof.ByLemma9.5wecanreplacebandb0byhomotopicmapssuchthattherightsquareoftheleftdiagramcommutesandtheleftsquareoftherightdiagramcommutes.Inotherwords,wehaveIm(bn)Im(An2!Bn2)andKer((b0)n)Im(An2!Bn2).Thenb0b=0asamapofcomplexes. DERIVEDCATEGORIES31Lemma9.13.014KLetAbeanadditivecategory.Letf1:K1!L1andf2:K2!L2bemorphismsofcomplexes.Let(a;b;c):(K1;L1;C(f1);f1;i1;p1)�!(K2;L2;C(f2);f2;i2;p2)beanymorphismoftrianglesofK(A).Ifaandbarehomotopyequivalencesthensoisc.Proof.Leta�1:K2!K1beamorphismofcomplexeswhichisinversetoainK(A).Letb�1:L2!L1beamorphismofcomplexeswhichisinversetobinK(A).Letc0:C(f2)!C(f1)bethemorphismfromLemma9.2ap
30 pliedtof1a�1=b�1f2.Ifwec
pliedtof1a�1=b�1f2.Ifwecanshowthatcc0andc0careisomorphismsinK(A)thenwewin.Henceitsu cestoprovethefollowing:Givenamorphismoftriangles(1;1;c):(K;L;C(f);f;i;p)inK(A)themorphismcisanisomorphisminK(A).ByassumptionthetwosquaresinthediagramL// 1 C(f)// c K[1]1 L// C(f)// K[1]commuteuptohomotopy.ByconstructionofC(f)therowsformtermwisesplitsequencesofcomplexes.Thusweseethat(c�1)2=0inK(A)byLemma9.12.HencecisanisomorphisminK(A)withinverse2�c.HenceifaandbarehomotopyequivalencesthentheresultingmorphismoftrianglesisanisomorphismoftrianglesinK(A).ItturnsoutthatthecollectionoftrianglesofK(A)givenbyconesandthecollectionoftrianglesofK(A)givenbytermwisesplitsequencesofcomplexesarethesameuptoisomorphisms,atleastuptosign!Lemma9.14.014LLetAbeanadditivecategory.(1)Givenatermwisesplitsequenceofcomplexes( :A!B;:B!C;sn;n)thereexistsahomotopyequivalenceC( )!CsuchthatthediagramA// B // C( )�p// A[1] A// B// C// A[1]de nesanisomorphismoftrianglesinK(A).(2)Givenamorphismofcomplexesf:K!LthereexistsanisomorphismoftrianglesK// ~L // M// K[1] K// L// C(f)�p// K[1]wheretheuppertriangleisthetriangleassociatedtoatermwisesplitexactsequenceK!~L!M. DERIVEDCATEGORIES32Proof.Proofof(1).WehaveC( )n=BnAn+1andwesimplyde neC( )n!CnviatheprojectionontoBnfollowedbyn.Thisde nesamorphismofcomplexesbecausethecompositionsAn+1!Bn+1!Cn+1arezero.TogetahomotopyinversewetakeC!C( )givenby(sn;�n)indegreen.ThisisamorphismofcomplexesbecausethemorphismncanbecharacterizedastheuniquemorphismCn!An+1suchthatdsn�sn+1d= n,seeproofofHomology,Lemma14.10.ThecompositionC!C( )!Cistheidentity.ThecompositionC( )!C!C( )isequaltothe
31 morphismsnn0�n
morphismsnn0�nn0Toseethatthisishomotopictotheidentitymapusethehomotopyhn:C( )n!C( )n�1givenbythematrix00n0:C( )n=BnAn+1!Bn�1An=C( )n�1Itistrivialtoverifythat1001�sn�n�n0=d n0�d00n0+00n+10d n+10�dTo nishtheproofof(1)wehavetoshowthatthemorphisms�p:C( )!A[1](seeDe nition9.1)andC( )!C!A[1]agreeuptohomotopy.Thisisclearfromtheabove.Namely,wecanusethehomotopyinverse(s;�):C!C( )andcheckinsteadthatthetwomapsC!A[1]agree.Andnotethatp(s;�)=�asdesired.Proofof(2).Welet~f:K!~L,s:L!~Land:~L!LbeasinLemma9.6.ByLemmas9.2and9.13thetriangles(K;L;C(f);i;p)and(K;~L;C(~f);~i;~p)areisomorphic.Notethatwecancomposeisomorphismsoftriangles.ThuswemayreplaceLby~Landfby~f.Inotherwordswemayassumethatfisatermwisesplitinjection.Inthiscasetheresultfollowsfrompart(1).Lemma9.15.014MLetAbeanadditivecategory.LetA1!A2!:::!Anbeasequenceofcomposablemorphismsofcomplexes.ThereexistsacommutativediagramA1// A2// :::// AnB1// OO B2// OO :::// BnOO suchthateachmorphismBi!Bi+1isasplitinjectionandeachBi!Aiisahomotopyequivalence.Moreover,ifallAiareinK+(A),K�(A),orKb(A),thensoaretheBi.Proof.Thecasen=1iswithoutcontent.Lemma9.6isthecasen=2.SupposewehaveconstructedthediagramexceptforBn.ApplyLemma9.6tothecompo-sitionBn�1!An�1!An.TheresultisafactorizationBn�1!Bn!Anasdesired.Lemma9.16.014NLetAbeanadditivecategory.Let( :A!B;:B!C;sn;n)beatermwisesplitsequenceofcomplexes.Let(A;B;C; ;;)be DERIVEDCATEGORIES33theassociatedtriangle.Thenthetriangle(C[�1];A;B;[�1]; ;)isisomorphictothetriangle(C[�1];A;C([�1]);[�
32 1];i;p).Proof.WewriteBn=AnCnandweide![1];i;p).Proof.WewriteBn=AnCnandweide](884125/1-i-p-proof-wewritebn-an-8-cnandweidenti.jpg "1];i;p).Proof.WewriteBn=AnCnandweide")
1];i;p).Proof.WewriteBn=AnCnandweidentify nandnwiththenaturalinclusionandprojectionmaps.ByconstructionofwehavednB=dnAn0dnCOntheotherhandtheconeof[�1]:C[�1]!AisgivenasC([�1])n=AnCnwithdierentialidenticalwiththematrixabove!Whencethelemma.Lemma9.17.014OLetAbeanadditivecategory.Letf:K!Lbeamorphismofcomplexes.Thetriangle(L;C(f);K[1];i;p;f[1])isthetriangleassociatedtothetermwisesplitsequence0!L!C(f)!K[1]!0comingfromthede nitionoftheconeoff.Proof.Immediatefromthede nitions.10.Distinguishedtrianglesinthehomotopycategory014PSincewewantourboundarymapsinlongexactsequencesofcohomologytobegivenbythemapsinthesnakelemmawithoutsignswede nedistinguishedtrianglesinthehomotopycategoryasfollows.De nition10.1.014QLetAbeanadditivecategory.Atriangle(X;Y;Z;f;g;h)ofK(A)iscalledadistinguishedtriangleofK(A)ifitisisomorphictothetriangleassociatedtoatermwisesplitexactsequenceofcomplexes,seeDe nition9.9.Samede nitionforK+(A),K�(A),andKb(A).NotethataccordingtoLemma9.14atriangleoftheform(K;L;C(f);f;i;�p)isadistinguishedtriangle.Thisdoesindeedleadtoatriangulatedcategory,seeProposition10.3.BeforewecanprovethepropositionweneedonemorelemmainordertobeabletoproveTR4.Lemma10.2.014RLetAbeanadditivecategory.Supposethat :A!Band:B!Caresplitinjectionsofcomplexes.Thenthereexistdistinguishedtriangles(A;B;Q1; ;p1;d1),(A;C;Q2; ;p2;d2)and(B;C;Q3;;p3;d3)forwhichTR4holds.Proof.Sayn1:Bn!An,andn3:Cn!Bnarethesplittings.ThenalsoA!Cisasplitinjectionwithsplittingsn2=n1n3.LetuswriteQ1,Q2andQ3forthequotientcomplexes.Inotherwords,Qn1=Ker(n1),Qn3=Ker(n3)andQn2=Ker(n2).Notethatthekernelsexist.ThenBn=AnQn1andCn=BnQn3,wherewethinkofAnasasubobjectofBnandsoon.ThisimpliesCn=AnQn1Qn3.Notethatn2=n1n3iszeroonbothQn1andQn3.HenceQn2=Qn1Qn3.Consid
33 erthecommutativediagram0!A!B!Q
erthecommutativediagram0!A!B!Q1!0###0!A!C!Q2!0###0!B!C!Q3!0 DERIVEDCATEGORIES34Therowsofthisdiagramaretermwisesplitexactsequences,andhencedeterminedistinguishedtrianglesbyde nition.Moreoverdownwardarrowsinthediagramabovearecompatiblewiththechosensplittingsandhencede nemorphismsoftriangles(A!B!Q1!A[1])�!(A!C!Q2!A[1])and(A!C!Q2!A[1])�!(B!C!Q3!B[1]):NotethatthesplittingsQn3!Cnofthebottomsplitsequenceinthediagramprovidesasplittingforthesplitsequence0!Q1!Q2!Q3!0uponcomposingwithCn!Qn2.Itfollowseasilyfromthisthatthemorphism:Q3!Q1[1]inthecorrespondingdistinguishedtriangle(Q1!Q2!Q3!Q1[1])isequaltothecompositionQ3!B[1]!Q1[1].HencewegetastructureasintheconclusionofaxiomTR4.Proposition10.3.014SLetAbeanadditivecategory.ThecategoryK(A)ofcom-plexesuptohomotopywithitsnaturaltranslationfunctorsanddistinguishedtrian-glesasde nedaboveisatriangulatedcategory.Proof.ProofofTR1.Byde nitioneverytriangleisomorphictoadistinguishedoneisdistinguished.Also,anytriangle(A;A;0;1;0;0)isdistinguishedsince0!A!A!0!0isatermwisesplitsequenceofcomplexes.Finally,givenanymorphismofcomplexesf:K!Lthetriangle(K;L;C(f);f;i;�p)isdistinguishedbyLemma9.14.ProofofTR2.Let(X;Y;Z;f;g;h)beatriangle.Assume(Y;Z;X[1];g;h;�f[1])isdistinguished.ThenthereexistsatermwisesplitsequenceofcomplexesA!B!Csuchthattheassociatedtriangle(A;B;C; ;;)isisomorphicto(Y;Z;X[1];g;h;�f[1]).Rotatingbackweseethat(X;Y;Z;f;g;h)isisomorphicto(C[�1];A;B;�[�1]; ;).ItfollowsfromLemma9.16thatthetriangle(C[�1];A;B;[�1]; ;)isisomorphicto(C[�1];A;C([�1]);[�1];i;p).Pre-composingthepreviousisomorphismoftriangleswith�1onYitfollowsthat(X;Y;Z;f;g;h)isisomorphicto(C[�1];A;C([�1]);[�1];i;�p).Henceitisdisti
34 nguishedbyLemma9.14.Ontheotherhand,suppo
nguishedbyLemma9.14.Ontheotherhand,supposethat(X;Y;Z;f;g;h)isdistinguished.ByLemma9.14thismeansthatitisisomorphictoatriangleoftheform(K;L;C(f);f;i;�p)forsomemorphismofcomplexesf.Thenthero-tatedtriangle(Y;Z;X[1];g;h;�f[1])isisomorphicto(L;C(f);K[1];i;�p;�f[1])whichisisomorphictothetriangle(L;C(f);K[1];i;p;f[1]).ByLemma9.17thistriangleisdistinguished.Hence(Y;Z;X[1];g;h;�f[1])isdistinguishedasdesired.ProofofTR3.Let(X;Y;Z;f;g;h)and(X0;Y0;Z0;f0;g0;h0)bedistinguishedtrian-glesofK(A)andleta:X!X0andb:Y!Y0bemorphismssuchthatf0a=bf.ByLemma9.14wemayassumethat(X;Y;Z;f;g;h)=(X;Y;C(f);f;i;�p)and(X0;Y0;Z0;f0;g0;h0)=(X0;Y0;C(f0);f0;i0;�p0).Atthispointwesimplyap-plyLemma9.2tothecommutativediagramgivenbyf;f0;a;b.ProofofTR4.AtthispointweknowthatK(A)isapre-triangulatedcategory.HencewecanuseLemma4.15.LetA!BandB!CbecomposablemorphismsofK(A).ByLemma9.15wemayassumethatA!BandB!Caresplitinjectivemorphisms.InthiscasetheresultfollowsfromLemma10.2. DERIVEDCATEGORIES35Remark10.4.05RPLetAbeanadditivecategory.ExactlythesameproofastheproofofProposition10.3showsthatthecategoriesK+(A),K�(A),andKb(A)aretriangulatedcategories.Namely,theconeofamorphismbetweenbounded(above,below)isbounded(above,below).ButweprovebelowthatthesearetriangulatedsubcategoriesofK(A)whichgivesanotherproof.Lemma10.5.05RQLetAbeanadditivecategory.ThecategoriesK+(A),K�(A),andKb(A)arefulltriangulatedsubcategoriesofK(A).Proof.Eachofthecategoriesmentionedisafulladditivesubcategory.WeusethecriterionofLemma4.16toshowthattheyaretriangulatedsubcategories.ItisclearthateachofthecategoriesK+(A),K�(A),andKb(A)ispreservedundertheshiftfunctors[1];[�1].Finally,supposethatf:A!BisamorphisminK+(A),K�(A),orKb(A).Then(A;B;C(f);f;i;�p)isadistinguishedtriangleofK(A)withC(f)2K+(A),K�(A),orKb(A)asisclearfromtheconstructionofthecone.Thusthelemmaisproved.(Alternatively,K!LisisomorphictoantermwisesplitinjectionofcomplexesinK+(A),K�(A),orKb(A),seeLemma9.6andthenonecandirectlyt
35 aketheassociateddistinguishedtriangle.)&
aketheassociateddistinguishedtriangle.)Lemma10.6.014XLetA,Bbeadditivecategories.LetF:A!Bbeanadditivefunctor.TheinducedfunctorsF:K(A)�!K(B)F:K+(A)�!K+(B)F:K�(A)�!K�(B)F:Kb(A)�!Kb(B)areexactfunctorsoftriangulatedcategories.Proof.SupposeA!B!CisatermwisesplitsequenceofcomplexesofAwithsplittings(sn;n)andassociatedmorphism:C!A[1],seeDe nition9.9.ThenF(A)!F(B)!F(C)isatermwisesplitsequenceofcomplexeswithsplittings(F(sn);F(n))andassociatedmorphismF():F(C)!F(A)[1].ThusFtransformsdistinguishedtrianglesintodistinguishedtriangles.Lemma10.7.0G6CLetAbeanadditivecategory.Let(A;B;C;a;b;c)beadistin-guishedtriangleinK(A).Thenthereexistsanisomorphicdistinguishedtriangle(A;(B0);C;a0;b0;c)suchthat0!An!(B0)n!Cn!0isasplitshortexactsequenceforalln.Proof.WewillusethatK(A)isatriangulatedcategorybyProposition10.3.LetWbetheconeonc:C!A[1]withitsmapsi:A[1]!Wandp:W!C[1].Then(C;A[1];W;c;i;�p)isadistinguishedtrianglebyLemma9.14.Rotatingbackwardstwiceweseethat(A;W[�1];C;�i[�1];p[�1];c)isadistinguishedtriangle.ByTR3thereisamorphismofdistinguishedtriangles(id;;id):(A;B;C;a;b;c)!(A;W[�1];C;�i[�1];p[�1];c)whichmustbeanisomorphismbyLemma4.3.This nishestheproofbecause0!A!W[�1]!C!0isatermwisesplitshortexactsequenceofcomplexesbytheveryconstructionofconesinSection9.Remark10.8.0G6DLetAbeanadditivecategorywithcountabledirectsums.LetDoubleComp(A)denotethecategoryofdoublecomplexesinA,seeHomology,Section18.Wecanusethiscategorytoconstructtwotriangulatedcategories. DERIVEDCATEGORIES36(1)WecanconsideranobjectA;ofDoubleComp(A)asacomplexofcom-plexesasfollows:::!A;�1!A;0!A;1!:::andtakethehomotopycategoryKfirst(DoubleComp(A))withthecorre-spondingtriangulatedstructuregivenbyProposition10.3.ByHomology,Remark18.6thefunctorTot:Kfirst(DoubleComp(A))�!K(A)isanexactfunctoroftriangulatedcategories.(2)Wec
36 anconsideranobjectA;ofDoubleCo
anconsideranobjectA;ofDoubleComp(A)asacomplexofcom-plexesasfollows:::!A�1;!A0;!A1;!:::andtakethehomotopycategoryKsecond(DoubleComp(A))withthecorre-spondingtriangulatedstructuregivenbyProposition10.3.ByHomology,Remark18.7thefunctorTot:Ksecond(DoubleComp(A))�!K(A)isanexactfunctoroftriangulatedcategories.Remark10.9.0G6ELetA,B,CbeadditivecategoriesandassumeChascountabledirectsums.Supposethat :AB�!C;(X;Y)7�!X Yisafunctorwhichisbilinearonmorphisms.ThisdeterminesafunctorComp(A)Comp(B)�!DoubleComp(C);(X;Y)7�!X YSeeHomology,Example18.2.(1)Fora xedobjectXofComp(A)thefunctorK(B)�!K(C);Y7�!Tot(X Y)isanexactfunctoroftriangulatedcategories.(2)Fora xedobjectYofComp(B)thefunctorK(A)�!K(C);X7�!Tot(X Y)isanexactfunctoroftriangulatedcategories.ThisfollowsfromRemark10.8sincethefunctorsComp(A)!DoubleComp(C),Y7!X YandComp(B)!DoubleComp(C),X7!X YareimmediatelyseentobecompatiblewithhomotopiesandtermwisesplitshortexactsequencesandhenceinduceexactfunctorsoftriangulatedcategoriesK(B)!Kfirst(DoubleComp(C))andK(A)!Ksecond(DoubleComp(C))Observethatforthe rstofthetwotheisomorphismTot(X Y[1])=Tot(X Y)[1]involvessigns(thisgoesbacktothesignschoseninHomology,Remark18.5). DERIVEDCATEGORIES3711.Derivedcategories05RRInthissectionweconstructthederivedcategoryofanabeliancategoryAbyinvert-ingthequasi-isomorphismsinK(A).BeforewedothisrecallthatthefunctorsHi:Comp(A)!AfactorthroughK(A),seeHomology,Lemma13.11.Moreover,inHomology,De nition14.8wehavede nedidenti cationsHi(K[n])=Hi+n(K).Atthispointitmakessensetorede neHi(K)=H0(K[i])inordertoavoidconfusionandpossiblesignerrors.Lemma11.1.05RSLetAbeanabeliancategory.ThefunctorH0:K(A)�!Aishomological.Proof.BecauseH0isafunctor,andbyourde nitionofdistinguishedtrianglesitsu cestoprovethatgivenatermwisesplitshortexactsequenceofcomplexes0!A!B!C!0thesequenceH0(A)!H0(B)!H0(C)isexact.Thisfollowsf
37 romHomology,Lemma13.12.Inparticular,
romHomology,Lemma13.12.Inparticular,thislemmaimpliesthatadistinguishedtriangle(X;Y;Z;f;g;h)inK(A)givesrisetoalongexactcohomologysequence(11.1.1)05ST:::// Hi(X)Hi(f)// Hi(Y)Hi(g)// Hi(Z)Hi(h)// Hi+1(X)// :::see(3.5.1).Moreover,thereisacompatibilitywiththelongexactsequenceofco-homologyassociatedtoashortexactsequenceofcomplexes(insertfuturereferencehere).Forexample,if(A;B;C; ;;)isthedistinguishedtriangleassociatedtoatermwisesplitexactsequenceofcomplexes(seeDe nition9.9),thentheco-homologysequenceaboveagreeswiththeonede nedusingthesnakelemma,seeHomology,Lemma13.12andforagreementofsequences,seeHomology,Lemma14.11.RecallthatacomplexKisacyclicifHi(K)=0foralli2Z.Moreover,recallthatamorphismofcomplexesf:K!Lisaquasi-isomorphismifandonlyifHi(f)isanisomorphismforalli.SeeHomology,De nition13.10.Lemma11.2.05RTLetAbeanabeliancategory.ThefullsubcategoryAc(A)ofK(A)consistingofacycliccomplexesisastrictlyfullsaturatedtriangulatedsubcategoryofK(A).Thecorrespondingsaturatedmultiplicativesystem(seeLemma6.10)ofK(A)isthesetQis(A)ofquasi-isomorphisms.Inparticular,thekernelofthelocalizationfunctorQ:K(A)!Qis(A)�1K(A)isAc(A)andthefunctorH0factorsthroughQ.Proof.WeknowthatH0isahomologicalfunctorbyLemma11.1.ThusthislemmaisaspecialcaseofLemma6.11.De nition11.3.05RULetAbeanabeliancategory.LetAc(A)andQis(A)beasinLemma11.2.ThederivedcategoryofAisthetriangulatedcategoryD(A)=K(A)=Ac(A)=Qis(A)�1K(A):WedenoteH0:D(A)!AtheuniquefunctorwhosecompositionwiththequotientfunctorgivesbackthefunctorH0de nedabove.UsingLemma6.4weintroduce DERIVEDCATEGORIES38thestrictlyfullsaturatedtriangulatedsubcategoriesD+(A);D�(A);Db(A)whosesetsofobjectsareOb(D+(A))=fX2Ob(D(A))jHn(X)=0foralln 0gOb(D�(A))=fX2Ob(D(A))jHn(X)=0foralln 0gOb(Db(A))=fX2Ob(D(A))jHn(X)=0foralljnj 0gThecategoryDb(A)iscalledtheboundedderivedcategoryofA.IfKandLarecomplexesofAthenwesometimessayKisquasi-isomorphictoLtoindicatethatKandLareisomorphicobjectsofD(A).Remark11.4.09PAInthischapter,weconsistentlyw
38 orkwithsmallabeliancate-gories
orkwithsmallabeliancate-gories(asistheconventionintheStacksproject).ForabigabeliancategoryA,itisn'tclearthatthederivedcategoryD(A)exists,becauseitisn'tclearthatmorphismsinthederivedcategoryaresets.Infact,ingeneraltheyaren't,seeEx-amples,Lemma60.1.However,ifAisaGrothendieckabeliancategory,andgivenK;LinK(A),thenbyInjectives,Theorem12.6thereexistsaquasi-isomorphismL!ItoaK-injectivecomplexIandLemma31.2showsthatHomD(A)(K;L)=HomK(A)(K;I)whichisaset.SomeexamplesofGrothendieckabeliancategoriesarethecategoryofmodulesoveraring,ormoregenerallythecategoryofsheavesofmodulesonaringedsite.EachofthevariantsD+(A);D�(A);Db(A)canbeconstructedasalocalizationofthecorrespondinghomotopycategory.Thisreliesonthefollowingsimplelemma.Lemma11.5.05RVLetAbeanabeliancategory.LetKbeacomplex.(1)IfHn(K)=0foralln 0,thenthereexistsaquasi-isomorphismK!LwithLboundedbelow.(2)IfHn(K)=0foralln 0,thenthereexistsaquasi-isomorphismM!KwithMboundedabove.(3)IfHn(K)=0foralljnj 0,thenthereexistsacommutativediagramofmorphismsofcomplexesK// LMOO // NOO whereallthearrowsarequasi-isomorphisms,Lboundedbelow,Mboundedabove,andNaboundedcomplex.Proof.Picka 0 bandsetM= bK,L= aK,andN= bL= aM.SeeHomology,Section15forthetruncationfunctors.TostatethefollowinglemmadenoteAc+(A),Ac�(A),resp.Acb(A)theintersectionofK+(A),K�(A),resp.Kb(A)withAc(A).DenoteQis+(A),Qis�(A),resp.Qisb(A)theintersectionofK+(A),K�(A),resp.Kb(A)withQis(A).Lemma11.6.05RWLetAbeanabeliancategory.ThesubcategoriesAc+(A),Ac�(A),resp.Acb(A)arestrictlyfullsaturatedtriangulatedsubcategoriesofK+(A),K�(A),resp.Kb(A).Thecorrespondingsaturatedmultiplicativesystems(seeLemma6.10)arethesetsQis+(A),Qis�(A),resp.Qisb(A). DERIVEDCATEGORIES39(1)ThekernelofthefunctorK+(A)!D+(A)isAc+(A)andthisinducesanequivalenceoftriangulatedcategoriesK+(A)=Ac+(A)=Qis+(A)�1K+(A)�!D+(A)(2)ThekernelofthefunctorK
39 �(A)!D�(A)isAc�(A)andthisinduce
�(A)!D�(A)isAc�(A)andthisinducesanequivalenceoftriangulatedcategoriesK�(A)=Ac�(A)=Qis�(A)�1K�(A)�!D�(A)(3)ThekernelofthefunctorKb(A)!Db(A)isAcb(A)andthisinducesanequivalenceoftriangulatedcategoriesKb(A)=Acb(A)=Qisb(A)�1Kb(A)�!Db(A)Proof.TheinitialstatementsfollowfromLemma6.11byconsideringtherestric-tionofthehomologicalfunctorH0.Thestatementonkernelsin(1),(2),(3)isaconsequenceofthede nitionsineachcase.EachofthefunctorsisessentiallysurjectivebyLemma11.5.To nishtheproofwehavetoshowthefunctorsarefullyfaithful.We rstdothisfortheboundedbelowversion.SupposethatK;Lareboundedabovecomplexes.AmorphismbetweentheseinD(A)isoftheforms�1fforapairf:K!(L0),s:L!(L0)wheresisaquasi-isomorphism.Thisimpliesthat(L0)hascohomologyboundedbelow.HencebyLemma11.5wecanchooseaquasi-isomorphisms0:(L0)!(L00)with(L00)boundedbelow.Thenthepair(s0f;s0s)de nesamorphisminQis+(A)�1K+(A).Hencethefunctorisfull.Finally,supposethatthepairf:K!(L0),s:L!(L0)de nesamorphisminQis+(A)�1K+(A)whichiszeroinD(A).Thismeansthatthereexistsaquasi-isomorphisms0:(L0)!(L00)suchthats0f=0.UsingLemma11.5oncemoreweobtainaquasi-isomorphisms00:(L00)!(L000)with(L000)boundedbelow.Thusweseethats00s0f=0whichimpliesthats�1fiszeroinQis+(A)�1K+(A).This nishestheproofthatthefunctorin(1)isanequivalence.Theproofof(2)isdualtotheproofof(1).Toprove(3)wemayusetheresultof(2).Henceitsu cestoprovethatthefunctorQisb(A)�1Kb(A)!Qis�(A)�1K�(A)isfullyfaithful.Theargumentgiveninthepreviousparagraphappliesdirectlytoshowthiswhereweconsistentlyworkwithcomplexeswhicharealreadyboundedabove.12.Thecanonicaldelta-functor014ZThederivedcategoryshouldbethereceptaclefortheuniversalcohomologyfunctor.Inordertostatetheresultweusethenotionofa-functorfromanabeliancategoryintoatriangulatedcategory,seeDe nition3.6.ConsiderthefunctorComp(A)!K(A).Thisfunctorisnota-functoringeneral.Theeasiestwaytoseethisistoconsideranonsplits
40 hortexactsequence0!A!B!C!0ofobjectsofA.S
hortexactsequence0!A!B!C!0ofobjectsofA.SinceHomK(A)(C[0];A[1])=0weseethatanydistinguishedtrianglearisingfromthisshortexactsequencewouldlooklike(A[0];B[0];C[0];a;b;0).ButtheexistenceofsuchadistinguishedtriangleinK(A)impliesthattheextensionissplit.Acontradiction. DERIVEDCATEGORIES40ItturnsoutthatthefunctorComp(A)!D(A)isa-functor.Inordertoseethiswehavetode nethemorphismsassociatedtoashortexactsequence0!Aa�!Bb�!C!0ofcomplexesintheabeliancategoryA.ConsidertheconeC(a)ofthemorphisma.WehaveC(a)n=BnAn+1andwede neqn:C(a)n!CnviatheprojectiontoBnfollowedbybn.Henceamorphismofcomplexesq:C(a)�!C:Itisclearthatqi=bwhereiisasinDe nition9.1.Notethat,asaisinjectiveineachdegree,thekernelofqisidenti edwiththeconeofidAwhichisacyclic.Henceweseethatqisaquasi-isomorphism.AccordingtoLemma9.14thetriangle(A;B;C(a);a;i;�p)isadistinguishedtriangleinK(A).AsthelocalizationfunctorK(A)!D(A)isexactweseethat(A;B;C(a);a;i;�p)isadistinguishedtriangleinD(A).Sinceqisaquasi-isomorphismweseethatqisanisomorphisminD(A).Hencewededucethat(A;B;C;a;b;�pq�1)isadistinguishedtriangleofD(A).Thissuggeststhefollowinglemma.Lemma12.1.0152LetAbeanabeliancategory.ThefunctorComp(A)!D(A)de nedhasthenaturalstructureofa-functor,withA!B!C=�pq�1withpandqasexplainedabove.ThesameconstructionturnsthefunctorsComp+(A)!D+(A),Comp�(A)!D�(A),andCompb(A)!Db(A)into-functors.Proof.Wehavealreadyseenthatthischoiceleadstoadistinguishedtrianglewhenevergivenashortexactsequenceofcomplexes.Wehavetoshowthatgivenacommutativediagram0// Aa// f Bb// g C// h 00// (A0)a0// (B0)b0// (C0)// 0wegetthedesiredcommutativediagramofDe nition3.6(2).ByLemma9.2thepair(f;g)inducesacanonicalmorphismc:C(a)!C(a0).Itisasimplecomputationtoshowthatq0c=hqandf[1]p=p0c.Fromthistheresultfollowsdirectly.Lemma12.2.0153LetAbeanabeliancategory.Let0// A// B// C// 00// D&#
41 15;// E// F// 0beacommutatived
15;// E// F// 0beacommutativediagramofmorphismsofcomplexessuchthattherowsareshortexactsequencesofcomplexes,andtheverticalarrowsarequasi-isomorphisms.The-functorofLemma12.1abovemapstheshortexactsequences0!A!B!C!0and0!D!E!F!0toisomorphicdistinguishedtriangles. DERIVEDCATEGORIES41Proof.TrivialfromthefactthatK(A)!D(A)transformsquasi-isomorphismsintoisomorphismsandthattheassociateddistinguishedtrianglesarefunctorial.Lemma12.3.0154LetAbeanabeliancategory.Let0// A// B// C// 0beashortexactsequencesofcomplexes.Assumethisshortexactsequenceistermwisesplit.Let(A;B;C; ;;)bethedistinguishedtriangleofK(A)asso-ciatedtothesequence.The-functorofLemma12.1abovemapstheshortexactsequences0!A!B!C!0toatriangleisomorphictothedistinguishedtriangle(A;B;C; ;;):Proof.FollowsfromLemma9.14.Remark12.4.08J5LetAbeanabeliancategory.LetKbeacomplexofA.Leta2Z.Weclaimthereisacanonicaldistinguishedtriangle aK!K! a+1K!( aK)[1]inD(A).Herewehaveusedthecanonicaltruncationfunctors fromHomology,Section15.Namely,we rsttakethedistinguishedtriangleassociatedbyour-functor(Lemma12.1)totheshortexactsequenceofcomplexes0! aK!K!K= aK!0Next,weusethatthemapK! a+1Kfactorsthroughaquasi-isomorphismK= aK! a+1KbythedescriptionofcohomologygroupsinHomology,Section15.Inasimilarwayweobtaincanonicaldistinguishedtriangles aK! a+1K!Ha+1(K)[�a�1]!( aK)[1]andHa(K)[�a]! aK! a+1K!Ha(K)[�a+1]Lemma12.5.08Q2LetAbeanabeliancategory.LetK0!K1!:::!Knbemapsofcomplexessuchthat(1)Hi(K0)=0fori�0,(2)H�j(Kj)!H�j(Kj+1)iszero.ThenthecompositionK0!Knfactorsthrough �nKn!KninD(A).Dually,givenmapsofcomplexesKn!Kn�1!:::!K0suchthat(1)Hi(K0)=0fori0,(2)Hj(K
42 j+1)!Hj(Kj)iszero,thenthecompo
j+1)!Hj(Kj)iszero,thenthecompositionKn!K0factorsthroughKn! nKninD(A). DERIVEDCATEGORIES42Proof.Thecasen=1.Since 0K0=K0inD(A)wecanreplaceK0by 0K0andK1by 0K1.Considerthedistinguishedtriangle �1K1!K1!H0(K1)[0]!( �1K1)[1](Remark12.4).ThecompositionK0!K1!H0(K1)[0]iszeroasitisequaltoK0!H0(K0)[0]!H0(K1)[0]whichiszerobyassumption.ThefactthatHomD(A)(K0;�)isahomologicalfunctor(Lemma4.2),allowsusto ndthedesiredfactorization.Forn=2wegetafactorizationK0! �1K1bythecasen=1andwecanapplythecasen=1tothemapofcomplexes �1K1! �1K2togetafactorization �1K1! �2K2.Thegeneralcaseisprovedinexactlythesamemanner.13.Filteredderivedcategories05RXAreferenceforthissectionis[Ill72,I,ChapterV].LetAbeanabeliancategory.Inthissectionwewillde nethe lteredderivedcategoryDF(A)ofA.Inshort,wewillde neitasthederivedcategoryoftheexactcategoryofobjectsofAendowedwitha nite ltration.(Thusourconstructionisaspecialcaseofamoregeneralconstructionofthederivedcategoryofanexactcategory,seeforexample[Büh10],[Kel90].)Illusie's lteredderivedcategoryisthefullsubcategoryofoursconsistingofthoseobjectswhose ltrationis nite.(Inourcategorythe ltrationisstill niteineachdegree,butmaynotbeuniformlybounded.)TherationaleforourchoiceisthatitisnotharderanditallowsustoapplythediscussiontothespectralsequencesofLemma21.3,seealsoRemark21.4.Wewillusethenotationregarding lteredobjectsintroducedinHomology,Section19.Thecategoryof lteredobjectsofAisdenotedFil(A).All ltrationswillbedecreasingby at.De nition13.1.05RYLetAbeanabeliancategory.Thecategoryof nite lteredobjectsofAisthecategoryof lteredobjects(A;F)ofAwhose ltrationFis nite.WedenoteitFilf(A).ThusFilf(A)isafullsubcategoryofFil(A).Foreachp2Zthereisafunctorgrp:Filf(A)!A.Thereisafunctorgr=Mp2Zgrp:Filf(A)!Gr(A)whereGr(A)isthe
43 categoryofgradedobjectsofA,seeHomology,D
categoryofgradedobjectsofA,seeHomology,De nition16.1.Finally,thereisafunctor(forgetF):Filf(A)�!Awhichassociatestothe lteredobject(A;F)theunderlyingobjectofA.ThecategoryFilf(A)isanadditivecategory,butnotabelianingeneral,seeHomology,Example3.13.Becausethefunctorsgrp,gr,(forgetF)areadditivetheyinduceexactfunctorsoftriangulatedcategoriesgrp;(forgetF):K(Filf(A))!K(A)andgr:K(Filf(A))!K(Gr(A))byLemma10.6.Byanalogywiththecaseofthehomotopycategoryofanabeliancategorywemakethefollowingde nitions. DERIVEDCATEGORIES43De nition13.2.05RZLetAbeanabeliancategory.(1)Let :K!LbeamorphismofK(Filf(A)).Wesaythat isa lteredquasi-isomorphismifthemorphismgr( )isaquasi-isomorphism.(2)LetKbeanobjectofK(Filf(A)).WesaythatKis lteredacyclicifthecomplexgr(K)isacyclic.Notethat :K!Lisa lteredquasi-isomorphismifandonlyifeachgrp( )isaquasi-isomorphism.SimilarlyacomplexKis lteredacyclicifandonlyifeachgrp(K)isacyclic.Lemma13.3.05S0LetAbeanabeliancategory.(1)ThefunctorK(Filf(A))�!Gr(A),K7�!H0(gr(K))ishomological.(2)ThefunctorK(Filf(A))!A,K7�!H0(grp(K))ishomological.(3)ThefunctorK(Filf(A))�!A,K7�!H0((forgetF)K)ishomological.Proof.ThisfollowsfromthefactthatH0:K(A)!Aishomological,seeLemma11.1andthefactthatthefunctorsgr;grp;(forgetF)areexactfunctorsoftriangu-latedcategories.SeeLemma4.20.Lemma13.4.05S1LetAbeanabeliancategory.ThefullsubcategoryFAc(A)ofK(Filf(A))consistingof lteredacycliccomplexesisastrictlyfullsaturatedtrian-gulatedsubcategoryofK(Filf(A)).Thecorrespondingsaturatedmultiplicativesys-tem(seeLemma6.10)ofK(Filf(A))isthesetFQis(A)of lteredquasi-isomorphisms.Inparticular,thekernelofthelocalizationfunctorQ:K(Filf(A))�!FQis(A)�1K(Filf(A))isFAc(A)andthefunctorH0grfactorsthroughQ.Proof.WeknowthatH0grisahomologicalfunctorbyLemma13.3.ThusthislemmaisaspecialcaseofLemma6.11.De nition13.5.05S2LetAbeanabeliancategory.LetFAc(A)andFQis(A)beasinLemma13.4.The lteredderivedcategoryofAisthetriangulatedcategoryDF(A)=K(
44 Filf(A))=FAc(A)=FQis(A)�1K(Filf(A)):L
Filf(A))=FAc(A)=FQis(A)�1K(Filf(A)):Lemma13.6.05S3Thefunctorsgrp;gr;(forgetF)inducecanonicalexactfunctorsgrp;gr;(forgetF):DF(A)�!D(A)whichcommutewiththelocalizationfunctors.Proof.Thisfollowsfromtheuniversalpropertyoflocalization,seeLemma5.6,providedwecanshowthata lteredquasi-isomorphismisturnedintoaquasi-isomorphismbyeachofthefunctorsgrp;gr;(forgetF).Thisistruebyde nitionforthe rsttwo.Forthelastonethestatementwehavetodoalittlebitofwork.Letf:K!Lbea lteredquasi-isomorphisminK(Filf(A)).Chooseadistinguishedtriangle(K;L;M;f;g;h)whichcontainsf.ThenMis lteredacyclic,seeLemma13.4.HencebythecorrespondinglemmaforK(A)itsu cestoshowthata lteredacycliccomplexisanacycliccomplexifweforgetthe ltration.ThisfollowsfromHomology,Lemma19.15. DERIVEDCATEGORIES44De nition13.7.05S4LetAbeanabeliancategory.Thebounded lteredderivedcategoryDFb(A)isthefullsubcategoryofDF(A)withobjectsthoseXsuchthatgr(X)2Db(A).Similarlyfortheboundedbelow lteredderivedcategoryDF+(A)andtheboundedabove lteredderivedcategoryDF�(A).Lemma13.8.05S5LetAbeanabeliancategory.LetK2K(Filf(A)).(1)IfHn(gr(K))=0forallna,thenthereexistsa lteredquasi-isomorphismK!LwithLn=0forallna.(2)IfHn(gr(K))=0foralln&#x-278;b,thenthereexistsa lteredquasi-isomorphismM!KwithMn=0foralln&#x-278;b.(3)IfHn(gr(K))=0foralljnj 0,thenthereexistsacommutativediagramofmorphismsofcomplexesK// LMOO // NOO whereallthearrowsare lteredquasi-isomorphisms,Lboundedbelow,Mboundedabove,andNaboundedcomplex.Proof.SupposethatHn(gr(K))=0forallna.ByHomology,Lemma19.15thesequenceKa�1da�2���!Ka�1da�1���!KaisanexactsequenceofobjectsofAandthemorphismsda�2andda�1arestrict.HenceCoim(da�1)=Im(da�1)inFilf(A)andthemapgr(Im(da�1))!gr(Ka)isinjectivewithimageequaltotheimageofgr(Ka�1)!gr(Ka),seeHomology,Lemma19.13.ThismeansthatthemapK! aKintothetruncation aK=(:::!0!Ka=Im(da�1)!Ka+1!:::)isa ltered
45 quasi-isomorphism.Thisproves(1).Theproof
quasi-isomorphism.Thisproves(1).Theproofof(2)isdualtotheproofof(1).Part(3)followsformallyfrom(1)and(2).TostatethefollowinglemmadenoteFAc+(A),FAc�(A),resp.FAcb(A)theinter-sectionofK+(FilfA),K�(FilfA),resp.Kb(FilfA)withFAc(A).DenoteFQis+(A),FQis�(A),resp.FQisb(A)theintersectionofK+(FilfA),K�(FilfA),resp.Kb(FilfA)withFQis(A).Lemma13.9.05S6LetAbeanabeliancategory.ThesubcategoriesFAc+(A),FAc�(A),resp.FAcb(A)arestrictlyfullsaturatedtriangulatedsubcategoriesofK+(FilfA),K�(FilfA),resp.Kb(FilfA).Thecorrespondingsaturatedmultiplicativesystems(seeLemma6.10)arethesetsFQis+(A),FQis�(A),resp.FQisb(A).(1)ThekernelofthefunctorK+(FilfA)!DF+(A)isFAc+(A)andthisinducesanequivalenceoftriangulatedcategoriesK+(FilfA)=FAc+(A)=FQis+(A)�1K+(FilfA)�!DF+(A)(2)ThekernelofthefunctorK�(FilfA)!DF�(A)isFAc�(A)andthisinducesanequivalenceoftriangulatedcategoriesK�(FilfA)=FAc�(A)=FQis�(A)�1K�(FilfA)�!DF�(A) DERIVEDCATEGORIES45(3)ThekernelofthefunctorKb(FilfA)!DFb(A)isFAcb(A)andthisin-ducesanequivalenceoftriangulatedcategoriesKb(FilfA)=FAcb(A)=FQisb(A)�1Kb(FilfA)�!DFb(A)Proof.Thisfollowsfromtheresultsabove,inparticularLemma13.8,byexactlythesameargumentsasusedintheproofofLemma11.6.14.Derivedfunctorsingeneral05S7AreferenceforthissectionisDeligne'sexposéXVIIin[AGV71].Averygeneralnotionofrightandleftderivedfunctorsexistswherewehaveanexactfunctorbetweentriangulatedcategories,amultiplicativesysteminthesourcecategoryandwewantto ndthecorrectextensionoftheexactfunctortothelocalizedcategory.Situation14.1.05S8HereF:D!D0isanexactfunctoroftriangulatedcategoriesandSisasaturatedmultiplicativesysteminDcompatiblewiththestructureoftriangulatedcategoryonD.LetX2Ob(D).RecallfromCategories,Remark27.7the lteredcategoryX=Sofarrowss:X!X0inSwithsourceX.Dually,inCategories,Remark27.15wede nedtheco lteredcategoryS=Xofarrowss:X0!XinSwithtargetX.De nition14.2.05S9AssumptionsandnotationasinSituation14.1.LetX2Ob(D).(1)wesaytherightderivedfunctorRFisde nedatXiftheind-object(X=S)�!D0;(s:X!X0)7�!F(X0)isessenti
46 allyconstant4;inthiscasethevalueYinD0isc
allyconstant4;inthiscasethevalueYinD0iscalledthevalueofRFatX.(2)wesaytheleftderivedfunctorLFisde nedatXifthepro-object(S=X)�!D0;(s:X0!X)7�!F(X0)isessentiallyconstant;inthiscasethevalueYinD0iscalledthevalueofLFatX.ByabuseofnotationweoftendenotethevaluessimplyRF(X)orLF(X).ItwillturnoutthatthefullsubcategoryofDconsistingofobjectswhereRFisde- nedisatriangulatedsubcategory,andRFwillde neafunctoronthissubcategorywhichtransformsmorphismsofSintoisomorphisms.Lemma14.3.05SAAssumptionsandnotationasinSituation14.1.Letf:X!YbeamorphismofD.(1)IfRFisde nedatXandYthenthereexistsauniquemorphismRF(f):RF(X)!RF(Y)betweenthevaluessuchthatforanycommutativedia-gramXf s// X0f0 Ys0// Y0 4Foradiscussionofwhenanind-objectorpro-objectofacategoryisessentiallyconstantwerefertoCategories,Section22. DERIVEDCATEGORIES46withs;s02SthediagramF(X) // F(X0) // RF(X) F(Y)// F(Y0)// RF(Y)commutes.(2)IfLFisde nedatXandYthenthereexistsauniquemorphismLF(f):LF(X)!LF(Y)betweenthevaluessuchthatforanycommutativedia-gramX0f0 s// Xf Y0s0// Ywiths;s0inSthediagramLF(X) // F(X0) // F(X) LF(Y)// F(Y0)// F(Y)commutes.Proof.Part(1)holdsifweonlyassumethatthecolimitsRF(X)=colims:X!X0F(X0)andRF(Y)=colims0:Y!Y0F(Y0)exist.Namely,togiveamorphismRF(X)!RF(Y)betweenthecolimitsisthesamethingasgivingforeachs:X!X0inOb(X=S)amorphismF(X0)!RF(Y)compatiblewithmorphismsinthecategoryX=S.TogetthemorphismwechooseacommutativediagramXf s// X0f0 Ys0// Y0withs;s0inSasispossiblebyMS2andwesetF(X0)!RF(Y)equaltothecompositionF(X0)!F(Y0)!RF(Y).ToseethatthisisindependentofthechoiceofthediagramaboveuseMS3.Detailsomitted.Theproofof(2)isdual.Lemma14.4.05SBAssumptionsandnotationasinSituation14.1.Lets:X!YbeanelementofS.(1)RFisde nedatXifandonlyifitisde nedatY.InthiscasethemapRF(s):RF(X)!RF(Y)betweenvaluesisanisomorphism.(2)LFisde nedatXifandonlyifitisde nedatY.InthiscasethemapLF(s):LF(X)!LF(Y)betweenvaluesisanisomorphism.Proof.Omitted.Lemma14.5.05SUAssumptionsandnotationa
47 sinSituation14.1.LetXbeanobjectofDandn2Z
sinSituation14.1.LetXbeanobjectofDandn2Z. DERIVEDCATEGORIES47(1)RFisde nedatXifandonlyifitisde nedatX[n].InthiscasethereisacanonicalisomorphismRF(X)[n]=RF(X[n])betweenvalues.(2)LFisde nedatXifandonlyifitisde nedatX[n].InthiscasethereisacanonicalisomorphismLF(X)[n]!LF(X[n])betweenvalues.Proof.Omitted.Lemma14.6.05SCAssumptionsandnotationasinSituation14.1.Let(X;Y;Z;f;g;h)beadistinguishedtriangleofD.IfRFisde nedattwooutofthreeofX;Y;Z,thenitisde nedatthethird.Moreover,inthiscase(RF(X);RF(Y);RF(Z);RF(f);RF(g);RF(h))isadistinguishedtriangleinD0.SimilarlyforLF.Proof.SayRFisde nedatX;YwithvaluesA;B.LetRF(f):A!Bbetheinducedmorphism,seeLemma14.3.Wemaychooseadistinguishedtriangle(A;B;C;RF(f);b;c)inD0.WeclaimthatCisavalueofRFatZ.Toseethispicks:X!X0inSsuchthatthereexistsamorphism :A!F(X0)asinCategories,De nition22.1.WemaychooseacommutativediagramXf s// X0f0 Ys0// Y0withs02SbyMS2.UsingthatY=Sis lteredwecan(afterreplacings0bysomes00:Y!Y00inS)assumethatthereexistsamorphism:B!F(Y0)asinCategories,De nition22.1.PictureARF(f) // F(X0)// F(f0) ARF(f) B// F(Y0)// BItmaynotbetruethattheleftsquarecommutes,buttheouterandrightsquarescommute.Theassumptionthattheind-objectfF(Y0)gs0:Y0!Yisessentiallycon-stantmeansthatthereexistsas00:Y!Y00inSandamorphismh:Y0!Y00suchthats00=hs0andsuchthatF(h)equaltoF(Y0)!B!F(Y0)!F(Y00).HenceafterreplacingY0byY00andbyF(h)thediagramwillcommute(bydirectcomputationwitharrows).UsingMS6chooseamorphismoftriangles(s;s0;s00):(X;Y;Z;f;g;h)�!(X0;Y0;Z0;f0;g0;h0)withs002S.ByTR3chooseamorphismoftriangles( ;; ):(A;B;C;RF(f);b;c)�!(F(X0);F(Y0);F(Z0);F(f0);F(g0);F(h0))ByLemma14.4itsu cestoprovethatRF(Z0)isde nedandhasvalueC.ConsiderthecategoryIofLemma5.8oftrianglesI=f(t;t0;t00):(X0;Y0;Z0;f0;g0;h0)!(X00;Y00;Z00;f00;g00;h00)j(t;t0;t00)2SgToshowthatthesystemF(Z00)isessentiallyconstantoverthecategoryZ0=SisequivalenttoshowingthatthesystemofF(Z00)isessentiallyconstantoverI DERIVEDCATEGORIES48becauseI!Z0=Sisco nal
48 ,seeCategories,Lemma22.11(co nalityi
,seeCategories,Lemma22.11(co nalityisproveninLemma5.8).ForanyobjectWinD0weconsiderthediagramcolimIMorD0(W;F(X00))MorD0(W;A)oo colimIMorD0(W;F(Y00))OO MorD0(W;B)OO oo colimIMorD0(W;F(Z00))OO MorD0(W;C)OO oo colimIMorD0(W;F(X00[1]))OO MorD0(W;A[1])OO oo colimIMorD0(W;F(Y00[1]))OO MorD0(W;B[1])OO oo wherethehorizontalarrowsaregivenbycomposingwith( ;; ).Since lteredcolimitsareexact(Algebra,Lemma8.8)theleftcolumnisanexactsequence.Thusthe5lemma(Homology,Lemma5.20)tellsusthecolimIMorD0(W;F(Z00))�!MorD0(W;C)isbijective.Chooseanobject(t;t0;t00):(X0;Y0;Z0)!(X00;Y00;Z00)ofI.ApplyingwhatwejustshowedtoW=F(Z00)andtheelementidF(X00)ofthecolimitwe ndauniquemorphismc(X00;Y00;Z00):F(Z00)!Csuchthatforsome(X00;Y00;Z00)!(X000;Y000;Z00)inIF(Z00)c(X00;Y00;Z00)��������!C �!F(Z0)!F(Z00)!F(Z000)equalsF(Z00)!F(Z000)Thefamilyofmorphismsc(X00;Y00;Z00)formanelementcoflimIMorD0(F(Z00);C)byuniqueness(computationomitted).Finally,weshowthatcolimIF(Z00)=Cviathemorphismsc(X00;Y00;Z00)whichwill nishtheproofbyCategories,Lemma22.9.Namely,letWbeanobjectofD0andletd(X00;Y00;Z00):F(Z00)!WbeafamilyofmapscorrespondingtoanelementoflimIMorD0(F(Z00);W).Ifd(X0;Y0;Z0) =0,thenforeveryobject(X00;Y00;Z00)ofIthemorphismd(X00;Y00;Z00)iszerobytheexistenceofc(X00;Y00;Z00)andthemorphism(X00;Y00;Z00)!(X000;Y000;Z00)inIsatisfyingthedisplayedequalityabove.HencethemaplimIMorD0(F(Z00);W)�!MorD0(C;W)(comingfromprecomposingby )isinjective.However,itisalsosurjectivebecausetheelementcgivesaleftinverse.WeconcludethatCisthecolimitbyCategories,Remark14.4.Lemma14.7.05SDAssumptionsandnotationasinSituation14.1.LetX;YbeobjectsofD.(1)IfRFisde nedatXandY,thenRFisde nedatXY.(2)IfD0isKaroubianandRFisde nedatXY,thenRFisde nedatbothXandY.IneithercasewehaveRF(XY)=RF(X)RF(Y).SimilarlyforLF. DERIVEDCATEGORIES49Proof.IfRFisde nedatXandY,thenthedistinguishedtriangleX!XY!Y!X[1](Lemma4.11)andLemma14.6showsthatRFisde nedatXYandthatwehaveadistinguishedtriangleRF(X)!RF(XY)!RF(Y)!RF(X)[1].ApplyingLemma4.11tothisoncemorewe
49 ndthatRF(XY)=RF(X)RF(Y).Thi
ndthatRF(XY)=RF(X)RF(Y).Thisproves(1)andthe nalassertion.Conversely,assumethatRFisde nedatXYandthatD0isKaroubian.SinceSisasaturatedsystemSisthesetofarrowswhichbecomeinvertibleundertheadditivelocalizationfunctorQ:D!S�1D,seeCategories,Lemma27.21.Thusforanys:X!X0ands0:Y!Y0inSthemorphismss0:XY!X0Y0isanelementofS.InthiswayweobtainafunctorX=SY=S�!(XY)=SRecallthatthecategoriesX=S;Y=S;(XY)=Sare ltered(Categories,Remark27.7).ByCategories,Lemma22.12X=SY=Sis lteredandFjX=S:X=S!D0(resp.GjY=S:Y=S!D0)isessentiallyconstantifandonlyifFjX=Spr1:X=SY=S!D0(resp.GjY=Spr2:X=SY=S!D0)isessentiallyconstant.Belowwewillshowthatthedisplayedfunctorisco nal,hencebyCategories,Lemma22.11.weseethatFj(XY)=SisessentiallyconstantimpliesthatFjX=Spr1FjY=Spr2:X=SY=S!D0isessentiallyconstant.ByHomology,Lemma30.3(andthisiswhereweusethatD0isKaroubian)weseethatFjX=Spr1FjY=Spr2beingessentiallyconstantimpliesFjX=Spr1andFjY=Spr2areessentiallyconstantprovingthatRFisde nedatXandY.Proofthatthedisplayedfunctorisco nal.Todothispickanyt:XY!ZinS.UsingMS2wecan ndmorphismsZ!X0,Z!Y0ands:X!X0,s0:Y!Y0inSsuchthatXs XY oo // Ys0 X0Zoo // Y0commutes.ThisprovesthereisamapZ!X0Y0in(XY)=S,i.e.,wegetpart(1)ofCategories,De nition17.1.Toprovepart(2)itsu cestoprovethatgivent:XY!Zandmorphismssis0i:Z!X0iY0i,i=1;2in(XY)=Swecan ndmorphismsa:X01!X0,b:X02!X0,c:Y01!Y0,d:Y02!Y0inSsuchthatas1=bs2andcs01=ds02.Todothiswe rstchooseanyX0andY0andmapsa;b;c;dinS;thisispossibleasX=SandY=Sare ltered.Thenthetwomapsas1;bs2:Z!X0becomeequalinS�1D.Hencewecan ndamorphismX0!X00inSequalizingthem.Similarlywe ndY0!Y00inSequalizingcs01andds02.ReplacingX0byX00andY0byY00wegetas1=bs2andcs01=ds02.TheproofofthecorrespondingstatementsforLFaredual.Proposition14.8.05SEAssumptionsandnotationasinSituation14.1.(1)ThefullsubcategoryEofDconsistingofobjectsatwhi
50 chRFisde nedisastrictlyfulltriangula
chRFisde nedisastrictlyfulltriangulatedsubcategoryofD.(2)WeobtainanexactfunctorRF:E�!D0oftriangulatedcategories.(3)ElementsofSwitheithersourceortargetinEaremorphismsofE.(4)ThefunctorS�1EE!S�1Disafullyfaithfulexactfunctoroftriangulatedcategories. DERIVEDCATEGORIES50(5)AnyelementofSE=Arrows(E)\SismappedtoanisomorphismbyRF.(6)WeobtainanexactfunctorRF:S�1EE�!D0:(7)IfD0isKaroubian,thenEisasaturatedtriangulatedsubcategoryofD.AsimilarresultholdsforLF.Proof.SinceSissaturateditcontainsallisomorphisms(seeremarkfollowingCategories,De nition27.20).Hence(1)followsfromLemmas14.4,14.6,and14.5.Weget(2)fromLemmas14.3,14.5,and14.6.Weget(3)fromLemma14.4.Thefullyfaithfulnessin(4)followsfrom(3)andthede nitions.ThefactthatS�1EE!S�1DisexactfollowsfromthefactthatatriangleinS�1EEisdistinguishedifandonlyifitisisomorphictotheimageofadistinguishedtriangleinE,seeproofofProposition5.5.Part(5)followsfromLemma14.4.ThefactorizationofRF:E!D0throughanexactfunctorS�1EE!D0followsfromLemma5.6.Part(7)followsfromLemma14.7.Proposition14.8tellsusthatRFlivesonamaximalstrictlyfulltriangulatedsub-categoryofS�1Dandisanexactfunctoronthistriangulatedcategory.Picture:DQ F// D0S�1DS�1EEfullyfaithfulexactoo RF De nition14.9.05SVInSituation14.1.WesayFisrightderivable,orthatRFeverywherede nedifRFisde nedateveryobjectofD.WesayFisleftderivable,orthatLFeverywherede nedifLFisde nedateveryobjectofD.Inthiscaseweobtainaright(resp.left)derivedfunctor(14.9.1)05SWRF:S�1D�!D0;(resp.LF:S�1D�!D0);seeProposition14.8.InmostinterestingsituationsitisnotthecasethatRFQisequaltoF.Infact,itmighthappenthatthecanonicalmapF(X)!RF(X)isneveranisomorphism.Inpracticethisdoesnothappen,becauseinpracticeweonlyknowhowtoproveFisrightderivablebyshowingthatRFcanbecomputedbyevaluatingFatjudiciouslychosenobjectsofthetriangulatedcategoryD.Thiswarrantsade nition.De nition14.10.05SXInSituation14.1.(1)AnobjectXofDcomputesRFifRFisde nedatXandthecanonicalmapF(X)!RF(X)isanisomorphism.(2)AnobjectXofDcomputesLFifLFisde nedatXandtheca
51 nonicalmapLF(X)!F(X)isanisomorphism.Lemm
nonicalmapLF(X)!F(X)isanisomorphism.Lemma14.11.05SYAssumptionsandnotationasinSituation14.1.LetXbeanobjectofDandn2Z.(1)XcomputesRFifandonlyifX[n]computesRF.(2)XcomputesLFifandonlyifX[n]computesLF.Proof.Omitted. DERIVEDCATEGORIES51Lemma14.12.05SZAssumptionsandnotationasinSituation14.1.Let(X;Y;Z;f;g;h)beadistinguishedtriangleofD.IfX;YcomputeRFthensodoesZ.SimilarforLF.Proof.ByLemma14.6weknowthatRFisde nedatZandthatRFappliedtothetriangleproducesadistinguishedtriangle.Considerthemorphismofdistin-guishedtriangles(F(X);F(Y);F(Z);F(f);F(g);F(h)) (RF(X);RF(Y);RF(Z);RF(f);RF(g);RF(h))Twooutofthreemapsareisomorphisms,hencesoisthethird.Lemma14.13.05T0AssumptionsandnotationasinSituation14.1.LetX;YbeobjectsofD.IfXYcomputesRF,thenXandYcomputeRF.SimilarlyforLF.Proof.IfXYcomputesRF,thenRF(XY)=F(X)F(Y).IntheproofofLemma14.7wehaveseenthatthefunctorX=SY=S!(XY)=S,(s;s0)7!ss0isco nal.Wewillusethiswithoutfurthermention.Lets:X!X0beanelementofS.ThenF(X)!F(X0)hasasection,namely,F(X0)!F(X0Y)!RF(X0Y)=RF(XY)=F(X)F(Y)!F(X):wherewehaveusedLemma14.4.HenceF(X0)=F(X)EforsomeobjectEofD0suchthatE!F(X0Y)!RF(X0Y)=RF(XY)iszero(Lemma4.12).BecauseRFisde nedatX0YwithvalueF(X)F(Y)wecan ndamorphismt:X0Y!ZofSsuchthatF(t)annihilatesE.WemayassumeZ=X00Y00andt=t0t00witht0;t002S.ThenF(t0)annihilatesE.ItfollowsthatFisessentiallyconstantonX=SwithvalueF(X)asdesired.Lemma14.14.05T1AssumptionsandnotationasinSituation14.1.(1)IfforeveryobjectX2Ob(D)thereexistsanarrows:X!X0inSsuchthatX0computesRF,thenRFiseverywherede ned.(2)IfforeveryobjectX2Ob(D)thereexistsanarrows:X0!XinSsuchthatX0computesLF,thenLFiseverywherede ned.Proof.Thisisclearfromthede nitions.Lemma14.15.06XNAssumptionsandnotationasinSituation14.1.IfthereexistsasubsetIOb(D)suchthat(1)forallX2Ob(D)thereexistss:X!X0inSwithX02I,and(2)foreveryarrows:X!X0inSwithX;X02IthemapF(s):F(X)!F(X0)isanisomorphism,thenRFiseverywherede nedandeveryX2IcomputesRF.Dually,ifthereexistsasubsetPOb(D)suchthat(1)forallX2Ob(D)the
52 reexistss:X0!XinSwithX02P,and(2)forevery
reexistss:X0!XinSwithX02P,and(2)foreveryarrows:X!X0inSwithX;X02PthemapF(s):F(X)!F(X0)isanisomorphism,thenLFiseverywherede nedandeveryX2PcomputesLF. DERIVEDCATEGORIES52Proof.LetXbeanobjectofD.Assumption(1)impliesthatthearrowss:X!X0inSwithX02Iareco nalinthecategoryX=S.Assumption(2)impliesthatFisconstantonthisco nalsubcategory.ClearlythisimpliesthatF:(X=S)!D0isessentiallyconstantwithvalueF(X0)foranys:X!X0inSwithX02I.Lemma14.16.05T2LetA;B;Cbetriangulatedcategories.LetS,resp.S0beasatu-ratedmultiplicativesysteminA,resp.Bcompatiblewiththetriangulatedstructure.LetF:A!BandG:B!Cbeexactfunctors.DenoteF0:A!(S0)�1BthecompositionofFwiththelocalizationfunctor.(1)IfRF0,RG,R(GF)areeverywherede ned,thenthereisacanonicaltransformationoffunctorst:R(GF)�!RGRF0.(2)IfLF0,LG,L(GF)areeverywherede ned,thenthereisacanonicaltransformationoffunctorst:LGLF0!L(GF).Proof.Inthisproofwetrytobecareful.HenceletusthinkofthederivedfunctorsasthefunctorsRF0:S�1A!(S0)�1B;R(GF):S�1A!C;RG:(S0)�1B!C:LetusdenoteQA:A!S�1AandQB:B!(S0)�1Bthelocalizationfunctors.ThenF0=QBF.NotethatforeveryobjectYofBthereisacanonicalmapG(Y)�!RG(QB(Y))inotherwords,thereisatransformationoffunctorst0:G!RGQB.LetXbeanobjectofA.WehaveR(GF)(QA(X))=colims:X!X02SG(F(X0))t0�!colims:X!X02SRG(QB(F(X0)))=colims:X!X02SRG(F0(X0))=RG(colims:X!X02SF0(X0))=RG(RF0(X)):ThesystemF0(X0)isessentiallyconstantinthecategory(S0)�1B.HencewemaypullthecolimitinsidethefunctorRGinthethirdequalityofthediagramabove,seeCategories,Lemma22.8anditsproof.Weomittheproofthisde nesatrans-formationoffunctors.Thecaseofleftderivedfunctorsissimilar.15.Derivedfunctorsonderivedcategories05T3Inpracticederivedfunctorscomeaboutmostoftenwhengivenanadditivefunctorbetweenabeliancategories.Situation15.1.05T4HereF:A!Bisanadditivefunctorbetweenabeliancategories.ThisinducesexactfunctorsF:K(A)!K(B);K+(A)!K+(B);K�(A)!K�(B):SeeLemma10.6.WealsodenoteFthecompositionK(A)!D(B),K+(A)!D+(B),andK�(A)!D�(B)ofFwiththelocalizationfunctorK(B)!D(B),etc.Thissitua
53 tionleadstofourderivedfunctorswewillcons
tionleadstofourderivedfunctorswewillconsiderinthefollowing.(1)TherightderivedfunctorofF:K(A)!D(B)relativetothemultiplicativesystemQis(A). DERIVEDCATEGORIES53(2)TherightderivedfunctorofF:K+(A)!D+(B)relativetothemulti-plicativesystemQis+(A).(3)TheleftderivedfunctorofF:K(A)!D(B)relativetothemultiplicativesystemQis(A).(4)TheleftderivedfunctorofF:K�(A)!D�(B)relativetothemultiplica-tivesystemQis�(A).EachofthesecasesisanexampleofSituation14.1.Someoftheambiguitythatmayariseisalleviatedbythefollowing.Lemma15.2.05T5InSituation15.1.(1)LetXbeanobjectofK+(A).TherightderivedfunctorofK(A)!D(B)isde nedatXifandonlyiftherightderivedfunctorofK+(A)!D+(B)isde nedatX.Moreover,thevaluesarecanonicallyisomorphic.(2)LetXbeanobjectofK+(A).ThenXcomputestherightderivedfunctorofK(A)!D(B)ifandonlyifXcomputestherightderivedfunctorofK+(A)!D+(B).(3)LetXbeanobjectofK�(A).TheleftderivedfunctorofK(A)!D(B)isde nedatXifandonlyiftheleftderivedfunctorofK�(A)!D�(B)isde nedatX.Moreover,thevaluesarecanonicallyisomorphic.(4)LetXbeanobjectofK�(A).ThenXcomputestheleftderivedfunctorofK(A)!D(B)ifandonlyifXcomputestheleftderivedfunctorofK�(A)!D�(B).Proof.LetXbeanobjectofK+(A).Consideraquasi-isomorphisms:X!X0inK(A).ByLemma11.5thereexistsquasi-isomorphismX0!X00withX00boundedbelow.HenceweseethatX=Qis+(A)isco nalinX=Qis(A).Thusitisclearthat(1)holds.Part(2)followsdirectlyfrompart(1).Parts(3)and(4)aredualtoparts(1)and(2).GivenanobjectAofanabeliancategoryAwegetacomplexA[0]=(:::!0!A!0!:::)whereAisplacedindegreezero.HenceafunctorA!K(A),A7!A[0].Letustemporarilysaythatapartialfunctorisonethatisde nedonasubcategory.De nition15.3.0157InSituation15.1.(1)TherightderivedfunctorsofFarethepartialfunctorsRFassociatedtocases(1)and(2)ofSituation15.1.(2)TheleftderivedfunctorsofFarethepartialfunctorsLFassociatedtocases(3)and(4)ofSituation15.1.(3)AnobjectAofAissaidtoberightacyclicforF,oracyclicforRFifA[0]computesRF.(4)AnobjectAofAissaidtobeleftacyclicforF,oracyclicforLFifA[0]computesLF.Thefollowingfewlemmasgivesomecriteriafortheexistenceofenoughacyclics.Lemma15.4.
54 05T7LetAbeanabeliancategory.LetPOb(
05T7LetAbeanabeliancategory.LetPOb(A)beasubsetcontaining0suchthateveryobjectofAisaquotientofanelementofP.Leta2Z.(1)GivenKwithKn=0forn�athereexistsaquasi-isomorphismP!KwithPn2PandPn!KnsurjectiveforallnandPn=0forn�a. DERIVEDCATEGORIES54(2)GivenKwithHn(K)=0forn�athereexistsaquasi-isomorphismP!KwithPn2PforallnandPn=0forn�a.Proof.Proofofpart(1).ConsiderthefollowinginductionhypothesisIHn:TherearePj2P,jn,withPj=0forj�a,mapsdj:Pj!Pj+1forjn,andsurjectivemaps j:Pj!KjforjnsuchthatthediagramPn // Pn+1 // Pn+2 // ::::::// Kn�1// Kn// Kn+1// Kn+2// :::iscommutative,suchthatdj+1dj=0forjn,suchthat inducesisomorphismsHj(K)!Ker(dj)=Im(dj�1)forj�n,andsuchthat :Ker(dn)!Ker(dnK)issurjective.ThenwechooseasurjectionPn�1�!Kn�1KnKer(dn)=Kn�1Ker(dnK)Ker(dn)withPn�1inP.ThisallowsustoextendthediagramabovetoPn�1 // Pn // Pn+1 // Pn+2 // ::::::// Kn�1// Kn// Kn+1// Kn+2// :::ThereadereasilychecksthatIHn�1holdswiththischoice.We nishtheproofof(1)asfollows.FirstwenotethatIHnistrueforn=a+1sincewecanjusttakePj=0forj�a.HenceweseethatproceedingbydescendinginductionweproduceacomplexPwithPn=0forn�aconsistingofobjectsfromP,andatermwisesurjectivequasi-isomorphism :P!Kasdesired.Proofofpart(2).Theassumptionimpliesthatthemorphism aK!K(Ho-mology,Section15)isaquasi-isomorphism.Applypart(1)to ndP! aK.ThecompositionP!Kisthedesiredquasi-isomorphism.Lemma15.5.05T6LetAbeanabeliancategory.LetIOb(A)beasubsetcontaining0suchthateveryobjectofAisasubobjectofanelementofI.Leta2Z.(1)GivenKwithKn=0fornathereexistsaquasi-isomorphismK!IwithKn!IninjectiveandIn2IforallnandIn=0forna,(2)GivenKwithHn(K)=0fornathereexistsaquasi-isomorphismK!IwithIn2IandIn=0forna.Proof.ThislemmaisdualtoLemma15.4.Lemma15.6.05T8InSituation1
55 5.1.LetIOb(A)beasubsetwiththefollow
5.1.LetIOb(A)beasubsetwiththefollowingproperties:(1)everyobjectofAisasubobjectofanelementofI,(2)foranyshortexactsequence0!P!Q!R!0ofAwithP;Q2I,thenR2I,and0!F(P)!F(Q)!F(R)!0isexact.TheneveryobjectofIisacyclicforRF. DERIVEDCATEGORIES55Proof.Wemayadd0toIifnecessary.PickA2I.LetA[0]!Kbeaquasi-isomorphismwithKboundedbelow.Thenwecan ndaquasi-isomorphismK!IwithIboundedbelowandeachIn2I,seeLemma15.5.Henceweseethattheseresolutionsareco nalinthecategoryA[0]=Qis+(A).To nishtheproofitthereforesu cestoshowthatforanyquasi-isomorphismA[0]!IwithIboundedaboveandIn2IwehaveF(A)[0]!F(I)isaquasi-isomorphism.ToseethissupposethatIn=0fornn0.Ofcoursewemayassumethatn00.Startingwithn=n0weproveinductivelythatIm(dn�1)=Ker(dn)andIm(d�1)areelementsofIusingproperty(2)andtheexactsequences0!Ker(dn)!In!Im(dn)!0:Moreover,property(2)alsoguaranteesthatthecomplex0!F(In0)!F(In0+1)!:::!F(I�1)!F(Im(d�1))!0isexact.Theexactsequence0!Im(d�1)!I0!I0=Im(d�1)!0impliesthatI0=Im(d�1)isanelementofI.Theexactsequence0!A!I0=Im(d�1)!Im(d0)!0thenimpliesthatIm(d0)=Ker(d1)isanelementsofIandfromthenononecontinuesasbeforetoshowthatIm(dn�1)=Ker(dn)isanelementofIforalln&#x]TJ/;དྷ ; .96;& T; 10;&#x.516;&#x 0 T; [0;0.ApplyingFtoeachoftheshortexactsequencesmentionedaboveandusing(2)weobservethatF(A)[0]!F(I)isanisomorphismasdesired.Lemma15.7.05T9InSituation15.1.LetPOb(A)beasubsetwiththefollowingproperties:(1)everyobjectofAisaquotientofanelementofP,(2)foranyshortexactsequence0!P!Q!R!0ofAwithQ;R2P,thenP2P,and0!F(P)!F(Q)!F(R)!0isexact.TheneveryobjectofPisacyclicforLF.Proof.DualtotheproofofLemma15.6.16.Higherderivedfunctors05TBThefollowingsimplelemmashowsthatrightderivedfunctorsmovetotheright.Lemma16.1.05TCLetF:A!Bbeanadditivefunctorbetweenabeliancategories.LetKbeacomplexofAanda2Z.(1)IfHi(K)=0foralliaandRFisde nedatK,thenHi(RF(K))=0forallia.(2)IfRFisde nedatKand aK,thenHi(RF( aK))=Hi(RF(K))forallia.Proof.AssumeKsati
56 s estheassumptionsof(1).LetK!L&
s estheassumptionsof(1).LetK!Lbeanyquasi-isomorphism.ThenitisalsotruethatK! aLisaquasi-isomorphismbyourassumptiononK.HenceinthecategoryK=Qis+(A)thequasi-isomorphismss:K!LwithLn=0fornaareco nal.ThusRFisthevalueoftheessentiallyconstantind-objectF(L)forthesesitfollowsthatHi(RF(K))=0foria.Toprove(2)weusethedistinguishedtriangle aK!K! a+1K!( aK)[1] DERIVEDCATEGORIES56ofRemark12.4toconcludeviaLemma14.6thatRFisde nedat a+1KaswellandthatwehaveadistinguishedtriangleRF( aK)!RF(K)!RF( a+1K)!RF( aK)[1]inD(B).Bypart(1)weseethatRF( a+1K)hasvanishingcohomologyindegreesa+1.Thelongexactcohomologysequenceofthisdistinguishedtrianglethenshowswhatwewant.De nition16.2.015ALetF:A!Bbeanadditivefunctorbetweenabeliancate-gories.AssumeRF:D+(A)!D+(B)iseverywherede ned.Leti2Z.TheithrightderivedfunctorRiFofFisthefunctorRiF=HiRF:A�!BThefollowinglemmashowsthatitreallydoesnotmakealotofsensetotaketherightderivedfunctorunlessthefunctorisleftexact.Lemma16.3.05TDLetF:A!BbeanadditivefunctorbetweenabeliancategoriesandassumeRF:D+(A)!D+(B)iseverywherede ned.(1)WehaveRiF=0fori0,(2)R0Fisleftexact,(3)themapF!R0FisanisomorphismifandonlyifFisleftexact.Proof.LetAbeanobjectofA.LetA[0]!Kbeanyquasi-isomorphism.ThenitisalsotruethatA[0]! 0Kisaquasi-isomorphism.HenceinthecategoryA[0]=Qis+(A)thequasi-isomorphismss:A[0]!KwithKn=0forn0areco nal.ThusitisclearthatHi(RF(A[0]))=0fori0.Moreover,forsuchansthesequence0!A!K0!K1isexact.HenceifFisleftexact,then0!F(A)!F(K0)!F(K1)isexactaswell,andweseethatF(A)!H0(F(K))isanisomorphismforeverys:A[0]!KasabovewhichimpliesthatH0(RF(A[0]))=F(A).Let0!A!B!C!0beashortexactsequenceofA.ByLemma12.1weobtainadistinguishedtriangle(A[0];B[0];C[0];a;b;c)inK+(A).Fromthelongexactcohomologysequence(andthevanishingfori0provedabove)wededucethat0!R0F(A)!R0F(B)!R0F(C)isexact.HenceR0Fisleftexact.OfcoursethisalsoprovesthatifF!R0Fisanisomorph
57 ism,thenFisleftexact.Lemma16.4.015CL
ism,thenFisleftexact.Lemma16.4.015CLetF:A!BbeanadditivefunctorbetweenabeliancategoriesandassumeRF:D+(A)!D+(B)iseverywherede ned.LetAbeanobjectofA.(1)AisrightacyclicforFifandonlyifF(A)!R0F(A)isanisomorphismandRiF(A)=0foralli&#x]TJ/;དྷ ; .96;& T; 17;&#x.359;&#x 0 T; [0;0,(2)ifFisleftexact,thenAisrightacyclicforFifandonlyifRiF(A)=0foralli&#x]TJ/;དྷ ; .96;& T; 17;&#x.359;&#x 0 T; [0;0.Proof.IfAisrightacyclicforF,thenRF(A[0])=F(A)[0]andinparticularF(A)!R0F(A)isanisomorphismandRiF(A)=0fori6=0.Conversely,ifF(A)!R0F(A)isanisomorphismandRiF(A)=0foralli&#x]TJ/;དྷ ; .96;& T; 17;&#x.359;&#x 0 T; [0;0thenF(A[0])!RF(A[0])isaquasi-isomorphismbyLemma16.3part(1)andhenceAisacyclic.IfFisleftexactthenF=R0F,seeLemma16.3. DERIVEDCATEGORIES57Lemma16.5.015DLetF:A!BbealeftexactfunctorbetweenabeliancategoriesandassumeRF:D+(A)!D+(B)iseverywherede ned.Let0!A!B!C!0beashortexactsequenceofA.(1)IfAandCarerightacyclicforFthensoisB.(2)IfAandBarerightacyclicforFthensoisC.(3)IfBandCarerightacyclicforFandF(B)!F(C)issurjectivethenAisrightacyclicforF.Ineachofthethreecases0!F(A)!F(B)!F(C)!0isashortexactsequenceofB.Proof.ByLemma12.1weobtainadistinguishedtriangle(A[0];B[0];C[0];a;b;c)inK+(A).AsRFisanexactfunctorandsinceRiF=0fori0andR0F=F(Lemma16.3)weobtainanexactcohomologysequence0!F(A)!F(B)!F(C)!R1F(A)!:::intheabeliancategoryB.ThusthelemmafollowsfromthecharacterizationofacyclicobjectsinLemma16.4.Lemma16.6.05TELetF:A!BbeanadditivefunctorbetweenabeliancategoriesandassumeRF:D+(A)!D+(B)iseverywherede ned.(1)ThefunctorsRiF,i0comeequippedwithacanonicalstructureofa-functorfromA!B,seeHomology,De nition12.1.(2)IfeveryobjectofAisasubobjectofarightacyclicobjectforF,thenfRiF;gi0isauniversal-functor,seeHomology,De nition12.3.Proof.ThefunctorA!Comp+(A),A7!A[0]isexact.ThefunctorComp+(A)!D+(A)isa-functor,seeLemma12.1.ThefunctorRF:D+(A)!D+(B)isex-act.Finally,thefunctorH0:D+(B)!Bisahomologicalfunctor,seeDe nition11.3.Hencewegetthestructureofa-functorfromLemma4.22andLemma4.21.Part(2
58 )followsfromHomology,Lemma12.4andthedesc
)followsfromHomology,Lemma12.4andthedescriptionofacyclicsinLemma16.4.Lemma16.7(Leray'sacyclicitylemma).015ELetF:A!Bbeanadditivefunctorbetweenabeliancategories.LetAbeaboundedbelowcomplexofrightF-acyclicobjectssuchthatRFisde nedatA5.ThecanonicalmapF(A)�!RF(A)isanisomorphisminD+(B),i.e.,AcomputesRF.Proof.LetAbeaboundedcomplexofrightF-acyclicobjects.WeclaimthatRFisde nedatAandthatF(A)!RF(A)isanisomorphisminD+(B).Namely,itholdsforcomplexeswithatmostonenonzerorightF-acyclicobjectforexamplebyLemma16.4.Next,supposethatAn=0forn62[a;b].Usingthestupidtruncationsweobtainatermwisesplitshortexactsequenceofcomplexes0!a+1A!A!aA!0 5ForexamplethisholdsifRF:D+(A)!D+(B)iseverywherede ned. DERIVEDCATEGORIES58seeHomology,Section15.Thusadistinguishedtriangle(a+1A;A;aA).ByinductionhypothesisRFisde nedforthetwooutercomplexesandthesecomplexescomputeRF.ThenthesameistrueforthemiddleonebyLemma14.12.SupposethatAisaboundedbelowcomplexofacyclicobjectssuchthatRFisde nedatA.ToshowthatF(A)!RF(A)isanisomorphisminD+(B)itsu cestoshowthatHi(F(A))!Hi(RF(A))isanisomorphismforalli.Picki.Considerthetermwisesplitshortexactsequenceofcomplexes0!i+2A!A!i+1A!0:Notethatthisinducesatermwisesplitshortexactsequence0!i+2F(A)!F(A)!i+1F(A)!0:Hencewegetdistinguishedtriangles(i+2A;A;i+1A)and(i+2F(A);F(A);i+1F(A))SinceRFisde nedatA(byassumption)andati+1A(bythe rstparagraph)weseethatRFisde nedati+1Aandwegetadistinghuishedtriangle(RF(i+2A);RF(A);RF(i+1A))SeeLemma14.6.UsingthesedistinguishedtrianglesweobtainamapofexactsequencesHi(i+2F(A))// Hi(F(A))// Hi(i+1F(A))// Hi+1(i+2F(A)) Hi(RF(i+2A))//
59 Hi(RF(A))// Hi(RF(i+1A&#
Hi(RF(A))// Hi(RF(i+1A))// Hi+1(RF(i+2A))Bytheresultsofthe rstparagraphthemapisanisomorphism.Byinspectiontheobjectsontheupperleftandtheupperrightarezero.Henceto nishtheproofitsu cestoshowthatHi(RF(i+2A))=0andHi+1(RF(i+2A))=0.ThisfollowsimmediatelyfromLemma16.1.Proposition16.8.05TALetF:A!Bbeanadditivefunctorofabeliancategories.(1)IfeveryobjectofAinjectsintoanobjectacyclicforRF,thenRFisde nedonallofK+(A)andweobtainanexactfunctorRF:D+(A)�!D+(B)see(14.9.1).Moreover,anyboundedbelowcomplexAwhosetermsareacyclicforRFcomputesRF.(2)IfeveryobjectofAisquotientofanobjectacyclicforLF,thenLFisde nedonallofK�(A)andweobtainanexactfunctorLF:D�(A)�!D�(B)see(14.9.1).Moreover,anyboundedabovecomplexAwhosetermsareacyclicforLFcomputesLF.Proof.AssumeeveryobjectofAinjectsintoanobjectacyclicforRF.LetIbethesetofobjectsacyclicforRF.LetKbeaboundedbelowcomplexinA.ByLemma15.5thereexistsaquasi-isomorphism :K!IwithIboundedbelowandIn2I.Henceinordertoprove(1)itsu cestoshowthatF(I)!F((I0))isaquasi-isomorphismwhens:I!(I0)isaquasi-isomorphismofboundedbelow DERIVEDCATEGORIES59complexesofobjectsfromI,seeLemma14.15.NotethattheconeC(s)isanacyclicboundedbelowcomplexallofwhosetermsareinI.Henceitsu cestoshow:givenanacyclicboundedbelowcomplexIallofwhosetermsareinIthecomplexF(I)isacyclic.SayIn=0fornn0.SettingJn=Im(dn)webreakIintoshortexactse-quences0!Jn!In+1!Jn+1!0fornn0.Thesesequencesinducedistinguishedtriangles(Jn;In+1;Jn+1)inD+(A)byLemma12.1.Foreachk2ZdenoteHktheassertion:ForallnktherightderivedfunctorRFisde nedatJnandRiF(Jn)=0fori6=0.ThenHkholdstriviallyforkn0.IfHnholds,then,usingProposition14.8,weseethatRFisde nedatJn+1and(RF(Jn);RF(In+1);RF(Jn+1))isadistinguishedtriangleofD+(B).Thusthelongexactcohomologysequence(11.1.1)associatedtothistrianglegivesanexactsequence0!R�1F(Jn+1)!R0F(Jn)!F(In+1)!R0F(Jn+1)!0andgivesthatRiF(Jn+1)=0fori62f�1;0g.ByLemma16.1weseethatR�1F(Jn+1)=0.T
60 hisprovesthatHn+1istruehenceHkholdsforal
hisprovesthatHn+1istruehenceHkholdsforallk.Wealsoconcludethat0!R0F(Jn)!F(In+1)!R0F(Jn+1)!0isshortexactforalln.ThisinturnprovesthatF(I)isexact.TheproofinthecaseofLFisdual.Lemma16.9.015FLetF:A!Bbeanexactfunctorofabeliancategories.Then(1)everyobjectofAisrightacyclicforF,(2)RF:D+(A)!D+(B)iseverywherede ned,(3)RF:D(A)!D(B)iseverywherede ned,(4)everycomplexcomputesRF,inotherwords,thecanonicalmapF(K)!RF(K)isanisomorphismforallcomplexes,and(5)RiF=0fori6=0.Proof.ThisistruebecauseFtransformsacycliccomplexesintoacycliccomplexesandquasi-isomorphismsintoquasi-isomorphisms.Detailsomitted.17.Triangulatedsubcategoriesofthederivedcategory06UPLetAbeanabeliancategory.InthissectionwelookatcertainstrictlyfullsaturatedtriangulatedsubcategoriesD0D(A).LetBAbeaweakSerresubcategory,seeHomology,De nition10.1andLemma10.3.WeletDB(A)thefullsubcategoryofD(A)whoseobjectsareOb(DB(A))=fX2Ob(D(A))jHn(X)isanobjectofBforallngWealsode neD+B(A)=D+(A)\DB(A)andsimilarlyfortheotherboundedversions.Lemma17.1.06UQLetAbeanabeliancategory.LetBAbeaweakSerresubcat-egory.ThecategoryDB(A)isastrictlyfullsaturatedtriangulatedsubcategoryofD(A).Similarlyfortheboundedversions. DERIVEDCATEGORIES60Proof.ItisclearthatDB(A)isanadditivesubcategorypreservedunderthetrans-lationfunctors.IfXYisinDB(A),thenbothHn(X)andHn(Y)arekernelsofmapsbetweenmapsofobjectsofBasHn(XY)=Hn(X)Hn(Y).HencebothXandYareinDB(A).ByLemma4.16itthereforesu cestoshowthatgivenadistinguishedtriangle(X;Y;Z;f;g;h)suchthatXandYareinDB(A)thenZisanobjectofDB(A).Thelongexactcohomologysequence(11.1.1)andthede nitionofaweakSerresubcategory(seeHomology,De nition10.1)showthatHn(Z)isanobjectofBforalln.ThusZisanobjectofDB(A).WecontinuetoassumethatBisaweakSerresubcategoryoftheabeliancategoryA.ThenBisanabeliancategoryandtheinclusionfunctorB!Aisexact.HenceweobtainaderivedfunctorD(B)!D(A),seeLemma16.9.ClearlythefunctorD(B)!D(A)factorsthroughacanonicalexactfunctor(17.1.1)06URD(B)�!DB(A)AfterallacomplexmadefromobjectsofBcertainlygivesrisetoanobjectofDB(A)andasdistinguishedtrian
61 glesinDB(A)areexactlythedistinguishedtri
glesinDB(A)areexactlythedistinguishedtrianglesofD(A)whoseverticesareinDB(A)weseethatthefunctorisexactsinceD(B)!D(A)isexact.SimilarlyweobtainfunctorsD+(B)!D+B(A),D�(B)!D�B(A),andDb(B)!DbB(A)fortheboundedversions.Akeyquestioninmanycasesiswhetherthedisplayedfunctorisanequivalence.Now,supposethatBisaSerresubcategoryofA.InthiscasewehavethequotientfunctorA!A=B,seeHomology,Lemma10.6.InthiscaseDB(A)isthekernelofthefunctorD(A)!D(A=B).ThusweobtainacanonicalfunctorD(A)=DB(A)�!D(A=B)byLemma6.8.Similarlyfortheboundedversions.Lemma17.2.06XLLetAbeanabeliancategory.LetBAbeaSerresubcategory.ThenD(A)!D(A=B)isessentiallysurjective.Proof.WewillusethedescriptionofthecategoryA=BintheproofofHomology,Lemma10.6.Let(X;d)beacomplexofA=B.ThismeansthatXiisanobjectofAanddi:Xi!Xi+1isamorphisminA=Bsuchthatdidi�1=0inA=B.Fori0wemaywritedi=(si;fi)wheresi:Yi!XiisamorphismofAwhosekernelandcokernelareinB(equivalentlysibecomesanisomorphisminthequotientcategory)andfi:Yi!Xi+1isamorphismofA.Byinductionwewillconstructacommutativediagram(X0)1// (X0)2// :::X0 X1OO X2OO :::Y0s0OO f0 Y1s1OO f1:: Y2s2OO f2 :::wheretheverticalarrowsXi!(X0)ibecomeisomorphismsinthequotientcate-gory.Namely,we rstlet(X0)1=Coker(Y0!X0X1)(orratherthepushoutofthediagramwitharrowss0andf0)whichgivesthe rstcommutativediagram. DERIVEDCATEGORIES61Next,wetake(X0)2=Coker(Y1!(X0)1X2).Andsoon.Settingadditionally(X0)n=Xnforn0weseethatthemap(X;d)!((X0);(d0))isanisomor-phismofcomplexesinA=B.Hencewemayassumedn:Xn!Xn+1isgivenbyamapXn!Xn+1inAforn0.Dually,fori0wemaywritedi=(gi;ti+1)whereti+1:Xi+1!Zi+1isanisomorphisminthequotientcategoryandgi:Xi!Zi+1isamorphism.Byinductionwewillconstructacommutativediagram:::Z�2Z�1Z0:::X�2t�2OO g�299 X�1t�1OO g�1;; X0t0OO :::(X0)�2OO // (X0)�1OO ;; wheretheverticalarrows(X0)i!Xibecomeisomorphismsinthequotientcate-gory.Namely,wetake(X0)�1=X�1Z0X0.Thenwetake(X0)�2=X�2Z�1(X0)�1.Andsoon.Settingadditionally(X0)n=Xnforn0weseethatthemap((X0);(d0))!(X;d
62 5;)isanisomorphismofcomplexesinA=B.Hence
5;)isanisomorphismofcomplexesinA=B.Hencewemayassumedn:Xn!Xn+1isgivenbyamapdn:Xn!Xn+1inAforalln2Z.Inthiscaseweknowthecompositionsdndn�1arezeroinA=B.Ifforn�0wereplaceXnby(X0)n=Xn=X0nIm(Im(Xk�2!Xk)!Xn)thenthecompositionsdndn�1arezeroforn0.(Similarlytothesecondparagraphaboveweobtainanisomorphismofcomplexes(X;d)!((X0);(d0)).)Finally,forn0wereplaceXnby(X0)n=\nk0(Xn!Xk)�1Ker(Xk!Xk+2)andweargueinthesamemannertogetacomplexinAwhoseimageinA=Bisisomorphictothegivenone.Lemma17.3.06XMLetAbeanabeliancategory.LetBAbeaSerresubcategory.Supposethatthefunctorv:A!A=Bhasaleftadjointu:A=B!Asuchthatvu=id.ThenD(A)=DB(A)=D(A=B)andsimilarlyfortheboundedversions.Proof.ThefunctorD(v):D(A)!D(A=B)isessentiallysurjectivebyLemma17.2.ForanobjectXofD(A)theadjunctionmappingcX:uvX!XmapstoanisomorphisminD(A=B)becausevuv=vbytheassumptionthatvu=id.Thusinadistinguishedtriangle(uvX;X;Z;cX;g;h)theobjectZisanobjectofDB(A)asweseebylookingatthelongexactcohomologysequence.HencecXisanelementofthemultiplicativesystemusedtode nethequotientcategoryD(A)=DB(A).ThusuvX=XinD(A)=DB(A).ForX;Y2Ob(A))themapHomD(A)=DB(A)(X;Y)�!HomD(A=B)(vX;vY) DERIVEDCATEGORIES62isbijectivebecauseugivesaninverse(bytheremarksabove).ForcertainSerresubcategoriesBAwecanprovethatthefunctorD(B)!DB(A)isfullyfaithful.Lemma17.4.0FCLLetAbeanabeliancategory.LetBAbeaSerresubcategory.AssumethatforeverysurjectionX!YwithX2Ob(A)andY2Ob(B)thereexistsX0X,X02Ob(B)whichsurjectsontoY.ThenthefunctorD�(B)!D�B(A)of(17.1.1)isanequivalence.Proof.LetXbeaboundedabovecomplexofAsuchthatHi(X)2Ob(B)foralli2Z.Moreover,supposewearegivenBiXi,Bi2Ob(B)foralli2Z.Claim:thereexistsasubcomplexYXsuchthat(1)Y!Xisaquasi-isomorphism,(2)Yi2Ob(B)foralli2Z,and(3)BiYiforalli2Z.Toprovetheclaim,usingtheassumptionofthelemmawe rstchooseCiKer(di:Xi!Xi+1),Ci2Ob(B)surjectingontoHi(X).SettingDi=Ci+di�1(Bi�1)+Biwe ndasubcomplexDsatisfying(2)and(3)suchthatHi(D)!Hi(X)issurjectiv
63 eforalli2Z.ForanychoiceofEiXiwithEi
eforalli2Z.ForanychoiceofEiXiwithEi2Ob(B)anddi(Ei)Di+1+Ei+1weseethatsettingYi=Di+EigivesasubcomplexwhosetermsareinBandwhosecohomologysurjectsontothecohomologyofX.Clearly,ifdi(Ei)=(Di+1+Ei+1)\Im(di)thenweseethatthemaponcohomologyisalsoinjective.Forn 0wecantakeEnequalto0.BydescendinginductionwecanchooseEiforalliwiththedesiredproperty.Namely,givenEi+1;Ei+2;:::wechooseEiXisuchthatdi(Ei)=(Di+1+Ei+1)\Im(di).Thisispossiblebyourassumptioninthelemmacombinedwiththefactthat(Di+1+Ei+1)\Im(di)isinBasBisaSerresubcategoryofA.Theclaimaboveimpliesthelemma.Essentialsurjectivityisimmediatefromtheclaim.Letusprovefaithfulness.Namely,supposewehaveamorphismf:U!VofboundedabovecomplexesofBwhoseimageinD(A)iszero.Thenthereexistsaquasi-isomorphisms:V!XintoaboundedabovecomplexofAsuchthatsfishomotopictozero.Chooseahomotopyhi:Ui!Xi�1between0andsf.ApplytheclaimwithBi=hi+1(Ui+1)+si(Vi).Theresultingmaps0:V!Yisaquasi-isomorphismaswellands0fishomotopictozeroasisclearfromthefactthathifactorsthroughYi�1.Thisprovesfaithfulness.Fullyfaithfulnessisprovedintheexactsamemanner.18.Injectiveresolutions013GInthissectionweprovesomelemmasregardingtheexistenceofinjectiveresolutionsinabeliancategorieshavingenoughinjectives.De nition18.1.013ILetAbeanabeliancategory.LetA2Ob(A).AninjectiveresolutionofAisacomplexItogetherwithamapA!I0suchthat:(1)WehaveIn=0forn0.(2)EachInisaninjectiveobjectofA.(3)ThemapA!I0isanisomorphismontoKer(d0).(4)WehaveHi(I)=0fori&#x]TJ/;དྷ ; .96;& T; 19;&#x.263;&#x 0 T; [0;0. DERIVEDCATEGORIES63HenceA[0]!Iisaquasi-isomorphism.Inotherwordsthecomplex:::!0!A!I0!I1!:::isacyclic.LetKbeacomplexinA.AninjectiveresolutionofKisacomplexItogetherwithamap :K!Iofcomplexessuchthat(1)WehaveIn=0forn 0,i.e.,Iisboundedbelow.(2)EachInisaninjectiveobjectofA.(3)Themap :K!Iisaquasi-isomorphism.InotherwordsaninjectiveresolutionK!Igivesrisetoadiagram:::// Kn�1 // Kn // Kn+1 // ::::::// In�1// In
64 // In+1// :::whichinducesanisomorphismon
// In+1// :::whichinducesanisomorphismoncohomologyobjectsineachdegree.AninjectiveresolutionofanobjectAofAisalmostthesamethingasaninjectiveresolutionofthecomplexA[0].Lemma18.2.013JLetAbeanabeliancategory.LetKbeacomplexofA.(1)IfKhasaninjectiveresolutionthenHn(K)=0forn 0.(2)IfHn(K)=0foralln 0thenthereexistsaquasi-isomorphismK!LwithLboundedbelow.Proof.Omitted.ForthesecondstatementuseL= nKforsomen 0.SeeHomology,Section15forthede nitionofthetruncation n.Lemma18.3.013KLetAbeanabeliancategory.AssumeAhasenoughinjectives.(1)AnyobjectofAhasaninjectiveresolution.(2)IfHn(K)=0foralln 0thenKhasaninjectiveresolution.(3)IfKisacomplexwithKn=0forna,thenthereexistsaninjectiveresolution :K!IwithIn=0fornasuchthateach n:Kn!Inisinjective.Proof.Proofof(1).FirstchooseaninjectionA!I0ofAintoaninjectiveobjectofA.Next,chooseaninjectionI0=A!I1intoaninjectiveobjectofA.Denoted0theinducedmapI0!I1.Next,chooseaninjectionI1=Im(d0)!I2intoaninjectiveobjectofA.Denoted1theinducedmapI1!I2.Andsoon.ByLemma18.2part(2)followsfrompart(3).Part(3)isaspecialcaseofLemma15.5.Lemma18.4.013RLetAbeanabeliancategory.LetKbeanacycliccomplex.LetIbeboundedbelowandconsistingofinjectiveobjects.AnymorphismK!Iishomotopictozero.Proof.Let :K!Ibeamorphismofcomplexes.Assumethat j=0forjn.Wewillshowthatthereexistsamorphismh:Kn+1!Insuchthat n=hd.Thus willbehomotopictothemorphismofcomplexesde nedbyj=8:0ifjn n+1�dhifj=n+1 jifj&#x]TJ ;� -1;.93; Td;&#x [00;n+1 DERIVEDCATEGORIES64Thiswillclearlyprovethelemma(byinduction).Toprovetheexistenceofhnotethat njdn�1(Kn�1)=0since n�1=0.SinceKisacyclicwehavedn�1(Kn�1)=Ker(Kn!Kn+1).Hencewecanthinkof nasamapintoInde nedonthesubobjectIm(Kn!Kn+1)ofKn+1.ByinjectivityoftheobjectInwecanextendthistoamaph:Kn+1!Inasdesired.Remark18.5.05TFLetAbeanabeliancategory.UsingthefactthatK(A)isatriangulatedcategorywemayuseLemma18.4toobtainproofsofs
65 omeofthelemmasbelowwhichareusuallyproved
omeofthelemmasbelowwhichareusuallyprovedbychasingthroughdiagrams.Namely,supposethat :K!Lisaquasi-isomorphismofcomplexes.Then(K;L;C( ); ;i;�p)isadistinguishedtriangleinK(A)(Lemma9.14)andC( )isanacycliccomplex(Lemma11.2).Next,letIbeaboundedbelowcomplexofinjectiveobjects.ThenHomK(A)(C( );I)// HomK(A)(L;I)// HomK(A)(K;I) rr HomK(A)(C( )[�1];I)isanexactsequenceofabeliangroups,seeLemma4.2.AtthispointLemma18.4guaranteesthattheoutertwogroupsarezeroandhenceHomK(A)(L;I)=HomK(A)(K;I).Lemma18.6.013PLetAbeanabeliancategory.ConsiderasoliddiagramK // L}} IwhereIisboundedbelowandconsistsofinjectiveobjects,and isaquasi-isomorphism.(1)Thereexistsamapofcomplexesmakingthediagramcommuteuptohomotopy.(2)If isinjectiveineverydegreethenwecan ndawhichmakesthediagramcommute.Proof.Thecorrectproofofpart(1)isexplainedinRemark18.5.Wealsogiveadirectproofhere.We rstshowthat(2)implies(1).Namely,let~ :K!~L,,sbeasinLemma9.6.Since~ isinjectiveby(2)thereexistsamorphism~:~L!Isuchthat =~~ .Set=~s.Thenwehave =~s~ ~~ = asdesired.Assumethat :K!Lisinjective.Supposewehavealreadyde nedinalldegreesn�1compatiblewithdierentialsandsuchthat j=j jforall DERIVEDCATEGORIES65jn�1.ConsiderthecommutativesoliddiagramKn�1// Kn Ln�1// Ln In�1// InThusweseethatthedottedarrowisprescribedonthesubobjects (Kn)anddn�1(Ln�1).Moreover,thesetwoarrowsagreeon (dn�1(Kn�1)).Henceif(18.6.1)013Q (dn�1(Kn�1))= (Kn)\dn�1(Ln�1)thenthesemorphismsgluetoamorphism (Kn)+dn�1(Ln�1)!Inand,usingtheinjectivityofIn,wecanextendthistoamorphismfromallofLnintoIn.Afterthisbyinductionwegetthemorphismforallnsimultaneously(notethatw
66 ecansetn=0foralln 0sinceIi
ecansetn=0foralln 0sinceIisboundedbelowinthiswaystartingtheinduction).Itremainstoprovetheequality(18.6.1).Thereaderisencouragedtoarguethisforthemselveswithasuitablediagramchase.Nonethelesshereisourargument.Notethattheinclusion (dn�1(Kn�1)) (Kn)\dn�1(Ln�1)isobvious.TakeanobjectTofAandamorphismx:T!Lnwhoseimageiscontainedinthesubobject (Kn)\dn�1(Ln�1).Since isinjectiveweseethatx= x0forsomex0:T!Kn.Moreover,sincexliesindn�1(Ln�1)weseethatdnx=0.Henceusinginjectivityof againweseethatdnx0=0.Thusx0givesamorphism[x0]:T!Hn(K).Ontheotherhandthecorrespondingmap[x]:T!Hn(L)inducedbyxiszerobyassumption.Since isaquasi-isomorphismweconcludethat[x0]=0.Thisofcoursemeansexactlythattheimageofx0iscontainedindn�1(Kn�1)andwewin.Lemma18.7.013SLetAbeanabeliancategory.ConsiderasoliddiagramK // Li}} IwhereIisboundedbelowandconsistsofinjectiveobjects,and isaquasi-isomorphism.Anytwomorphisms1;2makingthediagramcommuteuptoho-motopyarehomotopic.Proof.ThisfollowsfromRemark18.5.Wealsogiveadirectargumenthere.Let~ :K!~L,,sbeasinLemma9.6.Ifwecanshowthat1ishomotopicto2,thenwededucethat12becausesistheidentity.Hencewemayassume n:Kn!Lnistheinclusionofadirectsummandforalln.Thuswegetashortexactsequenceofcomplexes0!K!L!M!0 DERIVEDCATEGORIES66whichistermwisesplitandsuchthatMisacyclic.WechoosesplittingsLn=KnMn,sowehaveni:KnMn!Inand n:Kn!In.Inthiscasetheconditiononiisthattherearemorphismshni:Kn!In�1suchthat n�nijKn=dhni+hn+1idThusweseethatn1jKn�n2jKn=d(hn1�hn2)+(hn+11�hn+12)dConsiderthemaphn:KnMn!In�1whichequalshn1�hn2onthe rstsummandandzeroonthesecond.Thenweseethatn1�n2�(dhn+hn+1)disamorphismofcomplexesL!IwhichisidenticallyzeroonthesubcomplexK.HenceitfactorsasL!M!I.ThustheresultofthelemmafollowsfromLemma18.
67 4.Lemma18.8.05TGLetAbeanabeliancateg
4.Lemma18.8.05TGLetAbeanabeliancategory.LetIbeboundedbelowcomplexconsistingofinjectiveobjects.LetL2K(A).ThenMorK(A)(L;I)=MorD(A)(L;I):Proof.Letabeanelementoftherighthandside.Wemayrepresenta= �1where :K!Lisaquasi-isomorphismand :K!Iisamapofcomplexes.ByLemma18.6wecan ndamorphism:L!Isuchthat ishomotopicto .Thisprovesthatthemapissurjective.Letbbeanelementofthelefthandsidewhichmapstozerointherighthandside.Thenbisthehomotopyclassofamorphism:L!Isuchthatthereexistsaquasi-isomorphism :K!Lwith homotopictozero.ThenLemma18.7showsthatishomotopictozeroalso,i.e.,b=0.Lemma18.9.013TLetAbeanabeliancategory.AssumeAhasenoughinjectives.Foranyshortexactsequence0!A!B!C!0ofComp+(A)thereexistsacommutativediagraminComp+(A)0// A// B// C// 00// I1// I2// I3// 0wheretheverticalarrowsareinjectiveresolutionsandtherowsareshortexactsequencesofcomplexes.Infact,givenanyinjectiveresolutionA!IwemayassumeI1=I.Proof.Step1.ChooseaninjectiveresolutionA!I(seeLemma18.3)orusethegivenone.RecallthatComp+(A)isanabeliancategory,seeHomology,Lemma13.9.HencewemayformthepushoutalongtheinjectivemapA!Itoget0// A// B// C// 00// I// E// C// 0 DERIVEDCATEGORIES67Notethatthelowershortexactsequenceistermwisesplit,seeHomology,Lemma27.2.Henceitsu cestoprovethelemmawhen0!A!B!C!0istermwisesplit.Step2.Choosesplittings.Inotherwords,writeBn=AnCn.Denote:C!A[1]themorphismasinHomology,Lemma14.10.Chooseinjectiveresolutionsf1:A!I1andf3:C!I3.(IfAisacomplexofinjectives,thenuseI1=A.)Wemayassumef3isinjectiveineverydegree.ByLemma18.6wemay ndamorphism0:I3!I1[1]suchthat0f3=f1[1](equalityofmorphismsofcomplexes).SetIn2=In1In3.De nednI2=dnI1(0)n0dnI3andde nethemapsBn
68 !In2tobegivenasthesumofthemapsAn!In1andC
!In2tobegivenasthesumofthemapsAn!In1andCn!In3.Everythingisclear.19.Projectiveresolutions0643ThissectionisdualtoSection18.Wegivede nitionsandstateresults,butwedonotreprovethelemmas.De nition19.1.0644LetAbeanabeliancategory.LetA2Ob(A).AnprojectiveresolutionofAisacomplexPtogetherwithamapP0!Asuchthat:(1)WehavePn=0forn�0.(2)EachPnisanprojectiveobjectofA.(3)ThemapP0!AinducesanisomorphismCoker(d�1)!A.(4)WehaveHi(P)=0fori0.HenceP!A[0]isaquasi-isomorphism.Inotherwordsthecomplex:::!P�1!P0!A!0!:::isacyclic.LetKbeacomplexinA.AnprojectiveresolutionofKisacomplexPtogetherwithamap :P!Kofcomplexessuchthat(1)WehavePn=0forn 0,i.e.,Pisboundedabove.(2)EachPnisanprojectiveobjectofA.(3)Themap :P!Kisaquasi-isomorphism.Lemma19.2.0645LetAbeanabeliancategory.LetKbeacomplexofA.(1)IfKhasaprojectiveresolutionthenHn(K)=0forn 0.(2)IfHn(K)=0forn 0thenthereexistsaquasi-isomorphismL!KwithLboundedabove.Proof.DualtoLemma18.2.Lemma19.3.0646LetAbeanabeliancategory.AssumeAhasenoughprojectives.(1)AnyobjectofAhasaprojectiveresolution.(2)IfHn(K)=0foralln 0thenKhasaprojectiveresolution.(3)IfKisacomplexwithKn=0forn&#x]TJ/;དྷ ; .96;& T; 16;&#x.716;&#x 0 T; [0;a,thenthereexistsaprojectiveresolution :P!KwithPn=0forn&#x]TJ/;དྷ ; .96;& T; 16;&#x.716;&#x 0 T; [0;asuchthateach n:Pn!Knissurjective.Proof.DualtoLemma18.3. DERIVEDCATEGORIES68Lemma19.4.0647LetAbeanabeliancategory.LetKbeanacycliccomplex.LetPbeboundedaboveandconsistingofprojectiveobjects.AnymorphismP!Kishomotopictozero.Proof.DualtoLemma18.4.Remark19.5.0648LetAbeanabeliancategory.Supposethat :K!Lisaquasi-isomorphismofcomplexes.LetPbeaboundedabovecomplexofprojectives.ThenHomK(A)(P;K)�!HomK(A)(P;L)isanisomorphism.ThisisdualtoRemark18.5.Lemma19.6.0649LetAbeanabeliancategory.ConsiderasoliddiagramKL oo POO == wherePisboundedaboveandconsist
69 sofprojectiveobjects,and isaquasi-is
sofprojectiveobjects,and isaquasi-isomorphism.(1)Thereexistsamapofcomplexesmakingthediagramcommuteuptohomotopy.(2)If issurjectiveineverydegreethenwecan ndawhichmakesthediagramcommute.Proof.DualtoLemma18.6.Lemma19.7.064ALetAbeanabeliancategory.ConsiderasoliddiagramKL oo POO i== wherePisboundedaboveandconsistsofprojectiveobjects,and isaquasi-isomorphism.Anytwomorphisms1;2makingthediagramcommuteuptoho-motopyarehomotopic.Proof.DualtoLemma18.7.Lemma19.8.064BLetAbeanabeliancategory.LetPbeboundedabovecomplexconsistingofprojectiveobjects.LetL2K(A).ThenMorK(A)(P;L)=MorD(A)(P;L):Proof.DualtoLemma18.8.Lemma19.9.064CLetAbeanabeliancategory.AssumeAhasenoughprojectives.Foranyshortexactsequence0!A!B!C!0ofComp+(A)thereexistsacommutativediagraminComp+(A)0// P1// P2// P3// 00// A// B// C// 0 DERIVEDCATEGORIES69wheretheverticalarrowsareprojectiveresolutionsandtherowsareshortexactsequencesofcomplexes.Infact,givenanyprojectiveresolutionP!CwemayassumeP3=P.Proof.DualtoLemma18.9.Lemma19.10.064DLetAbeanabeliancategory.LetP,Kbecomplexes.Letn2Z.Assumethat(1)Pisaboundedcomplexconsistingofprojectiveobjects,(2)Pi=0forin,and(3)Hi(K)=0forin.ThenHomK(A)(P;K)=HomD(A)(P;K)=0.Proof.The rstequalityfollowsfromLemma19.8.Notethatthereisadistin-guishedtriangle( n�1K;K; nK;f;g;h)byRemark12.4.Hence,byLemma4.2itsu cestoproveHomK(A)(P; n�1K)=0andHomK(A)(P; nK)=0.The rstvanishingistrivialandthesecondisLemma19.4.Lemma19.11.064ELetAbeanabeliancategory.Let:P!Land :E!Lbemapsofcomplexes.Letn2Z.Assume(1)PisaboundedcomplexofprojectivesandPi=0forin,(2)Hi( )isanisomorphismfori&#x-277;nandsurjectivefori=n.Thenthereexistsamapofcomplexes :P!Esuchthat andarehomotopic.Proof.ConsidertheconeC=C( )&#
70 15;withmapi:L!C.Notethati
15;withmapi:L!C.NotethatiiszerobyLemma19.10.HencewecanlifttoEbyLemma4.2.20.Rightderivedfunctorsandinjectiveresolutions0156Atthispointwecanusethematerialabovetode netherightderivedfunctorsofanadditivefunctorbetweenanabeliancategoryhavingenoughinjectivesandageneralabeliancategory.Lemma20.1.05THLetAbeanabeliancategory.LetI2Ob(A)beaninjectiveobject.LetIbeaboundedbelowcomplexofinjectivesinA.(1)IcomputesRFrelativetoQis+(A)foranyexactfunctorF:K+(A)!DintoanytriangulatedcategoryD.(2)IisrightacyclicforanyadditivefunctorF:A!BintoanyabeliancategoryB.Proof.Part(2)isadirectconsequencesofpart(1)andDe nition15.3.Toprove(1)let :I!Kbeaquasi-isomorphismintoacomplex.ByLemma18.6weseethat hasaleftinverse.HencethecategoryI=Qis+(A)isessentiallyconstantwithvalueid:I!I.Thusalsotheind-objectI=Qis+(A)�!D;(I!K)7�!F(K)isessentiallyconstantwithvalueF(I).Thisproves(1),seeDe nitions14.2and14.10.Lemma20.2.05TILetAbeanabeliancategorywithenoughinjectives. DERIVEDCATEGORIES70(1)ForanyexactfunctorF:K+(A)!DintoatriangulatedcategoryDtherightderivedfunctorRF:D+(A)�!Diseverywherede ned.(2)ForanyadditivefunctorF:A!BintoanabeliancategoryBtherightderivedfunctorRF:D+(A)�!D+(B)iseverywherede ned.Proof.CombineLemma20.1andProposition16.8forthesecondassertion.Toseethe rstassertioncombineLemma18.3,Lemma20.1,Lemma14.14,andEquation(14.9.1).Lemma20.3.0159LetAbeanabeliancategorywithenoughinjectives.LetF:A!Bbeanadditivefunctor.(1)ThefunctorRFisanexactfunctorD+(A)!D+(B).(2)ThefunctorRFinducesanexactfunctorK+(A)!D+(B).(3)ThefunctorRFinducesa-functorComp+(A)!D+(B).(4)ThefunctorRFinducesa-functorA!D+(B).Proof.Thislemmasimplyreviewssomeoftheresultsobtainedsofar.NotethatbyLemma20.2RFiseverywherede ned.Herearesomereferences:(1)Thederivedfunctorisexact:ThisboilsdowntoLemma14.6.(2)ThisistruebecauseK+(A)!D+(A)isexactandcompositionsofexactfunctorsareexact.(3)ThisistruebecauseComp+(A)!D+(A)isa-functor,seeLemma12.1.(4)ThisistruebecauseA!Comp+(A)isexactand
71 precomposinga-functorbyanexactfunct
precomposinga-functorbyanexactfunctorgivesa-functor.Lemma20.4.015BLetAbeanabeliancategorywithenoughinjectives.LetF:A!Bbealeftexactfunctor.(1)Foranyshortexactsequence0!A!B!C!0ofcomplexesinComp+(A)thereisanassociatedlongexactsequence:::!Hi(RF(A))!Hi(RF(B))!Hi(RF(C))!Hi+1(RF(A))!:::(2)ThefunctorsRiF:A!Barezerofori0.AlsoR0F=F:A!B.(3)WehaveRiF(I)=0fori&#x]TJ/;དྷ ; .96;& T; 16;&#x.715;&#x 0 T; [0;0andIinjective.(4)Thesequence(RiF;)formsauniversal-functor(seeHomology,De ni-tion12.3)fromAtoB.Proof.Thislemmasimplyreviewssomeoftheresultsobtainedsofar.NotethatbyLemma20.2RFiseverywherede ned.Herearesomereferences:(1)ThisfollowsfromLemma20.3part(3)combinedwiththelongexactco-homologysequence(11.1.1)forD+(B).(2)ThisisLemma16.3.(3)Thisisthefactthatinjectiveobjectsareacyclic.(4)ThisisLemma16.6. DERIVEDCATEGORIES7121.Cartan-Eilenbergresolutions015GThissectioncanbeexpanded.Thematerialcanbegeneralizedandappliedinmorecases.Resolutionsneednotuseinjectivesandthemethodalsoworksintheunboundedcaseinsomesituations.De nition21.1.015HLetAbeanabeliancategory.LetKbeaboundedbelowcomplex.ACartan-EilenbergresolutionofKisgivenbyadoublecomplexI;andamorphismofcomplexes:K!I;0withthefollowingproperties:(1)Thereexistsai 0suchthatIp;q=0forallpiandallq.(2)WehaveIp;q=0ifq0.(3)ThecomplexIp;isaninjectiveresolutionofKp.(4)ThecomplexKer(dp;1)isaninjectiveresolutionofKer(dpK).(5)ThecomplexIm(dp;1)isaninjectiveresolutionofIm(dpK).(6)ThecomplexHpI(I;)isaninjectiveresolutionofHp(K).Lemma21.2.015ILetAbeanabeliancategorywithenoughinjectives.LetKbeaboundedbelowcomplex.ThereexistsaCartan-EilenbergresolutionofK.Proof.SupposethatKp=0forpn.DecomposeKintoshortexactsequencesasfollows:SetZp=Ker(dp),Bp=Im(dp�1),Hp=Zp=Bp,andconsider0!Zn!Kn!Bn+1!00!Bn+1!Zn+1!Hn+1!00!Zn+1!Kn+1!Bn+2!00!Bn+2!Zn+2!Hn+2!0:::SetIp;q=0forpn.Inductivelywechooseinjectiveresolutionsasfollows:(1)ChooseaninjectiveresolutionZn!Jn;Z.(2)UsingLemma18.9chooseinje
72 ctiveresolutionsKn!In;,Bn+1!Jn+1;&#
ctiveresolutionsKn!In;,Bn+1!Jn+1;B,andanexactsequenceofcomplexes0!Jn;Z!In;!Jn+1;B!0compatiblewiththeshortexactsequence0!Zn!Kn!Bn+1!0.(3)UsingLemma18.9chooseinjectiveresolutionsZn+1!Jn+1;Z,Hn+1!Jn+1;H,andanexactsequenceofcomplexes0!Jn+1;B!Jn+1;Z!Jn+1;H!0compatiblewiththeshortexactsequence0!Bn+1!Zn+1!Hn+1!0.(4)Etc.Takingasmapsd1:Ip;!Ip+1;thecompositionsIp;!Jp+1;B!Jp+1;Z!Ip+1;everythingisclear.Lemma21.3.015JLetF:A!Bbealeftexactfunctorofabeliancategories.LetKbeaboundedbelowcomplexofA.LetI;beaCartan-EilenbergresolutionforK.Thespectralsequences(0Er;0dr)r0and(00Er;00dr)r0associatedtothedoublecomplexF(I;)satisfytherelations0Ep;q1=RqF(Kp)and00Ep;q2=RpF(Hq(K))Moreover,thesespectralsequencesarebounded,convergetoH(RF(K)),andtheassociatedinduced ltrationsonHn(RF(K))are nite.Proof.Wewillusethefollowingremarkswithoutfurthermention: DERIVEDCATEGORIES72(1)AsIp;isaninjectiveresolutionofKpweseethatRFisde nedatKp[0]withvalueF(Ip;).(2)AsHpI(I;)isaninjectiveresolutionofHp(K)thederivedfunctorRFisde nedatHp(K)[0]withvalueF(HpI(I;)).(3)ByHomology,Lemma25.4thetotalcomplexTot(I;)isaninjectiveres-olutionofK.HenceRFisde nedatKwithvalueF(Tot(I;)).ConsiderthetwospectralsequencesassociatedtothedoublecomplexL;=F(I;),seeHomology,Lemma25.1.Thesearebothbounded,convergetoH(Tot(L;)),andinduce nite ltrationsonHn(Tot(L;)),seeHomology,Lemma25.3.SinceTot(L;)=Tot(F(I;))=F(Tot(I;))computesHn(RF(K))we ndthe nalassertionofthelemmaholdstrue.Computationofthe rstspectralsequence.Wehave0Ep;q1=Hq(Lp;)inotherwords0Ep;q1=Hq(F(Ip;))=RqF(Kp)asdesired.Observeforlaterusethatthemaps0dp;q1:0Ep;q1!0Ep+1;q1arethemapsRqF(Kp)!RqF(Kp+1)inducedbyKp!Kp+1andthefactthatRqFisafunctor.Computationofthesecondspectralsequence.Wehave00Ep;q1=Hq(L;p)=Hq(F(I;p)).Notethatthec
73 omplexI;pisboundedbelow,consistsofi
omplexI;pisboundedbelow,consistsofinjectives,andmoreovereachkernel,image,andcohomologygroupofthedierentialsisaninjectiveobjectofA.Hencewecansplitthedierentials,i.e.,eachdierentialisasplitsurjectionontoadirectsummand.ItfollowsthatthesameistrueafterapplyingF.Hence00Ep;q1=F(Hq(I;p))=F(HqI(I;p)).Thedierentialsonthisare(�1)qtimesFappliedtothedierentialofthecomplexHpI(I;)whichisaninjectiveresolutionofHp(K).HencethedescriptionoftheE2terms.Remark21.4.015KThespectralsequencesofLemma21.3arefunctorialinthecomplexK.ThisfollowsfromfunctorialitypropertiesofCartan-Eilenbergresolutions.Ontheotherhand,theyarebothexamplesofamoregeneralspectralsequencewhichmaybeassociatedtoa lteredcomplexofA.Thefunctorialitywillfollowfromitsconstruction.Wewillreturntothisinthesectiononthe lteredderivedcategory,seeRemark26.15.22.Compositionofrightderivedfunctors015LSometimeswecancomputetherightderivedfunctorofacomposition.SupposethatA;B;Cbeabeliancategories.LetF:A!BandG:B!Cbeleftexactfunctors.AssumethattherightderivedfunctorsRF:D+(A)!D+(B),RG:D+(B)!D+(C),andR(GF):D+(A)!D+(C)areeverywherede ned.Thenthereexistsacanonicaltransformationt:R(GF)�!RGRFoffunctorsfromD+(A)toD+(C),seeLemma14.16.Thistransformationneednotalwaysbeanisomorphism.Lemma22.1.015MLetA;B;Cbeabeliancategories.LetF:A!BandG:B!Cbeleftexactfunctors.AssumeA,Bhaveenoughinjectives.Thefollowingareequivalent(1)F(I)isrightacyclicforGforeachinjectiveobjectIofA,and DERIVEDCATEGORIES73(2)thecanonicalmapt:R(GF)�!RGRF:isisomorphismoffunctorsfromD+(A)toD+(C).Proof.If(2)holds,then(1)followsbyevaluatingtheisomorphismtonRF(I)=F(I).Conversely,assume(1)holds.LetAbeaboundedbelowcomplexofA.ChooseaninjectiveresolutionA!I.Themaptisgiven(seeproofofLemma14.16)bythemapsR(GF)(A)=(GF)(I)=G(F(I)))!RG(F(I))=RG(RF(A))wherethearrowisanisomorphismbyLemma16.7.Lemma22.2(Grothendieckspectralsequence).015NWithassumptionsasinLemma22.1andassumingtheequivalentconditions(1)and(2)hold.LetXbeanobjectofD+(A
74 ).Thereexistsaspectralsequence(Er;dr)r&#
).Thereexistsaspectralsequence(Er;dr)r0consistingofbigradedobjectsErofCwithdrofbidegree(r;�r+1)andwithEp;q2=RpG(Hq(RF(X)))Moreover,thisspectralsequenceisbounded,convergestoH(R(GF)(X)),andinducesa nite ltrationoneachHn(R(GF)(X)).ForanobjectAofAwegetEp;q2=RpG(RqF(A))convergingtoRp+q(GF)(A).Proof.WemayrepresentXbyaboundedbelowcomplexA.ChooseaninjectiveresolutionA!I.ChooseaCartan-EilenbergresolutionF(I)!I;usingLemma21.2.ApplythesecondspectralsequenceofLemma21.3.23.Resolutionfunctors013ULetAbeanabeliancategorywithenoughinjectives.DenoteIthefulladditivesubcategoryofAwhoseobjectsaretheinjectiveobjectsofA.ItturnsoutthatK+(I)andD+(A)areequivalentinthiscase(seeProposition23.1).FormanypurposesitthereforemakessensetothinkofD+(A)asthe(easiertogrok)categoryK+(I)inthiscase.Proposition23.1.013VLetAbeanabeliancategory.AssumeAhasenoughinjectives.DenoteIAthestrictlyfulladditivesubcategorywhoseobjectsaretheinjectiveobjectsofA.ThefunctorK+(I)�!D+(A)isexact,fullyfaithfulandessentiallysurjective,i.e.,anequivalenceoftriangulatedcategories.Proof.Itisclearthatthefunctorisexact.ItisessentiallysurjectivebyLemma18.3.FullyfaithfulnessisaconsequenceofLemma18.8.Proposition23.1impliesthatwecan ndresolutionfunctors.ItturnsoutthatwecanproveresolutionfunctorsexisteveninsomecaseswheretheabeliancategoryAisabigcategory,i.e.,hasaclassofobjects.De nition23.2.013WLetAbeanabeliancategorywithenoughinjectives.Aresolu-tionfunctor6forAisgivenbythefollowingdata: 6Thisislikelynonstandardterminology. DERIVEDCATEGORIES74(1)forallK2Ob(K+(A))aboundedbelowcomplexofinjectivesj(K),and(2)forallK2Ob(K+(A))aquasi-isomorphismiK:K!j(K).Lemma23.3.05TJLetAbeanabeliancategorywithenoughinjectives.Givenaresolutionfunctor(j;i)thereisauniquewaytoturnjintoafunctorandiintoa2-isomorphismproducinga2-commutativediagramK+(A) $$ j// K+(I) zz D+(A)whereIisthefulladditivesubcategoryofAconsistingofinjectiveobjects.Proof.Foreverymorphism :K!LofK+(A)thereisauniquemorphismj( ):j(K
75 ;)!j(L)inK+(I)suchthatK //
;)!j(L)inK+(I)suchthatK // iK LiL j(K)j( )// j(L)iscommutativeinK+(A).ToseethiseitheruseLemmas18.6and18.7ortheequivalentLemma18.8.Theuniquenessimpliesthatjisafunctor,andthecom-mutativityofthediagramimpliesthatigivesa2-morphismwhichwitnessesthe2-commutativityofthediagramofcategoriesinthestatementofthelemma.Lemma23.4.013XLetAbeanabeliancategory.AssumeAhasenoughinjectives.Thenaresolutionfunctorjexistsandisuniqueuptouniqueisomorphismoffunc-tors.Proof.ConsiderthesetofallobjectsKofK+(A).(Recallthatbyourconven-tionsanycategoryhasasetofobjectsunlessmentionedotherwise.)ByLemma18.3everyobjecthasaninjectiveresolution.BytheaxiomofchoicewecanchooseforeachKaninjectiveresolutioniK:K!j(K).Lemma23.5.014WLetAbeanabeliancategorywithenoughinjectives.Anyresolutionfunctorj:K+(A)!K+(I)isexact.Proof.DenoteiK:K!j(K)thecanonicalmapsofDe nition23.2.Firstwediscusstheexistenceofthefunctorialisomorphismj(K[1])!j(K)[1].ConsiderthediagramK[1]iK[1] K[1]iK[1] j(K[1])K// j(K)[1]ByLemmas18.6and18.7thereexistsauniquedottedarrowKinK+(I)mak-ingthediagramcommuteinK+(A).Weomittheveri cationthatthisgivesafunctorialisomorphism.(Hint:useLemma18.7again.)Let(K;L;M;f;g;h)beadistinguishedtriangleofK+(A).Wehavetoshowthat(j(K);j(L);j(M);j(f);j(g);Kj(h))isadistinguishedtriangleofK+(I). DERIVEDCATEGORIES75NotethatwehaveacommutativediagramKf// Lg// Mh// K[1] j(K)j(f)// j(L)j(g)// j(M)Kj(h)// j(K)[1]inK+(A)whoseverticalarrowsarethequasi-isomorphismsiK;iL;iM.Henceweseethattheimageof(j(K);j(L);j(M);j(f);j(g);Kj(h))inD+(A)isiso-morphictoadistinguishedtriangleandhenceadistinguishedtrianglebyTR1.ThusweseefromLemma4.18that(j(K);j(L);j(M);j(f);j(g);Kj(h))isadistinguishedtriangleinK+(I).Lemma23.6.05TKLetAbeanabe
76 liancategorywhichhasenoughinjectives.Let
liancategorywhichhasenoughinjectives.Letjbearesolutionfunctor.WriteQ:K+(A)!D+(A)forthenaturalfunctor.Thenj=j0Qforauniquefunctorj0:D+(A)!K+(I)whichisquasi-inversetothecanonicalfunctorK+(I)!D+(A).Proof.ByLemma11.6Qisalocalizationfunctor.Toprovetheexistenceofj0itsu cestoshowthatanyelementofQis+(A)ismappedtoanisomorphismunderthefunctorj,seeLemma5.6.ThisistruebytheremarksfollowingDe nition23.2.Remark23.7.013YSupposethatAisabigabeliancategorywithenoughinjectivessuchasthecategoryofabeliangroups.Inthiscasewehavetobeslightlymorecarefulinconstructingourresolutionfunctorsincewecannotusetheaxiomofchoicewithaquanti errangingoveraclass.Butnotethattheproofofthelemmadoesshowthatanytwolocalizationfunctorsarecanonicallyisomorphic.Namely,givenquasi-isomorphismsi:K!Iandi0:K!JofaboundedbelowcomplexKintoboundedbelowcomplexesofinjectivesthereexistsaunique(!)morphisma:I!JinK+(I)suchthati0=iaasmorphismsinK+(I).Hencetheonlyissueisexistence,andwewillseehowtodealwiththisinthenextsection.24.Functorialinjectiveembeddingsandresolutionfunctors0140InthissectionweredotheconstructionofaresolutionfunctorK+(A)!K+(I)incasethecategoryAhasfunctorialinjectiveembeddings.Therearetworeasonsforthis:(1)theproofiseasierand(2)theconstructionalsoworksifAisabigabeliancategory.SeeRemark24.3below.LetAbeanabeliancategory.AsbeforedenoteItheadditivefullsubcategoryofAconsistingofinjectiveobjects.ConsiderthecategoryInjRes(A)ofarrows :K!IwhereKisaboundedbelowcomplexofA,IisaboundedbelowcomplexofinjectivesofAand isaquasi-isomorphism.Inotherwords, isaninjectiveresolutionandKisboundedbelow.Thereisanobviousfunctors:InjRes(A)�!Comp+(A)de nedby( :K!I)7!K.Thereisalsoafunctort:InjRes(A)�!K+(I)de nedby( :K!I)7!I. DERIVEDCATEGORIES76Lemma24.1.0141LetAbeanabeliancategory.AssumeAhasfunctorialinjectiveembeddings,seeHomology,De nition27.5.(1)Thereexistsafunctorinj:Comp+(A)!InjRes(A)suchthatsinj=id.(2)Foranyfunctorinj:Comp+(A)!InjRes(A)suchthats&
77 #14;inj=idweobtainaresolutionfunctor,see
#14;inj=idweobtainaresolutionfunctor,seeDe nition23.2.Proof.LetA7!(A!J(A))beafunctorialinjectiveembedding,seeHomology,De nition27.5.We rstnotethatwemayassumeJ(0)=0.Namely,ifnotthenforanyobjectAwehave0!A!0whichgivesadirectsumdecompositionJ(A)=J(0)Ker(J(A)!J(0)).NotethatthefunctorialmorphismA!J(A)hastomapintothesecondsummand.HencewecanreplaceourfunctorbyJ0(A)=Ker(J(A)!J(0))ifneeded.LetKbeaboundedbelowcomplexofA.SayKp=0ifpB.WearegoingtoconstructadoublecomplexI;ofinjectives,togetherwithamap :K!I;0suchthat inducesaquasi-isomorphismofKwiththeassociatedtotalcomplexofI;.FirstwesetIp;q=0wheneverq0.Next,wesetIp;0=J(Kp)and p:Kp!Ip;0thefunctorialembedding.SinceJisafunctorweseethatI;0isacomplexandthat isamorphismofcomplexes.Each pisinjective.AndIp;0=0forpBbecauseJ(0)=0.Next,wesetIp;1=J(Coker(Kp!Ip;0)).AgainbyfunctorialityweseethatI;1isacomplex.AndagainwegetthatIp;1=0forpB.ItisalsoclearthatKpmapsisomorphicallyontoKer(Ip;0!Ip;1).AsourthirdstepwetakeIp;2=J(Coker(Ip;0!Ip;1)).Andsoonandsoforth.AtthispointwecanapplyHomology,Lemma25.4togetthatthemap :K�!Tot(I;)isaquasi-isomorphism.Toprovewegetafunctorinjitreststoshowthattheconstructionaboveisfunctorial.Thisveri cationisomitted.Supposewehaveafunctorinjsuchthatsinj=id.ForeveryobjectKofComp+(A)wecanwriteinj(K)=(iK:K!j(K))ThisprovidesuswitharesolutionfunctorasinDe nition23.2.Remark24.2.05TLSupposeinjisafunctorsuchthatsinj=idasinpart(2)ofLemma24.1.Writeinj(K)=(iK:K!j(K))asintheproofofthatlemma.Suppose :K!Lisamapofboundedbelowcomplexes.Considerthemapinj( )inthecategoryInjRes(A).ItinducesacommutativediagramK // iK LiL j(K)inj( )// j(L)ofmorphismsofcomplexes.Hence,lookingattheproofofLemma23.3weseethatthefunctorj:K+(A)!K+(I)isgivenbytherulej( uptohomotopy)=inj( )uptohomotopy2HomK+(I)(j(K);j(L)) DERIVEDCATEGORIES77Henceweseethatjmatchestinjinthiscase,i.e.,thediagramComp
78 +(A)tinj// && K+(I)K+(A) j:: iscomm
+(A)tinj// && K+(I)K+(A) j:: iscommutative.Remark24.3.0142LetMod(OX)bethecategoryofOX-modulesonaringedspace(X;OX)(ormoregenerallyonaringedsite).WewillseelaterthatMod(OX)hasenoughinjectivesandinfactfunctorialinjectiveembeddings,seeInjectives,Theorem8.4.NotethattheproofofLemma23.4doesnotapplytoMod(OX).ButtheproofofLemma24.1doesapplytoMod(OX).Thusweobtainj:K+(Mod(OX))�!K+(I)whichisaresolutionfunctorwhereIistheadditivecategoryofinjectiveOX-modules.Thisargumentalsoworksinthefollowingcases:(1)ThecategoryModRofR-modulesoveraringR.(2)ThecategoryPMod(O)ofpresheavesofO-modulesonasiteendowedwithapresheafofrings.(3)ThecategoryMod(O)ofsheavesofO-modulesonaringedsite.(4)Addmorehereasneeded.25.Rightderivedfunctorsviaresolutionfunctors05TMThecontentofthefollowinglemmaisthatwecansimplyde neRF(K)=F(j(K))ifwearegivenaresolutionfunctorj.Lemma25.1.05TNLetAbeanabeliancategorywithenoughinjectivesLetF:A!Bbeanadditivefunctorintoanabeliancategory.Let(i;j)bearesolutionfunctor,seeDe nition23.2.TherightderivedfunctorRFofF tsintothefollowing2-commutativediagramD+(A) RF$$ j0// K+(I) Fzz D+(B)wherej0isthefunctorfromLemma23.6.Proof.ByLemma20.1wehaveRF(K)=F(j(K)).Remark25.2.0158InthesituationofLemma25.1weseethatwehaveactuallyliftedtherightderivedfunctortoanexactfunctorFj0:D+(A)!K+(B).Itisoccasionallyusefultousesuchafactorization.26.Filteredderivedcategoryandinjectiveresolutions015OLetAbeanabeliancategory.InthissectionwewillshowthatifAhasenoughinjectives,thensodoesthecategoryFilf(A)insomesense.Onecanusethisobservationtocomputeinthe lteredderivedcategoryofA. DERIVEDCATEGORIES78ThecategoryFilf(A)isanexampleofanexactcategory,seeInjectives,Remark9.6.Aspecialroleisplayedbythestrictmorphisms,seeHomology,De nition19.3,i.e.,themorphismsfsuchthatCoim(f)=Im(f).WewillsaythatacomplexA!B!CinFilf(A)isexactifthesequencegr(A)!gr(B)!gr(C)isexactinA.ThisimpliesthatA!BandB!Carestrictmorphisms,seeHomology,Lemma19.15.De nition26.1.015PLetAbeanabeliancategory.WesayanobjectIofFilf(A)is lteredinjectiveifeachgrp(I)isaninjectiveobjectofA.Lemma26
79 .2.05TPLetAbeanabeliancategory.AnobjectI
.2.05TPLetAbeanabeliancategory.AnobjectIofFilf(A)is lteredinjectiveifandonlyifthereexistab,injectiveobjectsIn,anbofAandanisomorphismI=LanbInsuchthatFpI=LnpIn.Proof.FollowsfromthefactthatanyinjectionJ!MofAissplitifJisaninjectiveobject.Detailsomitted.Lemma26.3.05TQLetAbeanabeliancategory.Anystrictmonomorphismu:I!AofFilf(A)whereIisa lteredinjectiveobjectisasplitinjection.Proof.LetpbethelargestintegersuchthatFpI6=0.Inparticulargrp(I)=FpI.LetI0betheobjectofFilf(A)whoseunderlyingobjectofAisFpIandwith ltrationgivenbyFnI0=0forn�pandFnI0=I0=FpIfornp.NotethatI0!Iisastrictmonomorphismtoo.ThefactthatuisastrictmonomorphismimpliesthatFpI!A=Fp+1(A)isinjective,seeHomology,Lemma19.13.Chooseasplittings:A=Fp+1A!FpIinA.Theinducedmorphisms0:A!I0isastrictmorphismof lteredobjectssplittingthecompositionI0!I!A.HencewecanwriteA=I0Ker(s0)andI=I0Ker(s0jI).NotethatKer(s0jI)!Ker(s0)isastrictmonomorphismandthatKer(s0jI)isa lteredinjectiveobject.Byinductiononthelengthofthe ltrationonIthemapKer(s0jI)!Ker(s0)isasplitinjection.Thuswewin.Lemma26.4.05TRLetAbeanabeliancategory.Letu:A!Bbeastrictmonomor-phismofFilf(A)andf:A!IamorphismfromAintoa lteredinjectiveobjectinFilf(A).Thenthereexistsamorphismg:B!Isuchthatf=gu.Proof.Thepushoutf0:I!IqABoffbyuisastrictmonomorphism,seeHomology,Lemma19.10.HencetheresultfollowsformallyfromLemma26.3.Lemma26.5.05TSLetAbeanabeliancategorywithenoughinjectives.ForanyobjectAofFilf(A)thereexistsastrictmonomorphismA!IwhereIisa lteredinjectiveobject.Proof.Pickabsuchthatgrp(A)=0unlessp2fa;a+1;:::;bg.Foreachn2fa;a+1;:::;bgchooseaninjectionun:A=Fn+1A!InwithInaninjectiveobject.SetI=LanbInwith ltrationFpI=LnpInandsetu:A!Iequaltothedirectsumofthemapsun.Lemma26.6.05TTLetAbeanabeliancategorywithenoughinjectives.ForanyobjectAofFilf(A)thereexistsa lteredquasi-isomorphismA[0]!IwhereIisacomplexof lteredinjectiveobjectswithIn=0forn0. DERIVEDCATEGORIES79Proof.Firstchooseastrictmonomorphismu0:A!I0ofAintoa lteredinjec-tiveobjec
80 t,seeLemma26.5.Next,chooseastrictmonomor
t,seeLemma26.5.Next,chooseastrictmonomorphismu1:Coker(u0)!I1intoa lteredinjectiveobjectofA.Denoted0theinducedmapI0!I1.Next,chooseastrictmonomorphismu2:Coker(u1)!I2intoa lteredinjectiveobjectofA.Denoted1theinducedmapI1!I2.Andsoon.Thisworksbecauseeachofthesequences0!Coker(un)!In+1!Coker(un+1)!0isshortexact,i.e.,inducesashortexactsequenceonapplyinggr.ToseethisuseHomology,Lemma19.13.Lemma26.7.05TULetAbeanabeliancategorywithenoughinjectives.Letf:A!BbeamorphismofFilf(A).Given lteredquasi-isomorphismsA[0]!IandB[0]!JwhereI;Jarecomplexesof lteredinjectiveobjectswithIn=Jn=0forn0,thenthereexistsacommutativediagramA[0]// B[0] I// JProof.AsA[0]!IandC[0]!Jare lteredquasi-isomorphismsweconcludethata:A!I0,b:B!J0andallthemorphismsdnI,dnJarestrict,seeHo-mology,Lemma19.15.WewillinductivelyconstructthemapsfninthefollowingcommutativediagramAa// f I0// f0 I1// f1 I2// f2 :::Bb// J0// J1// J2// :::BecauseA!I0isastrictmonomorphismandbecauseJ0is lteredinjective,wecan ndamorphismf0:I0!J0suchthatf0a=bf,seeLemma26.4.Thecompositiond0Jbfiszero,henced0Jf0a=0,henced0Jf0factorsthroughauniquemorphismCoker(a)=Coim(d0I)=Im(d0I)�!J1:AsIm(d0I)!I1isastrictmonomorphismwecanextendthedisplayedarrowtoamorphismf1:I1!J1byLemma26.4again.Andsoon.Lemma26.8.05TVLetAbeanabeliancategorywithenoughinjectives.Let0!A!B!C!0beashortexactsequenceinFilf(A).Given lteredquasi-isomorphismsA[0]!IandC[0]!JwhereI;Jarecomplexesof lteredinjectiveobjectswithIn=Jn=0forn0,thenthereexistsacommutativediagram0// A[0]// B[0]// C[0]// 00// I// M// J// 0wherethelowerrowisatermwisesplitsequenceofcomplexes. DERIVEDCATEGORIES80Proof.AsA[0]!IandC[0]!Jare lteredquasi-isomorphismsweconcludethata:A!I0,c:C!J0andallthemorphismsdnI,dnJarestrict,seeHomology,Lemma13.4.Wearegoingtostepbystepconstructthesouth-eastandthesoutharrowsinthefollowingcommutativediagramB// b &
81 #31; Cc// b J00 // J1&#
#31; Cc// b J00 // J11 // :::A OO a// I0// I1// I2// :::AsA!Bisastrictmonomorphism,wecan ndamorphismb:B!I0suchthatb =a,seeLemma26.4.AsAisthekernelofthestrictmorphismI0!I1and=Coker( )weobtainauniquemorphism b:C!I1 ttingintothediagram.AscisastrictmonomorphismandI1is lteredinjectivewecan nd0:J0!I1,seeLemma26.4.BecauseB!CisastrictepimorphismandbecauseB!I0!I1!I2iszero,weseethatC!I1!I2iszero.Henced1I0iszeroonC=Im(c).Henced1I0factorsthroughauniquemorphismCoker(c)=Coim(d0J)=Im(d0J)�!I2:AsI2is lteredinjectiveandIm(d0J)!J1isastrictmonomorphismwecanextendthedisplayedmorphismtoamorphism1:J1!I2,seeLemma26.4.Andsoon.WesetM=IJwithdierentialdnM=dnI(�1)n+1n0dnJFinally,themapB[0]!Misgivenbybc:M!I0J0.Lemma26.9.05TWLetAbeanabeliancategorywithenoughinjectives.ForeveryK2K+(Filf(A))thereexistsa lteredquasi-isomorphismK!IwithIboundedbelow,eachIna lteredinjectiveobject,andeachKn!Inastrictmonomorphism.Proof.AfterreplacingKbyashift(whichisharmlessfortheproof)wemayassumethatKn=0forn0.Considertheshortexactsequences0!Ker(d0K)!K0!Coim(d0K)!00!Ker(d1K)!K1!Coim(d1K)!00!Ker(d2K)!K2!Coim(d2K)!0:::oftheexactcategoryFilf(A)andthemapsui:Coim(diK)!Ker(di+1K).Foreachi0wemaychoose lteredquasi-isomorphismsKer(diK)[0]!Iker;iCoim(diK)[0]!Icoim;iwithInker;i;Incoim;i lteredinjectiveandzeroforn0,seeLemma26.6.ByLemma26.7wemayliftuitoamorphismofcomplexesui:Icoim;i!Iker;i+1.Finally,foreachi0wemaycompletethediagrams0// Ker(diK)[0]// Ki[0]// Coim(diK)[0]// 00// Iker;i i// Iii// Icoim;i// 0 DERIVEDCATEGORIES81withthelowersequenceatermwisesplitexactsequence,seeLemma26.8.Fori0setdi:Ii!Ii+1equaltodi= i+1uii.Notethatdidi�1=0becausei i=0.HencewehaveconstructedacommutativediagramI0// I1// I2// :::K0[0]// OO K1[0]// OO K2[0]/
82 / OO :::Heretheverticalarrowsare lte
/ OO :::Heretheverticalarrowsare lteredquasi-isomorphisms.Theupperrowisacom-plexofcomplexesandeachcomplexconsistsof lteredinjectiveobjectswithnononzeroobjectsindegree0.ThusweobtainadoublecomplexbysettingIa;b=Ibaandusingda;b1:Ia;b=Iba!Iba+1=Ia+1;bthemapdbaandusingforda;b2:Ia;b=Iba!Ib+1a=Ia;b+1themapdbIa.DenoteTot(I;)thetotalcomplexassociatedtothisdoublecomplex,seeHomology,De nition18.3.ObservethatthemapsKn[0]!IncomefrommapsKn!In;0whichgiverisetoamapofcomplexesK�!Tot(I;)Weclaimthisisa lteredquasi-isomorphism.Asgr(�)isanadditivefunctor,weseethatgr(Tot(I;))=Tot(gr(I;)).ThuswecanuseHomology,Lemma25.4toconcludethatgr(K)!gr(Tot(I;))isaquasi-isomorphismasdesired.Lemma26.10.05TXLetAbeanabeliancategory.LetK;I2K(Filf(A)).AssumeKis lteredacyclicandIboundedbelowandconsistingof lteredinjectiveobjects.AnymorphismK!Iishomotopictozero:HomK(Filf(A))(K;I)=0.Proof.Let :K!Ibeamorphismofcomplexes.Assumethat j=0forjn.Wewillshowthatthereexistsamorphismh:Kn+1!Insuchthat n=hd.Thus willbehomotopictothemorphismofcomplexesde nedbyj=8:0ifjn n+1�dhifj=n+1 jifj&#x]TJ ;� -1;.93; Td;&#x [00;n+1Thiswillclearlyprovethelemma(byinduction).Toprovetheexistenceofhnotethat ndn�1K=0since n�1=0.SinceKis lteredacyclicweseethatdn�1KanddnKarestrictandthat0!Im(dn�1K)!Kn!Im(dnK)!0isanexactsequenceoftheexactcategoryFilf(A),seeHomology,Lemma19.15.Hencewecanthinkof nasamapintoInde nedonIm(dnK).UsingthatIm(dnK)!Kn+1isastrictmonomorphismandthatInis lteredinjectivewemayliftthismaptoamaph:Kn+1!Inasdesired,seeLemma26.4.Lemma26.11.05TYLetAbeanabeliancategory.LetI2K(Filf(A))beaboundedbelowcomplexconsistingof lteredinjectiveobjects. DERIVEDCATEGORIES82(1)Let :K!LinK(Filf(A))bea lteredquasi-isomorphism.ThenthemapHomK(Filf(A))(L;I)!HomK(Filf(A))(K;I)isbijective.(2)LetL2K(Filf(A)).ThenHomK(Filf(A))(
83 L;I)=HomDF(A)(L;I):P
L;I)=HomDF(A)(L;I):Proof.Proofof(1).Notethat(K;L;C( ); ;i;�p)isadistinguishedtriangleinK(Filf(A))(Lemma9.14)andC( )isa lteredacycliccomplex(Lemma13.4).ThenHomK(Filf(A))(C( );I)// HomK(Filf(A))(L;I)// HomK(Filf(A))(K;I) qq HomK(Filf(A))(C( )[�1];I)isanexactsequenceofabeliangroups,seeLemma4.2.AtthispointLemma26.10guaranteesthattheoutertwogroupsarezeroandhenceHomK(A)(L;I)=HomK(A)(K;I).Proofof(2).Letabeanelementoftherighthandside.Wemayrepresenta= �1where :K!Lisa lteredquasi-isomorphismand :K!Iisamapofcomplexes.Bypart(1)wecan ndamorphism:L!Isuchthat ishomotopicto .Thisprovesthatthemapissurjective.Letbbeanelementofthelefthandsidewhichmapstozerointherighthandside.Thenbisthehomotopyclassofamorphism:L!Isuchthatthereexistsa lteredquasi-isomorphism :K!Lwith homotopictozero.Thenpart(1)showsthatishomotopictozeroalso,i.e.,b=0.Lemma26.12.015QLetAbeanabeliancategorywithenoughinjectives.LetIfFilf(A)denotethestrictlyfulladditivesubcategorywhoseobjectsarethe lteredinjectiveobjects.ThecanonicalfunctorK+(If)�!DF+(A)isexact,fullyfaithfulandessentiallysurjective,i.e.,anequivalenceoftriangulatedcategories.FurthermorethediagramsK+(If)grp // DF+(A)grp K+(I)// D+(A)K+(If)forgetF // DF+(A)forgetF K+(I)// D+(A)arecommutative,whereIAisthestrictlyfulladditivesubcategorywhoseobjectsaretheinjectiveobjects.Proof.ThefunctorK+(If)!DF+(A)isessentiallysurjectivebyLemma26.9.ItisfullyfaithfulbyLemma26.11.Itisanexactfunctorbyourde nitionsregardingdistinguishedtriangles.Thecommutativityofthesquaresisimmediate. DERIVEDCATEGORIES83Remark26.13.015RWecaninvertthearrowofthelemmaonlyifAisacategoryinoursense,namelyifithasasetofobjects.However,supposegivenabigabeliancategoryAwithenoughinjectives,suchasMod(OX)forexample.ThenforanygivensetofobjectsfAigi2IthereisanabeliansubcategoryA0
84 6;Acontainingallofthemandhavingenoughinj
6;Acontainingallofthemandhavingenoughinjectives,seeSets,Lemma12.1.ThuswemayusethelemmaaboveforA0.Thisessentiallymeansthatifweuseasetworthofdiagrams,etcthenwewillneverrunintotroubleusingthelemma.LetA;Bbeabeliancategories.LetT:A!Bbealeftexactfunctor.(WecannotusetheletterFforthefunctorsincethiswouldcon icttoomuchwithouruseoftheletterFtoindicate ltrations.)NotethatTinducesanadditivefunctorT:Filf(A)!Filf(B)bytheruleT(A;F)=(T(A);F)whereFpT(A)=T(FpA)whichmakessenseasTisleftexact.(Warning:Itmaynotbethecasethatgr(T(A))=T(gr(A)).)Thisinducesfunctorsoftriangulatedcategories(26.13.1)05TZT:K+(Filf(A))�!K+(Filf(B))The lteredrightderivedfunctorofTistherightderivedfunctorofDe nition14.2forthisexactfunctorcomposedwiththeexactfunctorK+(Filf(B))!DF+(B)andthemultiplicativesetFQis+(A).AssumeAhasenoughinjectives.AtthispointwecanredothediscussionofSection20tode nethe lteredrightderivedfunctors(26.13.2)015SRT:DF+(A)�!DF+(B)ofourfunctorT.However,insteadwewillproceedasinSection25,anditwillturnoutthatwecande neRTevenifTisjustadditive.Namely,we rstchooseaquasi-inversej0:DF+(A)!K+(If)oftheequivalenceofLemma26.12.ByLemma4.18weseethatj0isanexactfunctoroftriangulatedcategories.Next,wenotethatfora lteredinjectiveobjectIwehavea(noncanonical)decomposition(26.13.3)015TI=Mp2ZIp;withFpI=MqpIqbyLemma26.2.HenceifTisanyadditivefunctorT:A!Bthenwegetanadditivefunctor(26.13.4)05U0Text:If!Filf(B)bysettingText(I)=LT(Ip)withFpText(I)=LqpT(Iq).Notethatwehavethepropertygr(Text(I))=T(gr(I))byconstruction.Henceweobtainafunctor(26.13.5)05U1Text:K+(If)!K+(Filf(B))whichcommuteswithgr.Thenwede ne(26.13.2)bythecomposition(26.13.6)05U2RT=Textj0:SinceRT:D+(A)!D+(B)iscomputedbyinjectiveresolutionsaswell,seeLemmas20.1,thecommutationofTwithgr,andthecommutativediagramsofLemma26.12implythat(26.13.7)015UgrpRT=RTgrp DERIVEDCATEGORIES84and(26.13.8)015V(forgetF)RT=RT(forgetF)asfunctorsDF+(A)!D+(B).The lteredderivedfunctorRT(26.13.2)inducesfunctorsRT:Filf(A)!DF+(B);RT:Comp+(Filf(A))!DF+(B);RT:KF+(A)!DF+(B):Notethatsin
85 ceFilf(A),andComp+(Filf(A))arenolongerab
ceFilf(A),andComp+(Filf(A))arenolongerabelianitdoesnotmakesensetosaythatRTrestrictstoa-functoronthem.(Thiscanberepairedbythinkingofthesecategoriesasexactcategoriesandformulatingthenotionofa-functorfromanexactcategoryintoatriangulatedcategory.)Butitdoesmakesense,anditistruebyconstruction,thatRTisanexactfunctoronthetriangulatedcategoryKF+(A).Lemma26.14.015WLetA;Bbeabeliancategories.LetT:A!Bbealeftexactfunc-tor.AssumeAhasenoughinjectives.Let(K;F)beanobjectofComp+(Filf(A)).Thereexistsaspectralsequence(Er;dr)r0consistingofbigradedobjectsErofBanddrofbidegree(r;�r+1)andwithEp;q1=Rp+qT(grp(K))Moreover,thisspectralsequenceisbounded,convergestoRT(K),andinducesa nite ltrationoneachRnT(K).TheconstructionofthisspectralsequenceisfunctorialintheobjectKofComp+(Filf(A))andtheterms(Er;dr)forr1donotdependonanychoices.Proof.Choosea lteredquasi-isomorphismK!IwithIaboundedbe-lowcomplexof lteredinjectiveobjects,seeLemma26.9.ConsiderthecomplexRT(K)=Text(I),see(26.13.6).Thuswecanconsiderthespectralsequence(Er;dr)r0associatedtothisasa lteredcomplexinB,seeHomology,Section24.ByHomology,Lemma24.2wehaveEp;q1=Hp+q(grp(T(I))).ByEqua-tion(26.13.3)wehaveEp;q1=Hp+q(T(grp(I))),andbyde nitionofa lteredinjectiveresolutionthemapgrp(K)!grp(I)isaninjectiveresolution.HenceEp;q1=Rp+qT(grp(K)).Ontheotherhand,eachInhasa nite ltrationandhenceeachT(In)hasa nite ltration.ThuswemayapplyHomology,Lemma24.11toconcludethatthespectralsequenceisbounded,convergestoHn(T(I))=RnT(K)moreoverinducing nite ltrationsoneachoftheterms.SupposethatK!LisamorphismofComp+(Filf(A)).Choosea lteredquasi-isomorphismL!JwithJaboundedbelowcomplexof lteredinjectiveobjects,seeLemma26.9.Byourresultsabove,forexampleLemma26.11,thereexistsadiagramK// L I// J DERIVEDCATEGORIES85whichcommutesuptohomotopy.Hencewegetamorphismof lteredcomplexesT(I)!T(J)whichgivesrisetothemorphismofsp
86 ectralsequences,seeHomol-ogy,Lemma24.4.T
ectralsequences,seeHomol-ogy,Lemma24.4.Thelaststatementfollowsfromthis.Remark26.15.015XAspromisedinRemark21.4wediscusstheconnectionofthelemmaabovewiththeconstructionsusingCartan-Eilenbergresolutions.Namely,letT:A!Bbealeftexactfunctorofabeliancategories,assumeAhasenoughinjectives,andletKbeaboundedbelowcomplexofA.Wegiveanalternativeconstructionofthespectralsequences0Eand00EofLemma21.3.Firstspectralsequence.Considerthestupid ltrationonKobtainedbyset-tingFp(K)=p(K),seeHomology,Section15.Notethatthisstupidinthesensethatd(Fp(K))Fp+1(K),compareHomology,Lemma24.3.Notethatgrp(K)=Kp[�p]withthis ltration.AccordingtoLemma26.14thereisaspec-tralsequencewithE1termEp;q1=Rp+qT(Kp[�p])=RqT(Kp)asinthespectralsequence0Er.ObservemoreoverthatthedierentialsEp;q1!Ep+1;q1agreewiththedierentialsin0E1,seeHomology,Lemma24.3part(2)andthedescriptionof0d1intheproofofLemma21.3.Secondspectralsequence.Considerthe ltrationonthecomplexKobtainedbysettingFp(K)= �p(K),seeHomology,Section15.Theminussignisnecessarytogetadecreasing ltration.Notethatgrp(K)isquasi-isomorphictoH�p(K)[p]withthis ltration.AccordingtoLemma26.14thereisaspectralsequencewithE1termEp;q1=Rp+qT(H�p(K)[p])=R2p+qT(H�p(K))=00Ei;j2withi=2p+qandj=�p.(Thislooksunnatural,butnotethatwecouldjusthavewelldevelopedthewholetheoryof lteredcomplexesusingincreasing ltrations,withtheendresultthatthisthenlooksnatural,buttheotheronedoesn't.)Weleaveittothereadertoseethatthedierentialsmatchup.Actually,givenaCartan-EilenbergresolutionK!I;theinducedmorphismK!Tot(I;)intotheassociatedtotalcomplexwillbea lteredinjectivereso-lutionforeither ltrationusingsuitable ltrationsonTot(I;).Thiscanbeusedtomatchupthespectralsequencesexactly.27.Extgroups06XPInthissectionwestartdescribingtheExtgroupsofobjectsofanabeliancategory.Firstwehavethefollowingverygeneralde nition.De nition27.1.06XQLetAbeanabeliancategory.Leti2Z.LetX;Ybeobj
87 ectsofD(A).TheithextensiongroupofXbyYist
ectsofD(A).TheithextensiongroupofXbyYisthegroupExtiA(X;Y)=HomD(A)(X;Y[i])=HomD(A)(X[�i];Y):IfA;B2Ob(A)wesetExtiA(A;B)=ExtiA(A[0];B[0]).SinceHomD(A)(X;�),resp.HomD(A)(�;Y)isahomological,resp.cohomologi-calfunctor,seeLemma4.2,weseethatadistinguishedtriangle(Y;Y0;Y00),resp.(X;X0;X00)leadstoalongexactsequence:::!ExtiA(X;Y)!ExtiA(X;Y0)!ExtiA(X;Y00)!Exti+1A(X;Y)!::: DERIVEDCATEGORIES86respectively:::!ExtiA(X00;Y)!ExtiA(X0;Y)!ExtiA(X;Y)!Exti+1A(X00;Y)!:::NotethatsinceD+(A),D�(A),Db(A)arefullsubcategorieswemaycomputetheExtgroupsbyHomgroupsinthesecategoriesprovidedX,Yarecontainedinthem.IncasethecategoryAhasenoughinjectivesorenoughprojectiveswecancomputetheExtgroupsusinginjectiveorprojectiveresolutions.Toavoidconfusion,recallthathavinganinjective(resp.projective)resolutionimpliesvanishingofhomologyinalllow(resp.high)degrees,seeLemmas18.2and19.2.Lemma27.2.06XRLetAbeanabeliancategory.LetX;Y2Ob(K(A)).(1)LetY!Ibeaninjectiveresolution(De nition18.1).ThenExtiA(X;Y)=HomK(A)(X;I[i]):(2)LetP!Xbeaprojectiveresolution(De nition19.1).ThenExtiA(X;Y)=HomK(A)(P[�i];Y):Proof.FollowsimmediatelyfromLemma18.8andLemma19.8.Intherestofthissectionwediscussextensionsofobjectsoftheabeliancategoryitself.Firstweobservethefollowing.Lemma27.3.06XSLetAbeanabeliancategory.(1)LetX,YbeobjectsofD(A).Givena;b2ZsuchthatHi(X)=0fori�aandHj(Y)=0forjb,wehaveExtnA(X;Y)=0fornb�aandExtb�aA(X;Y)=HomA(Ha(X);Hb(Y))(2)LetA;B2Ob(A).Fori0wehaveExtiA(B;A)=0.WehaveExt0A(B;A)=HomA(B;A).Proof.ChoosecomplexesXandYrepresentingXandY.SinceY! bYisaquasi-isomorphism,wemayassumethatYj=0forjb.LetL!Xbeanyquasi-isomorphism.Then aL!Xisaquasi-isomorphism.HenceamorphismX!Y[n]inD(A)canberepresentedasfs�1wheres:L!Xisaquasi-isomorphism,f:L!Y[n]amorphism,andLi=0foria.NotethatfmapsLitoYi+n.Thusf=0ifnb�abecausealwayseitherLiorYi+niszero.Ifn=b�a,thenfcorrespondsexactlytoamorphismHa(X)!Hb(Y).Part(2)isaspecialcaseof(1).LetAbeanabelianc
88 ategory.Supposethat0!A!A0!A00!0isashorte
ategory.Supposethat0!A!A0!A00!0isashortexactsequenceofobjectsofA.Then0!A[0]!A0[0]!A00[0]!0leadstoadistinguishedtriangleinD(A)(seeLemma12.1)hencealongexactsequenceofExtgroups0!Ext0A(B;A)!Ext0A(B;A0)!Ext0A(B;A00)!Ext1A(B;A)!:::Similarly,givenashortexactsequence0!B!B0!B00!0weobtainalongexactsequenceofExtgroups0!Ext0A(B00;A)!Ext0A(B0;A)!Ext0A(B;A)!Ext1A(B00;A)!:::WemayviewtheseExtgroupsasanapplicationoftheconstructionofthederivedcategory.Itshowsonecande neExtgroupsandconstructthelongexactsequence DERIVEDCATEGORIES87ofExtgroupswithoutneedingtheexistenceofenoughinjectivesorprojectives.ThereisanalternativeconstructionoftheExtgroupsduetoYonedawhichavoidstheuseofthederivedcategory,see[Yon60].De nition27.4.06XTLetAbeanabeliancategory.LetA;B2Ob(A).AdegreeiYonedaextensionofBbyAisanexactsequenceE:0!A!Zi�1!Zi�2!:::!Z0!B!0inA.WesaytwoYonedaextensionsEandE0ofthesamedegreeareequivalentifthereexistsacommutativediagram0// A// Zi�1// :::// Z0// B// 00// A// idOO id Z00i�1// OO :::// Z000// OO B// idOO id 00// A// Z0i�1// :::// Z00// B// 0wherethemiddlerowisaYonedaextensionaswell.Itisnotimmediatelyclearthattheequivalenceofthede nitionisanequivalencerelation.AlthoughitisinstructivetoprovethisdirectlythiswillalsofollowfromLemma27.5below.LetAbeanabeliancategorywithobjectsA,B.GivenaYonedaextensionE:0!A!Zi�1!Zi�2!:::!Z0!B!0wede neanassociatedelement(E)2Exti(B;A)asthemorphism(E)=fs�1:B[0]!A[i]wheresisthequasi-isomorphism(:::!0!A!Zi�1!:::!Z0!0!:::)�!B[0]andfisthemorphismofcomplexes(:::!0!A!Zi�1!:::!Z0!0!:::)�!A[i]Wecall(E)=fs�1theclassoftheYonedaextension.ItturnsoutthatthisclasscharacterizestheequivalenceclassoftheYonedaextension.Lemma27.5.06XULetAbeanabeliancategorywithobjectsA,B.AnyelementinExtiA(B;A)is(E)forsomedegreeiYonedaextensionofBbyA.GiventwoYonedaextensionsE,E0ofthesamedegreethenEisequivalenttoE0ifandonlyif(E)=(E0).Proof.Let:B[0]!A[i]beanelementofExtiA(B;A).Wemaywrite=fs�1forsomequasi-isomorphisms:L!B[0]andmapf:L!A[i].A
89 fterreplacingLby 0Lwe
fterreplacingLby 0LwemayassumethatLj=0forj�0.PictureL�i�1// L�i// :::// L0// B// 0AThensettingZi�1=(L�i+1A)=L�iandZj=L�jforj=i�2;:::;0weseethatweobtainadegreeiextensionEofBbyAwhoseclass(E)equals. DERIVEDCATEGORIES88Itisimmediatefromthede nitionsthatequivalentYonedaextensionshavethesameclass.SupposethatE:0!A!Zi�1!Zi�2!:::!Z0!B!0andE0:0!A!Z0i�1!Z0i�2!:::!Z00!B!0areYonedaextensionswiththesameclass.ByconstructionofD(A)asthelocalizationofK(A)atthesetofquasi-isomorphisms,thismeansthereexistsacomplexLandquasi-isomorphismst:L!(:::!0!A!Zi�1!:::!Z0!0!:::)andt0:L!(:::!0!A!Z0i�1!:::!Z00!0!:::)suchthatst=s0t0andft=f0t0,seeCategories,Section27.LetE00bethedegreeiextensionofBbyAconstructedfromthepairL!B[0]andL!A[i]inthe rstparagraphoftheproof.ThenthereaderseesreadilythatthereexistsmorphismsofdegreeiYonedaextensionsE00!EandE00!E0asinthede nitionofequivalentYonedaextensions(detailsomitted).This nishestheproof.Lemma27.6.06XVLetAbeanabeliancategory.LetA,BbeobjectsofA.ThenExt1A(B;A)isthegroupExtA(B;A)constructedinHomology,De nition6.2.Proof.Thisisthecasei=1ofLemma27.5.Lemma27.7.0EWWLetAbeanabeliancategoryandletp0.IfExtpA(B;A)=0foranypairofobjectsA,BofA,thenExtiA(B;A)=0foripandanypairofobjectsA,BofA.Proof.Fori�pwriteanyclassas(E)whereEisaYonedaextensionE:0!A!Zi�1!Zi�2!:::!Z0!B!0ThisispossiblebyLemma27.5.SetC=Ker(Zp�1!Zp)=Im(Zp!Zp�1).Then(E)isthecompositionof(E0)and(E00)whereE0:0!C!Zp�1!:::!Z0!B!0andE00:0!A!Zi�1!Zi�2!:::!Zp!C!0Since(E0)2ExtpA(B;C)=0weconclude.Lemma27.8.0GM4LetAbeanabeliancategory.LetKbeanobjectofDb(A)suchthatExtpA(Hi(K);Hj(K))=0forallp2andi�j.ThenKisisomorphictothedirectsumofitscohomologies:K=LHi(K)[�i].Proof.Choosea;bsuchthatHi(K)=0fori62[a;b].Wewillprovethelemmabyinductiononb�a.Ifb�a0,thentheresultisclear.Ifb�a�0,thenwelookatthedistinguishedtriangleoftruncations
90 0;b�1K!K!Hb(K)[�b]!( b�![0;b�1K!K!Hb(K)[�b]!( b�](884125/0-b-0-1k-k-hb-k-0-b-28-20-b-0-1k-1-seere.jpg "0;b�1K!K!Hb(K)[�b]!( b�")
0;b�1K!K!Hb(K)[�b]!( b�1K)[1]seeRemark12.4.ByLemma4.11ifthelastarrowiszero,thenK= b�1KHb(K)[�b]andwewinbyinduction.AgainusinginductionweseethatHomD(A)(Hb(K)[�b];( b�1K)[1])=MiExtb�i+1A(Hb(K);Hi(K))Byassumptionthedirectsumiszeroandtheproofiscomplete. DERIVEDCATEGORIES89Lemma27.9.0EWXLetAbeanabeliancategory.AssumeExt2A(B;A)=0foranypairofobjectsA,BofA.ThenanyobjectKofDb(A)isisomorphictothedirectsumofitscohomologies:K=LHi(K)[�i].Proof.TheassumptionimpliesthatExtiA(B;A)=0fori2andanypairofobjectsA;BofAbyLemma27.7.HencethislemmaisaspecialcaseofLemma27.8.28.K-groups0FCMAtinybitaboutK0ofatriangulatedcategory.De nition28.1.0FCNLetDbeatriangulatedcategory.WedenoteK0(D)thezerothK-groupofD.Itistheabeliangroupconstructedasfollows.TakethefreeabeliangroupontheobjectsonDandforeverydistinguishedtriangleX!Y!Zimposetherelation[Y]�[X]�[Z]=0.Observethatthisimpliesthat[X[n]]=(�1)n[X]becausewehavethedistinguishedtriangle(X;0;X[1];0;0;�id[1]).Lemma28.2.0FCPLetAbeanabeliancategory.Thenthereisacanonicalidenti ca-tionK0(Db(A))=K0(A)ofzerothK-groups.Proof.GivenanobjectAofAdenoteA[0]theobjectAviewedasacomplexsittingindegree0.If0!A!A0!A00!0isashortexactsequence,thenwegetadistinguishedtriangleA[0]!A0[0]!A00[0]!A[1],seeSection12.ThisshowsthatweobtainamapK0(A)!K0(Db(A))bysending[A]to[A[0]]withapologiesforthehorrendousnotation.Ontheotherhand,givenanobjectXofDb(A)wecanconsidertheelementc(X)=X(�1)i[Hi(X)]2K0(A)GivenadistinguishedtriangleX!Y!Zthelongexactsequenceofcohomology(11.1.1)andtherelationsinK0(A)showthatc(Y)=c(X)+c(Z).Thuscfactorsthroughamapc:K0(Db(A))!K0(A).Wewanttoshowthatthetwomapsabovearemutuallyinverse.ItisclearthatthecompositionK0(A)!K0(Db(A))!K0(A)istheidentity.SupposethatXisaboundedcomplexofA.Theexistenceofthedistinguishedtrianglesofstupidtruncations(seeHomology,Section15)nX!n�1X!Xn�1[�n+1]!(nX)[1]andinductionshowthat[X]=X(�1)i[Xi[0]]inK0(Db(A))(withagainapologiesforthenotation).Itfollowsthatt
91 hecomposi-tionK0(A)!K0(Db(A))issurjectiv
hecomposi-tionK0(A)!K0(Db(A))issurjectivewhich nishestheproof.Lemma28.3.0FCQLetF:D!D0beanexactfunctoroftriangulatedcategories.ThenFinducesagrouphomomorphismK0(D)!K0(D0).Proof.Omitted. DERIVEDCATEGORIES90Lemma28.4.0FCRLetH:D!Abeahomologicalfunctorfromatriangulatedcategorytoanabeliancategory.AssumethatforanyXinDonlya nitenumberoftheobjectsH(X[i])arenonzeroinA.ThenHinducesagrouphomomorphismK0(D)!K0(A)sending[X]toP(�1)i[H(X[i])].Proof.Omitted.Lemma28.5.0FCSLetBbeaweakSerresubcategoryoftheabeliancategoryA.ThentherearecanonicalmapsK0(B)�!K0(DbB(A))�!K0(B)whosecompositioniszero.Thesecondarrowsendstheclass[X]oftheobjectXtotheelementP(�1)i[Hi(X)]ofK0(B).Proof.Omitted.Lemma28.6.0FCTLetD,D0,D00betriangulatedcategories.Let :DD0�!D00beafunctorsuchthatfor xedXinDthefunctorX �:D0!D00isanexactfunctorandfor xedX0inD0thefunctor� X0:D!D00isanexactfunctor.Then inducesabilinearmapK0(D)K0(D0)!K0(D00)whichsends([X];[X0])to[X X0].Proof.Omitted.29.Unboundedcomplexes06XWAreferenceforthematerialinthissectionis[Spa88].Thefollowinglemmaisusefulto ndgoodleftresolutionsofunboundedcomplexes.Lemma29.1.06XXLetAbeanabeliancategory.LetPOb(A)beasubset.AssumePcontains0,isclosedunder( nite)directsums,andeveryobjectofAisaquotientofanelementofP.LetKbeacomplex.ThereexistsacommutativediagramP1 // P2 // ::: 1K// 2K// :::inthecategoryofcomplexessuchthat(1)theverticalarrowsarequasi-isomorphismsandtermwisesurjective,(2)PnisaboundedabovecomplexwithtermsinP,(3)thearrowsPn!Pn+1aretermwisesplitinjectionsandeachcokernelPin+1=PinisanelementofP.Proof.WearegoingtousethatthehomotopycategoryK(A)isatriangulatedcategory,seeProposition10.3.ByLemma15.4wecan ndatermwisesurjectivemapofcomplexesP1! 1Kwhichisaquasi-isomorphismsuchthatthetermsofP1areinP.Byinductionitsu ces,givenP1;:::;PntoconstructPn+1andthemapsPn!Pn+1andPn+1! n+1K.ChooseadistinguishedtrianglePn!
92 0;n+1K!C!Pn[1]inK(A).Appl![0;n+1K!C!Pn[1]inK(A).Appl](884125/0-n-1k-15-c-15-p-15-n-1-ink-a-applyingle.jpg "0;n+1K!C!Pn[1]inK(A).Appl")
0;n+1K!C!Pn[1]inK(A).ApplyingLemma15.4wechooseamapofcomplexesQ!Cwhichisaquasi-isomorphismsuchthatthetermsofQareinP.Bytheaxiomsoftriangulatedcategorieswe DERIVEDCATEGORIES91may tthecompositionQ!C!Pn[1]intoadistinguishedtrianglePn!Pn+1!Q!Pn[1]inK(A).ByLemma10.7wemayanddoassume0!Pn!Pn+1!Q!0isatermwisesplitshortexactsequence.ThisimpliesthatthetermsofPn+1areinPandthatPn!Pn+1isatermwisesplitinjectionwhosecokernelsareinP.BytheaxiomsoftriangulatedcategoriesweobtainamapofdistinguishedtrianglesPn// Pn+1// Q// Pn[1] Pn// n+1K// C// Pn[1]inthetriangulatedcategoryK(A).Chooseanactualmorphismofcomplexesf:Pn+1! n+1K.Theleftsquareofthediagramabovecommutesuptohomotopy,butasPn!Pn+1isatermwisesplitinjectionwecanliftthehomotopyandmodifyourchoiceofftomakeitcommute.Finally,fisaquasi-isomorphism,becausebothPn!PnandQ!Care.Atthispointwehaveallthepropertieswewant,exceptwedon'tknowthatthemapf:Pn+1! n+1Kistermwisesurjective.SincewehavethecommutativediagramPn // Pn+1 nK// n+1Kofcomplexes,byinductionhypothesisweseethatfissurjectiveontermsinalldegreesexceptpossiblynandn+1.ChooseanobjectP2Pandasurjectionq:P!Kn.Considerthemapg:P=(:::!0!P1�!P!0!:::)�! n+1Kwith rstcopyofPindegreenandmapsgivenbyqindegreenanddKqindegreen+1.Thisisasurjectionindegreenandthecokernelindegreen+1isHn+1( n+1K);toseethisrecallthat n+1KhasKer(dn+1K)indegreen+1.However,sincefisaquasi-isomorphismweknowthatHn+1(f)issurjective.Henceafterreplacingf:Pn+1! n+1Kbyfg:Pn+1P! n+1Kwewin.Insomecaseswecanusethelemmaabovetoshowthataleftderivedfunctoriseverywherede ned.Proposition29.2.0794LetF:A!Bbearightexactfunctorofabeliancategories.LetPOb(A)beasubset.Assume(1)Pcontains0,isclosedunder(&#
93 28;nite)directsums,andeveryobjectofAisaq
28;nite)directsums,andeveryobjectofAisaquotientofanelementofP,(2)foranyboundedaboveacycliccomplexPofAwithPn2PforallnthecomplexF(P)isexact,(3)AandBhavecolimitsofsystemsoverN,(4)colimitsoverNareexactinbothAandB,and(5)FcommuteswithcolimitsoverN.ThenLFisde nedonallofD(A). DERIVEDCATEGORIES92Proof.By(1)andLemma15.4foranyboundedabovecomplexKthereexistsaquasi-isomorphismP!KwithPboundedaboveandPn2Pforalln.Supposethats:P!(P0)isaquasi-isomorphismofboundedabovecomplexesconsistingofobjectsofP.ThenF(P)!F((P0))isaquasi-isomorphismbecauseF(C(s))isacyclicbyassumption(2).ThisalreadyshowsthatLFisde nedonD�(A)andthataboundedabovecomplexconsistingofobjectsofPcomputesLF,seeLemma14.15.Next,letKbeanarbitrarycomplexofA.ChooseadiagramP1 // P2 // ::: 1K// 2K// :::asinLemma29.1.NotethatthemapcolimPn!Kisaquasi-isomorphismbecausecolimitsoverNinAareexactandHi(Pn)=Hi(K)forn�i.WeclaimthatF(colimPn)=colimF(Pn)(termwisecolimits)isLF(K),i.e.,thatcolimPncomputesLF.Toseethis,byLemma14.15,itsu cestoprovethefollowingclaim.SupposethatcolimQn=Q ��!P=colimPnisaquasi-isomorphismofcomplexes,suchthateachPn,QnisaboundedabovecomplexwhosetermsareinPandthemapsPn! nPandQn! nQarequasi-isomorphisms.Claim:F( )isaquasi-isomorphism.Theproblemisthatwedonotassumethat isgivenasacolimitofmapsbetweenthecomplexesPnandQn.However,foreachnweknowthatthesolidarrowsinthediagramR Pn Loo // Qn nP n // nQarequasi-isomorphisms.Becausequasi-isomorphismsformamultiplicativesysteminK(A)(seeLemma11.2)wecan ndaquasi-isomorphismL!PnandmapofcomplexesL!Qnsuchthatthediagramabovecommutesuptohomotopy.Then nL!Lisaquasi-isomorphism.Hence(bythe rstpartoftheproof)wecan ndaboundedabovecomplexRwhosetermsareinPandaquasi
94 -isomorphismR!L(asindicatedint
-isomorphismR!L(asindicatedinthediagram).Usingtheresultofthe rstparagraphoftheproofweseethatF(R)!F(Pn)andF(R)!F(Qn)arequasi-isomorphisms.ThusweobtainaisomorphismsHi(F(Pn))!Hi(F(Qn)) tting DERIVEDCATEGORIES93intothecommutativediagramHi(F(Pn))// Hi(F(Qn)) Hi(F(P))// Hi(F(Q))Theexactsameargumentshowsthatthesemapsarealsocompatibleasnvaries.Sinceby(4)and(5)wehaveHi(F(P))=Hi(F(colimPn))=Hi(colimF(Pn))=colimHi(F(Pn))andsimilarlyforQweconcludethatHi( ):Hi(F(P)!Hi(F(Q)isaniso-morphismandtheclaimfollows.Lemma29.3.070FLetAbeanabeliancategory.LetIOb(A)beasubset.AssumeIcontains0,isclosedunder( nite)products,andeveryobjectofAisasubobjectofanelementofI.LetKbeacomplex.Thereexistsacommutativediagram:::// �2K// �1K :::// I2// I1inthecategoryofcomplexessuchthat(1)theverticalarrowsarequasi-isomorphismsandtermwiseinjective,(2)InisaboundedbelowcomplexwithtermsinI,(3)thearrowsIn+1!InaretermwisesplitsurjectionsandKer(Iin+1!Iin)isanelementofI.Proof.ThislemmaisdualtoLemma29.1.30.Derivingadjoints0FNCLetF:D!D0andG:D0!Dbeexactfunctorsoftriangulatedcategories.LetS,resp.S0beamultiplicativesystemforD,resp.D0compatiblewiththetriangulatedstructure.DenoteQ:D!S�1DandQ0:D0!(S0)�1D0thelocalizationfunctors.Inthissituation,byabuseofnotation,oneoftendenotesRFthepartiallyde nedrightderivedfunctorcorrespondingtoQ0F:D!(S0)�1D0andthemultiplicativesystemS.SimilarlyonedenotesLGthepartiallyde nedleftderivedfunctorcorrespondingtoQG:D0!S�1DandthemultiplicativesystemS0.PictureDF// Q D0Q0 S�1DRF// (S0)�1D0andD0G// Q0 DQ (S0)�1D0LG// S�1DLemma30.1.0FNDInthesituationaboveassumeFisrightadjointtoG.LetK2Ob(D)andM2Ob(D0).IfRFisde nedatKandLGisde nedatM,thenthereisacanonicalisomorphismHom(S0)�1D0(M;RF(K))=HomS�1D(LG(M);K) DERIVEDCATEGORIES94Thisisomorphismisfunctorialinbothvariablesont
95 hetriangulatedsubcategoriesofS�1Dand(
hetriangulatedsubcategoriesofS�1Dand(S0)�1D0whereRFandLGarede ned.Proof.SinceRFisde nedatK,weseethattherulewhichassignstoas:K!IinStheobjectF(I)isessentiallyconstantasanind-objectof(S0)�1D0withvalueRF(K).Similarly,therulewhichassignstoat:P!MinS0theobjectG(P)isessentiallyconstantasapro-objectofS�1DwithvalueLG(M).ThuswehaveHom(S0)�1D0(M;RF(K))=colims:K!IHom(S0)�1D0(M;F(I))=colims:K!Icolimt:P!MHomD0(P;F(I))=colimt:P!Mcolims:K!IHomD0(P;F(I))=colimt:P!Mcolims:K!IHomD(G(P);I)=colimt:P!MHomS�1D(G(P);K)=HomS�1D(LG(M);K)The rstequalityholdsbyCategories,Lemma22.9.Thesecondequalityholdsbythede nitionofmorphismsinD(B),seeCategories,Remark27.15.ThethirdequalityholdsbyCategories,Lemma14.10.ThefourthequalityholdsbecauseFandGareadjoint.The fthequalityholdsbyde nitionofmorphisminD(A),seeCategories,Remark27.7.ThesixthequalityholdsbyCategories,Lemma22.10.Weomittheproofoffunctoriality.Lemma30.2.0DVCLetF:A!BandG:B!AbefunctorsofabeliancategoriessuchthatFisarightadjointtoG.LetKbeacomplexofAandletMbeacomplexofB.IfRFisde nedatKandLGisde nedatM,thenthereisacanonicalisomorphismHomD(B)(M;RF(K))=HomD(A)(LG(M);K)ThisisomorphismisfunctorialinbothvariablesonthetriangulatedsubcategoriesofD(A)andD(B)whereRFandLGarede ned.Proof.ThisisaspecialcaseoftheverygeneralLemma30.1.Thefollowinglemmaisanexampleofwhyitiseasiertoworkwithunboundedderivedcategories.Namely,withouthavingtheunboundedderivedfunctors,thelemmacouldnotevenbestated.Lemma30.3.09T5LetF:A!BandG:B!AbefunctorsofabeliancategoriessuchthatFisarightadjointtoG.IfthederivedfunctorsRF:D(A)!D(B)andLG:D(B)!D(A)exist,thenRFisarightadjointtoLG.Proof.ImmediatefromLemma30.2.31.K-injectivecomplexes070GThefollowingtypesofcomplexescanbeusedtocomputerightderivedfunctorsontheunboundedderivedcategory.De nition31.1.070HLetAbeanabeliancategory.AcomplexIisK-injectiveifforeveryacycliccomplexMwehaveHomK(A)(M;I)=0.Inthesituationofthede nitionwehaveinfactHomK(A)(M[i];I)=0foralliasthetranslateofanacycliccompl
96 exisacyclic. DERIVEDCATEGORIES95Lemma31.
exisacyclic. DERIVEDCATEGORIES95Lemma31.2.070ILetAbeanabeliancategory.LetIbeacomplex.Thefollowingareequivalent(1)IisK-injective,(2)foreveryquasi-isomorphismM!NthemapHomK(A)(N;I)!HomK(A)(M;I)isbijective,and(3)foreverycomplexNthemapHomK(A)(N;I)!HomD(A)(N;I)isanisomorphism.Proof.Assume(1).Then(2)holdsbecausethefunctorHomK(A)(�;I)iscoho-mologicalandtheconeonaquasi-isomorphismisacyclic.Assume(2).AmorphismN!IinD(A)isoftheformfs�1:N!Iwheres:M!Nisaquasi-isomorphismandf:M!Iisamap.By(2)thiscorrespondstoauniquemorphismN!IinK(A),i.e.,(3)holds.Assume(3).IfMisacyclicthenMisisomorphictothezerocomplexinD(A)henceHomD(A)(M;I)=0,whenceHomK(A)(M;I)=0by(3),i.e.,(1)holds.Lemma31.3.090XLetAbeanabeliancategory.Let(K;L;M;f;g;h)beadistin-guishedtriangleofK(A).IftwooutofK,L,MareK-injectivecomplexes,thenthethirdistoo.Proof.Followsfromthede nition,Lemma4.2,andthefactthatK(A)isatrian-gulatedcategory(Proposition10.3).Lemma31.4.070JLetAbeanabeliancategory.AboundedbelowcomplexofinjectivesisK-injective.Proof.FollowsfromLemmas31.2and18.8.Lemma31.5.0BK6LetAbeanabeliancategory.LetTbeasetandforeacht2TletItbeaK-injectivecomplex.IfIn=QtIntexistsforalln,thenIisaK-injectivecomplex.Moreover,IrepresentstheproductoftheobjectsItinD(A).Proof.LetKbeancomplex.ObservethatthecomplexC:YbHom(K�b;Ib�1)!YbHom(K�b;Ib)!YbHom(K�b;Ib+1)hascohomologyHomK(A)(K;I)inthemiddle.Similarly,thecomplexCt:YbHom(K�b;Ib�1t)!YbHom(K�b;Ibt)!YbHom(K�b;Ib+1t)computesHomK(A)(K;It).Next,observethatwehaveC=Yt2TCtascomplexesofabeliangroupsbyourchoiceofI.Takingproductsisanexactfunc-toronthecategoryofabeliangroups.HenceifKisacyclic,thenHomK(A)(K;It)=0,henceCtisacyclic,henceCisacyclic,hencewegetHomK(A)(K;I)=0.Thus DERIVEDCATEGORIES96we ndthatIisK-injective.Havingsaidthis,wecanuseLemma31.2toconcludethatHomD(A)(K;I)=Yt2THomD(A)(K;It)andindeedI
97 representstheproductinthederivedcat
representstheproductinthederivedcategory.Lemma31.6.070YLetAbeanabeliancategory.LetF:K(A)!D0beanexactfunc-toroftriangulatedcategories.ThenRFisde nedateverycomplexinK(A)whichisquasi-isomorphictoaK-injectivecomplex.Infact,everyK-injectivecomplexcomputesRF.Proof.ByLemma14.4itsu cestoshowthatRFisde nedataK-injectivecomplex,i.e.,itsu cestoshowaK-injectivecomplexIcomputesRF.Anyquasi-isomorphismI!NisahomotopyequivalenceasithasaninversebyLemma31.2.ThusI!Iisa nalobjectofI=Qis(A)andwewin.Lemma31.7.070KLetAbeanabeliancategory.Assumeeverycomplexhasaquasi-isomorphismtowardsaK-injectivecomplex.ThenanyexactfunctorF:K(A)!D0oftriangulatedcategorieshasarightderivedfunctorRF:D(A)�!D0andRF(I)=F(I)forK-injectivecomplexesI.Proof.ToseethisweapplyLemma14.15withIthecollectionofK-injectivecomplexes.Since(1)holdsbyassumption,itsu cestoprovethatifI!Jisaquasi-isomorphismofK-injectivecomplexes,thenF(I)!F(J)isaniso-morphism.ThisisclearbecauseI!Jisahomotopyequivalence,i.e.,anisomorphisminK(A),byLemma31.2.Thefollowinglemmacanbegeneralizedtolimitsoverbiggerordinals.Lemma31.8.070LLetAbeanabeliancategory.Let:::!I3!I2!I1beaninversesystemofcomplexes.Assume(1)eachInisK-injective,(2)eachmapImn+1!Imnisasplitsurjection,(3)thelimitsIm=limImnexist.ThenthecomplexIisK-injective.Proof.Weurgethereadertoskiptheproofofthislemma.LetMbeanacycliccomplex.LetusabbreviateHn(a;b)=HomA(Ma;Ibn).Withthisnota-tionHomK(A)(M;I)isthecohomologyofthecomplexYmlimnHn(m;m�2)!YmlimnHn(m;m�1)!YmlimnHn(m;m)!YmlimnHn(m;m+1)inthethirdspotfromtheleft.WemayexchangetheorderofQandlimandeachofthecomplexesYmHn(m;m�2)!YmHn(m;m�1)!YmHn(m;m)!YmHn(m;m+1)isexactbyassumption(1).Byassumption(2)themapsinthesystems:::!YmH3(m;m�2)!YmH2(m;m�2)!YmH1(m;m�2) DERIVEDCATEGORIES97aresurjective.ThusthelemmafollowsfromHomology,Lemma31.4.ItappearsthatacombinationofLemmas29.3,31.4,and31.8producesenoughK-injectivesforanyabeliancategorywithenoughinjectivesandco
98 untableproducts.Actually,thismaynotwork!
untableproducts.Actually,thismaynotwork!SeeLemma34.4foranexplanation.Lemma31.9.08BJLetAandBbeabeliancategories.Letu:A!Bandv:B!Abeadditivefunctors.Assume(1)uisrightadjointtov,and(2)visexact.ThenutransformsK-injectivecomplexesintoK-injectivecomplexes.Proof.LetIbeaK-injectivecomplexofA.LetMbeaacycliccomplexofB.Asvisexactweseethatv(M)isanacycliccomplex.Byadjointnessweget0=HomK(A)(v(M);I)=HomK(B)(M;u(I))hencethelemmafollows.32.Boundedcohomologicaldimension07K5Thereisanothercasewheretheunboundedderivedfunctorexists.Namely,whenthefunctorhasboundedcohomologicaldimension.Lemma32.1.07K6LetAbeanabeliancategory.Letd:Ob(A)!f0;1;2;:::;1gbeafunction.Assumethat(1)everyobjectofAisasubobjectofanobjectAwithd(A)=0,(2)d(AB)maxfd(A);d(B)gforA;B2A,and(3)if0!A!B!C!0isshortexact,thend(C)maxfd(A)�1;d(B)g.LetKbeacomplexsuchthatn+d(Kn)tendsto�1asn!�1.Thenthereexistsaquasi-isomorphismK!Lwithd(Ln)=0foralln2Z.Proof.ByLemma15.5wecan ndaquasi-isomorphism0K!MwithMn=0forn0andd(Mn)=0forn0.ThenKisquasi-isomorphictothecomplex:::!K�2!K�1!M0!M1!:::Hencewemayassumethatd(Kn)=0forn 0.Notethattheconditionn+d(Kn)!�1asn!�1isnotviolatedbythisreplacement.WearegoingtoimproveKbyan(in nite)sequenceofelementaryreplacements.Anelementaryreplacementisthefollowing.Chooseanindexnsuchthatd(Kn)&#x]TJ/;དྷ ; .96;& T; 19;&#x.263;&#x 0 T; [0;0.ChooseaninjectionKn!Mwhered(M)=0.SetM0=Coker(Kn!MKn+1).ConsiderthemapofcomplexesK: Kn�1 // Kn // Kn+1 // Kn+2 (K0):Kn�1// M// M0// Kn+2ItisclearthatK!(K0)isaquasi-isomorphism.Moreover,itisclearthatd((K0)n)=0andd((K0)n+1)maxfd(Kn)�1;d(MKn+1)gmaxfd(Kn)�1;d(Kn+1)gandtheothervaluesareunchanged. DERIVEDCATEGORIES98To nishtheproofwecarefulychoosetheorderinwhichtodotheelementaryreplacementssothatforeveryintegermthecomplexmKischangedonlya nitenumberoftimes.Todothisset(K)=maxfn+d(Kn)jd(Kn)�0
99 gandI=fn2Zj(K)=n+d(Kn)andd(Kn)
gandI=fn2Zj(K)=n+d(Kn)andd(Kn)�0gOurassumptionthatn+d(Kn)tendsto�1asn!�1andthefactthatd(Kn)=0forn��0implies(K)+1andthatIisa niteset.Itisclearthat((K0))(K)foranelementarytransformationasabove.Anelementarytransformationchangesthecomplexindegrees(K)+1.Henceifwecan nd nitesequenceofelementarytransformationswhichdecrease(K),thenwewin.However,notethatifwedoanelementarytransformationstartingwiththesmallestelementn2I,thenweeitherdecreasethesizeofI,orweincreaseminI.SinceeveryelementofIis(K)weseethatwewinaftera nitenumberofsteps.Lemma32.2.07K7LetF:A!Bbealeftexactfunctorofabeliancategories.Assume(1)everyobjectofAisasubobjectofanobjectwhichisrightacyclicforF,(2)thereexistsanintegern0suchthatRnF=0,Then(1)RF:D(A)!D(B)exists,(2)anycomplexconsistingofrightacyclicobjectsforFcomputesRF,(3)anycomplexisthesourceofaquasi-isomorphismintoacomplexconsistingofrightacyclicobjectsforF,(4)forE2D(A)(a)Hi(RF( aE)!Hi(RF(E))isanisomorphismforia,(b)Hi(RF(E))!Hi(RF( b�n+1E))isanisomorphismforib,(c)ifHi(E)=0fori62[a;b]forsome�1ab1,thenHi(RF(E))=0fori62[a;b+n�1].Proof.Notethatthe rstassumptionimpliesthatRF:D+(A)!D+(B)exists,seeProposition16.8.LetAbeanobjectofA.ChooseaninjectionA!A0withA0acyclic.ThenweseethatRn+1F(A)=RnF(A0=A)=0bythelongexactcohomologysequence.HenceweconcludethatRn+1F=0.Continuinglikethisusinginductionwe ndthatRmF=0forallmn.WearegoingtouseLemma32.1withthefunctiond:Ob(A)!f0;1;2;:::ggivenbyd(A)=maxf0g[fijRiF(A)6=0g.The rstassumptionofLemma32.1isourassumption(1).ThesecondassumptionofLemma32.1followsfromthefactthatRF(AB)=RF(A)RF(B).ThethirdassumptionofLemma32.1followsfromthelongexactcohomologysequence.HenceforeverycomplexKthereexistsaquasi-isomorphismK!LintoacomplexofobjectsrightacyclicforF.Thisprovesstatement(3).WeclaimthatifL!Misaquasi-isomorphismofcomplexesofrightacyclicobjectsforF,thenF(L)!F(M)isaquasi-isomorphism.Ifweprovet
100 hisclaimthenwegetstatements(1)and(2)ofth
hisclaimthenwegetstatements(1)and(2)ofthelemmabyLemma14.15.Toprovetheclaimpickanintegeri2Z.Considerthedistinguishedtrianglei�n�1L!i�n�1M!Q; DERIVEDCATEGORIES99i.e.,letQbetheconeofthe rstmap.NotethatQisboundedbelowandthatHj(Q)iszeroexceptpossiblyforj=i�n�1orj=i�n�2.WemayapplyRFtoQ.UsingthesecondspectralsequenceofLemma21.3andtheassumedvanishingofcohomology(2)weconcludethatHj(RF(Q))iszeroexceptpossiblyforj2fi�n�2;:::;i�1g.HenceweseethatRF(i�n�1L)!RF(i�n�1M)inducesanisomorphismofcohomologyobjectsindegreesi.ByProposition16.8weknowthatRF(i�n�1L)=i�n�1F(L)andRF(i�n�1M)=i�n�1F(M).WeconcludethatF(L)!F(M)isanisomorphismindegreeiasdesired.Part(4)(a)followsfromLemma16.1.Forpart(4)(b)letEberepresentedbythecomplexLofobjectsrightacyclicforF.Bypart(2)RF(E)isrepresentedbythecomplexF(L)andRF(cL)isrepresentedbycF(L).ConsiderthedistinguishedtriangleHb�n(L)[n�b]! b�nL! b�n+1LofRemark12.4.ThevanishingestablishedabovegivesthatHi(RF( b�nL))agreeswithHi(RF( b�n+1L))forib.Considertheshortexactsequenceofcomplexes0!Im(Lb�n�1!Lb�n)[n�b]!b�nL! b�nL!0Usingthedistinguishedtriangleassociatedtothis(seeSection12)andthevanishingasbeforeweconcludethatHi(RF( b�nL))agreeswithHi(RF(b�nL))forib.SincethemapRF(b�nL)!RF(L)isrepresentedbyb�nF(L)!F(L)weconcludethatthisinturnagreeswithHi(RF(L))foribasdesired.Proofof(4)(c).UndertheassumptiononEwehave a�1E=0andwegetthevanishingofHi(RF(E))foria�1frompart(4)(a).Similarly,wehave b+1E=0andhencewegetthevanishingofHi(RF(E))forib+nfrompart(4)(b).Lemma32.3.07K8LetF:A!Bbearightexactfunctorofabeliancategories.If(1)everyobjec
101 tofAisaquotientofanobjectwhichisleftacyc
tofAisaquotientofanobjectwhichisleftacyclicforF,(2)thereexistsanintegern0suchthatLnF=0,Then(1)LF:D(A)!D(B)exists,(2)anycomplexconsistingofleftacyclicobjectsforFcomputesLF,(3)anycomplexisthetargetofaquasi-isomorphismfromacomplexconsistingofleftacyclicobjectsforF,(4)forE2D(A)(a)Hi(LF( a+n�1E)!Hi(LF(E))isanisomorphismforia,(b)Hi(LF(E))!Hi(LF( bE))isanisomorphismforib,(c)ifHi(E)=0fori62[a;b]forsome�1ab1,thenHi(LF(E))=0fori62[a�n+1;b].Proof.ThisisdualtoLemma32.2.33.Derivedcolimits0A5KInatriangulatedcategorythereisanotionofderivedcolimit. DERIVEDCATEGORIES100De nition33.1.090ZLetDbeatriangulatedcategory.Let(Kn;fn)beasystemofobjectsofD.WesayanobjectKisaderivedcolimit,orahomotopycolimitofthesystem(Kn)ifthedirectsumLKnexistsandthereisadistinguishedtriangleMKn!MKn!K!MKn[1]wherethemapLKn!LKnisgivenby1�fnindegreen.Ifthisisthecase,thenwesometimesindicatethisbythenotationK=hocolimKn.ByTR3aderivedcolimit,ifitexists,isuniqueupto(non-unique)isomorphism.Moreover,byTR1aderivedcolimitofKnexistsassoonasLKnexists.ThederivedcategoryD(Ab)ofthecategoryofabeliangroupsisanexampleofatrian-gulatedcategorywhereallhomotopycolimitsexist.Thenonuniquenessmakesithardtopindownthederivedcolimit.InMoreonAlgebra,Lemma85.4thereader ndsanexactsequence0!R1limHom(Kn;L[�1])!Hom(hocolimKn;L)!limHom(Kn;L)!0describingtheHomsoutofahomotopycolimitintermsoftheusualHoms.Remark33.2.0CRHLetDbeatriangulatedcategory.Let(Kn;fn)beasystemofobjectsofD.WemaythinkofaderivedcolimitasanobjectKofDendowedwithmorphismsin:Kn!Ksuchthatin+1fn=inandsuchthatthereexistsamorphismc:K!LKnwiththepropertythatMKn1�fn���!MKnin�!Kc�!MKn[1]isadistinguishedtriangle.If(K0;i0n;c0)isasecondderivedcolimit,thenthereexistsanisomorphism':K!K0suchthat'in=i0nandc0'=c.Theexistenceof'isTR3andthefactthat'isanisomorphismisLemma4.3.Remark33.3.0CRILetDbeatriangulatedcategory.Let(an):(Kn;fn)!(Ln;gn)beamorphismofsystemsofobjectsofD.Let(K;in;c)beaderivedcolimitofthe rstsystemandlet(L;jn;d)beaderivedcolimitofthesecondsystemwithnotationasi
102 nRemark33.2.Thenthereexistsamorphisma:K!
nRemark33.2.Thenthereexistsamorphisma:K!Lsuchthatain=jnandda=(an[1])c.ThisfollowsfromTR3appliedtothede ningdistinguishedtriangles.Lemma33.4.0CRJLetDbeatriangulatedcategory.Let(Kn;fn)beasystemofobjectsofD.Letn1n2n3:::beasequenceofintegers.AssumeLKnandLKniexist.ThenthereexistsanisomorphismhocolimKni!hocolimKnsuchthatKni// id hocolimKni Kni// hocolimKncommutesforalli. DERIVEDCATEGORIES101Proof.Letgi:Kni!Kni+1bethecompositionfni+1�1:::fni.WeconstructcommutativediagramsLiKni1�gi// b LiKnia LnKn1�fn// LnKnandLnKn1�fn// d LnKnc LiKni1�gi// LiKniasfollows.Letai=ajKnibetheinclusionofKniintothedirectsum.Inotherwords,aisthenaturalinclusion.Letbi=bjKnibethemapKni1;fni;fni+1fni;:::;fni+1�2:::fni�������������������������!KniKni+1:::Kni+1�1Ifni�1jni,thenweletcj=cjKjbethemapKjfni�1:::fj��������!KniWeletdj=djKjbezeroifj6=niforanyiandweletdnibethenaturalinclusionofKniintothedirectsum.Inotherwords,disthenaturalprojection.ByTR3thesediagramsde nemorphisms':hocolimKni!hocolimKnand :hocolimKn!hocolimKniSincecaanddbaretheidentitymapsweseethat' isanisomorphismbyLemma4.3.Theotherwayaroundwegetthemorphismsacandbd.Considerthemorphismh=(hj):LKn!LKngivenbytherule:forni�1jniwesethj:Kj1;fj;fj+1fj;:::;fni�1:::fj��������������������!Kj:::KniThenthereaderveri esthat(1�f)h=id�acandh(1�f)=id�bd.Thismeansthatid� 'hassquarezerobyLemma4.5(smallargumentomitted).Inotherwords, 'diersfromtheidentitybyanilpotentendomorphism,henceisanisomorphism.Thus'and areisomorphismsasdesired.Lemma33.5.0A5LLetAbeanabeliancategory.IfAhasexactcountabledirectsums,thenD(A)hascountabledirectsums.InfactgivenacollectionofcomplexesKiindexedbyacountableindexsetIthetermwisedirectsumLK
103 ;iisthedirectsumofKiinD(A).Proof.Le
;iisthedirectsumofKiinD(A).Proof.LetLbeacomplex.Supposegivenmaps i:Ki!LinD(A).Thismeansthereexistquasi-isomorphismssi:Mi!Kiofcomplexesandmapsofcomplexesfi:Mi!Lsuchthat i=fis�1i.Byassumptionthemapofcom-plexess:MMi�!MKiisaquasi-isomorphism.Hencesettingf=Lfiweseethat =fs�1isamapinD(A)whosecompositionwiththecoprojectionKi!LKiis i.Weomittheveri cationthat isunique.Lemma33.6.093WLetAbeanabeliancategory.AssumecolimitsoverNexistandareexact.Thencountabledirectsumsexistsandareexact.Moreover,if(An;fn)isasystemoverN,thenthereisashortexactsequence0!MAn!MAn!colimAn!0 DERIVEDCATEGORIES102wherethe rstmapindegreenisgivenby1�fn.Proof.The rststatementfollowsfromLAn=colim(A1:::An).Forthesecond,notethatforeachnwehavetheshortexactsequence0!A1:::An�1!A1:::An!An!0wherethe rstmapisgivenbythemaps1�fiandthesecondmapisthesumofthetransitionmaps.Takethecolimittogetthesequenceofthelemma.Lemma33.7.0949LetAbeanabeliancategory.LetLnbeasystemofcomplexesofA.AssumecolimitsoverNexistandareexactinA.ThenthetermwisecolimitL=colimLnisahomotopycolimitofthesysteminD(A).Proof.Wehaveanexactsequenceofcomplexes0!MLn!MLn!L!0byLemma33.6.ThedirectsumsaredirectsumsinD(A)byLemma33.5.Thustheresultfollowsfromthede nitionofderivedcolimitsinDe nition33.1andthefactthatashortexactsequenceofcomplexesgivesadistinguishedtriangle(Lemma12.1).Lemma33.8.0CRKLetDbeatriangulatedcategoryhavingcountabledirectsums.LetAbeanabeliancategorywithexactcolimitsoverN.LetH:D!Abeahomologicalfunctorcommutingwithcountabledirectsums.ThenH(hocolimKn)=colimH(Kn)foranysystemofobjectsofD.Proof.WriteK=hocolimKn.ApplyHtothede ningdistinguishedtriangletogetMH(Kn)!MH(Kn)!H(K)!MH(Kn[1])!MH(Kn[1])wherethe rstmapisgivenby1�H(fn)andthelastmapisgivenby1�H(fn[1]).ApplyLemma33.6toseethatthisprovesthelemma.Thefollowinglemmatellsusthattakingmapsoutofacompactobject(tobede nedlater)commuteswithderivedcolimits.Lemma33.9.094ALetDbeatriangulatedca
104 tegorywithcountabledirectsums.LetK2Dbean
tegorywithcountabledirectsums.LetK2DbeanobjectsuchthatforeverycountablesetofobjectsEn2DthecanonicalmapMHomD(K;En)�!HomD(K;MEn)isabijection.Then,givenanysystemLnofDoverNwhosederivedcolimitL=hocolimLnexistswehavethatcolimHomD(K;Ln)�!HomD(K;L)isabijection.Proof.Considerthede ningdistinguishedtriangleMLn!MLn!L!MLn[1]ApplythecohomologicalfunctorHomD(K;�)(seeLemma4.2).Byelementaryconsiderationsconcerningcolimitsofabeliangroupswegettheresult. DERIVEDCATEGORIES10334.Derivedlimits08TBInatriangulatedcategorythereisanotionofderivedlimit.De nition34.1.08TCLetDbeatriangulatedcategory.Let(Kn;fn)beaninversesystemofobjectsofD.WesayanobjectKisaderivedlimit,orahomotopylimitofthesystem(Kn)iftheproductQKnexistsandthereisadistinguishedtriangleK!YKn!YKn!K[1]wherethemapQKn!QKnisgivenby(kn)7!(kn�fn+1(kn+1)).Ifthisisthecase,thenwesometimesindicatethisbythenotationK=RlimKn.ByTR3aderivedlimit,ifitexists,isuniqueupto(non-unique)isomorphism.Moreover,byTR1aderivedlimitRlimKnexistsassoonasQKnexists.ThederivedcategoryD(Ab)ofthecategoryofabeliangroupsisanexampleofatrian-gulatedcategorywhereallderivedlimitsexist.Thenonuniquenessmakesithardtopindownthederivedlimit.InMoreonAlgebra,Lemma85.3thereader ndsanexactsequence0!R1limHom(L;Kn[�1])!Hom(L;RlimKn)!limHom(L;Kn)!0describingtheHomsintoaderivedlimitintermsoftheusualHoms.Lemma34.2.07KCLetAbeanabeliancategorywithexactcountableproducts.Then(1)D(A)hascountableproducts,(2)countableproductsQKiinD(A)areobtainedbytakingtermwiseproductsofanycomplexesrepresentingtheKi,and(3)Hp(QKi)=QHp(Ki).Proof.LetKibeacomplexrepresentingKiinD(A).LetLbeacomplex.Sup-posegivenmaps i:L!KiinD(A).Thismeansthereexistquasi-isomorphismssi:Ki!Miofcomplexesandmapsofcomplexesfi:L!Misuchthat i=s�1ifi.Byassumptionthemapofcomplexess:YKi�!YMiisaquasi-isomorphism.Hencesettingf=Qfiweseethat =s�1fisamapinD(A)whosecompositionwiththeprojectionQKi!Kiis i.Weomittheveri cationthat isunique.ThedualsofLemmas33.6,33.7,and33.9shouldbestatedhereandproved.How-eve
105 r,wedonotknowanyapplicationsoftheselemma
r,wedonotknowanyapplicationsoftheselemmasfornow.Lemma34.3.0BK7LetAbeanabeliancategorywithcountableproductsandenoughinjectives.Let(Kn)beaninversesystemofD+(A).ThenRlimKnexists.Proof.Itsu cestoshowthatQKnexistsinD(A).ForeverynwecanrepresentKnbyaboundedbelowcomplexInofinjectives(Lemma18.3).ThenQKnisrepresentedbyQIn,seeLemma31.5.Lemma34.4.070MLetAbeanabeliancategorywithcountableproductsandenoughinjectives.LetKbeacomplex.LetInbetheinversesystemofboundedbelowcomplexesofinjectivesproducedbyLemma29.3.ThenI=limInexists,isK-injective,andthefollowingareequivalent(1)themapK!Iisaquasi-isomorphism, DERIVEDCATEGORIES104(2)thecanonicalmapK!Rlim �nKisanisomorphisminD(A).Proof.ThestatementofthelemmamakessenseasRlim �nKexistsbyLemma34.3.EachcomplexInisK-injectivebyLemma31.4.Choosedirectsumdecom-positionsIpn+1=Cpn+1Ipnforalln1.SetCp1=Ip1.ThecomplexI=limInexistsbecausewecantakeIp=Qn1Cpn.Fixp2Z.Weclaimthereisasplitshortexactsequence0!Ip!YIpn!YIpn!0ofobjectsofA.Herethe rstmapisgivenbytheprojectionmapsIp!Ipnandthesecondmapby(xn)7!(xn�fpn+1(xn+1))wherefpn:Ipn!Ipn�1arethetransitionmaps.ThesplittingcomesfromthemapQIpn!QCpn=Ip.Weobtainatermwisesplitshortexactsequenceofcomplexes0!I!YIn!YIn!0HenceacorrespondingdistinguishedtriangleinK(A)andD(A).ByLemma31.5theproductsareK-injectiveandrepresentthecorrespondingproductsinD(A).ItfollowsthatIrepresentsRlimIn(De nition34.1).Moreover,itfollowsthatIisK-injectivebyLemma31.3.BythecommutativediagramofLemma29.3weobtainacorrespondingcommutativediagramK// Rlim �nK I// RlimIninD(A).Sincetherightverticalarrowisanisomorphism(asderivedlimitsarede nedonthelevelofthederivedcategoryandsince �nK!Inisaquasi-isomorphism),thelemmafollows.Lemma34.5.090YLetAbeanabeliancategoryhavingenoughinjectivesandexactcountableproducts.Thenforeverycomplexthereisaquasi-isomorphismtoaK-injectivecomplex.Proof.ByLemma34.4itsu cest
106 oshowthatK!Rlim �nKisanisomo
oshowthatK!Rlim �nKisanisomor-phismforallKinD(A).Considerthede ningdistinguishedtriangleRlim �nK!Y �nK!Y �nK!(Rlim �nK)[1]ByLemma34.2wehaveHp(Y �nK)=Yp�nHp(K)ItfollowsinastraightforwardmannerfromthelongexactcohomologysequenceofthedisplayeddistinguishedtrianglethatHp(Rlim �nK)=Hp(K).35.Operationsonfullsubcategories0FX0LetTbeatriangulatedcategory.WewillidentifyfullsubcategoriesofTwithsubsetsofOb(T).GivenfullsubcategoriesA;B;:::welet(1)A[a;b]for�1ab1bethefullsubcategoryofTconsistingofallobjectsA[�i]withi2[a;b]\ZwithA2Ob(A)(notetheminussign!),(2)smd(A)bethefullsubcategoryofTconsistingofallobjectswhichareisomorphictodirectsummandsofobjectsofA, DERIVEDCATEGORIES105(3)add(A)bethefullsubcategoryofTconsistingofallobjectswhichareisomorphicto nitedirectsumsofobjectsofA,(4)A?BbethefullsubcategoryofTconsistingofallobjectsXofTwhich tintoadistinguishedtriangleA!X!BwithA2Ob(A)andB2Ob(B),(5)A?n=A?:::?Awithn1factors(wewillsee?isassociativebelow),(6)smd(add(A)?n)=smd(add(A)?:::?add(A))withn1factors.IfEisanobjectofT,thenwethinkofEsometimesalsoasthefullsubcategoryofTwhosesingleobjectisE.Thenwecanconsiderthingslikeadd(E[�1;2])andsoonandsoforth.Wewarnthereaderthatthisnotationisnotuniversallyaccepted.Lemma35.1.0FX1LetTbeatriangulatedcategory.GivenfullsubcategoriesA,B,Cwehave(A?B)?C=A?(B?C).Proof.IfwehavedistinguishedtrianglesA!X!BandX!Y!CthenbyAxiomTR4wehavedistinguishedtrianglesA!Y!ZandB!Z!C.Lemma35.2.0FX2LetTbeatriangulatedcategory.GivenfullsubcategoriesA,Bwehavesmd(A)?smd(B)smd(A?B)andsmd(smd(A)?smd(B))=smd(A?B).Proof.SupposewehaveadistinguishedtriangleA1!X!B1whereA1A22Ob(A)andB1B22Ob(B).ThenweobtainadistinguishedtriangleA1A2!A2XB2!B1B2whichprovesthatXisinsmd(A?B).Thisprovestheinclusion.Theequalityfollowstriviallyfromthis.Lemma35.3.0FX3LetTbeatriangulatedcategory.GivenfullsubcategoriesA,Bthefullsubcategoriesadd(A)?add(B)andsmd(add(A))areclosedunderdirectsums.Proof.Namely,ifA!X!BandA0!X0!B0aredi
107 stinguishedtrianglesandA;A02add(A)andB;B
stinguishedtrianglesandA;A02add(A)andB;B02add(B)thenAA0!XX0!BB0isadistinguishedtrianglewithAA02add(A)andBB02add(B).Theresultforsmd(add(A))istrivial.Lemma35.4.0FX4LetTbeatriangulatedcategory.GivenafullsubcategoryAforn1thesubcategoryCn=smd(add(A)?n)=smd(add(A)?:::?add(A))de nedaboveisastrictlyfullsubcategoryofTclosedunderdirectsumsanddirectsummandsandCn+m=smd(Cn?Cm)foralln;m1.Proof.ImmediatefromLemmas35.1,35.2,and35.3.Remark35.5.0FX5LetF:T!T0beanexactfunctoroftriangulatedcategories.GivenafullsubcategoryAofTwedenoteF(A)thefullsubcategoryofT0whoseobjectsconsistsofallobjectsF(A)withA2Ob(A).WehaveF(A[a;b])=F(A)[a;b]F(smd(A))smd(F(A));F(add(A))add(F(A));F(A?B)F(A)?F(B);F(A?n)F(A)?n:Weomitthetrivialveri cations. DERIVEDCATEGORIES106Remark35.6.0FX6LetTbeatriangulatedcategory.GivenfullsubcategoriesA1A2A3:::andBofTwehave[Ai[a;b]=[Ai[a;b]smd[Ai=[smd(Ai);add[Ai=[add(Ai);[Ai?B=[Ai?B;B?[Ai=[B?Ai;[Ai?n=[A?ni:Weomitthetrivialveri cations.Lemma35.7.0FX7LetAbeanabeliancategory.LetD=D(A).LetEOb(A)beasubsetwhichweviewasasubsetofOb(D)also.LetKbeanobjectofD.(1)LetbaandassumeHi(K)iszerofori62[a;b]andHi(K)2Eifi2[a;b].ThenKisinsmd(add(E[a;b])?(b�a+1)).(2)LetbaandassumeHi(K)iszerofori62[a;b]andHi(K)2smd(add(E))ifi2[a;b].ThenKisinsmd(add(E[a;b])?(b�a+1)).(3)LetbaandassumeKcanberepresentedbyacomplexKwithKi=0fori62[a;b]andKi2Efori2[a;b].ThenKisinsmd(add(E[a;b])?(b�a+1)).(4)LetbaandassumeKcanberepresentedbyacomplexKwithKi=0fori62[a;b]andKi2smd(add(E))fori2[a;b].ThenKisinsmd(add(E[a;b])?(b�a+1)).Proof.WewilluseLemma35.4withoutfurthermention.Wewillprove(2)whichtriviallyimplies(1).Weuseinductiononb�a.Ifb�a=0,thenKisisomorphictoHi(K)[�a]inDandtheresultisimmediate.Ifb�a�0,thenweconsiderthedistinguishedtriangle b�1K!K!Kb[�b]andweconcludebyinductiononb�a.Weomittheproofof(3)and(4).Lemma35.8.0FX8LetTbeatriangulatedcategory.LetH:T!Abeahomologicalfunct
108 ortoanabeliancategoryA.LetabandE
ortoanabeliancategoryA.LetabandEOb(T)beasubsetsuchthatHi(E)=0forE2Eandi62[a;b].ThenforX2smd(add(E[�m;m])?n)wehaveHi(X)=0fori62[�m+na;m+nb].Proof.Omitted.Pleasantexerciseinthede nitions.36.Generatorsoftriangulatedcategories09SIInthissectionwebrie yintroduceafewofthedierentnotionsofageneratorforatriangulatedcategory.Ourterminologyistakenfrom[BV03](exceptthatweusesaturatedforwhattheycallépaisse,seeDe nition6.1,andourde nitionofadd(A)isdierent). DERIVEDCATEGORIES107LetDbeatriangulatedcategory.LetEbeanobjectofD.DenotehEi1thestrictlyfullsubcategoryofDconsistingofobjectsinDisomorphictodirectsummandsof nitedirectsumsMi=1;:::;rE[ni]ofshiftsofE.ItisclearthatinthenotationofSection35wehavehEi1=smd(add(E[�1;1]))Forn�1lethEindenotethefullsubcategoryofDconsistingofobjectsofDisomorphictodirectsummandsofobjectsXwhich tintoadistinguishedtriangleA!X!B!A[1]whereAisanobjectofhEi1andBanobjectofhEin�1.InthenotationofSection35wehavehEin=smd(hEi1?hEin�1)EachofthecategorieshEinisastrictlyfulladditive(byLemma35.3)subcategoryofDpreservedundershiftsandundertakingsummands.But,hEinisnotnecessarilyclosedundertakingconesorextensions,hencenotnecessarilyatriangulatedsubcategory.ThiswillbetrueforthesubcategoryhEi=[nhEinaswillbeshowninthelemmasbelow.Lemma36.1.0FX9LetTbeatriangulatedcategory.LetEbeanobjectofT.Forn1wehavehEin=smd(hEi1?:::?hEi1)=smd(hEi1?n)=[m1smd(add(E[�m;m])?n)Forn;n01wehavehEin+n0=smd(hEin?hEin0).Proof.TheleftequalityinthedisplayedformulafollowsfromLemmas35.1and35.2andinduction.Themiddleequalityisamatterofnotation.SincehEi1=smd(add(E[�1;1])])andsinceE[�1;1]=Sm1E[�m;m]weseefromRe-mark35.6andLemma35.2thatwegettheequalityontheright.Thenthe nalstatementfollowsfromtheremarkandthecorrespondingstatementofLemma35.4.Lemma36.2.0ATGLetDbeatriangulatedcategory.LetEbeanobjectofD.ThesubcategoryhEi=[nhEin=[n;m1smd(add(E[�m;m])?n)isastrictlyfull,saturated,triangulatedsubcategoryofDanditisthesmallestsuchsubcategoryofDcontainingth
109 eobjectE.Proof.Theequalityontherightfoll
eobjectE.Proof.TheequalityontherightfollowsfromLemma36.1.ItisclearthathEi=ShEincontainsE,ispreservedundershifts,directsums,directsummands.IfA2hEiaandB2hEibandifA!X!B!A[1]isadistinguishedtriangle,thenX2hEia+bbyLemma36.1.HenceShEinisalsopreservedunderextensionsanditfollowsthatitisatriangulatedsubcategory.Finally,letD0Dbeastrictlyfull,saturated,triangulatedsubcategoryofDcontainingE.ThenD0[�1;1]D0,add(D)D0,smd(D0)D0,andD0?D0D0.Inotherwords,alltheoperationsweusedtoconstructhEioutofEpreserveD0.HencehEiD0andthis nishestheproof. DERIVEDCATEGORIES108De nition36.3.09SJLetDbeatriangulatedcategory.LetEbeanobjectofD.(1)WesayEisaclassicalgeneratorofDifthesmalleststrictlyfull,saturated,triangulatedsubcategoryofDcontainingEisequaltoD,inotherwords,ifhEi=D.(2)WesayEisastronggeneratorofDifhEin=Dforsomen1.(3)WesayEisaweakgeneratororageneratorofDifforanynonzeroobjectKofDthereexistsanintegernandanonzeromapE!K[n].Thisde nitioncanbegeneralizedtothecaseofafamilyofobjects.Lemma36.4.09SKLetDbeatriangulatedcategory.LetE;KbeobjectsofD.Thefollowingareequivalent(1)Hom(E;K[i])=0foralli2Z,(2)Hom(E0;K)=0forallE02hEi.Proof.Theimplication(2))(1)isimmediate.Conversely,assume(1).ThenHom(X;K)=0forallXinhEi1.ArguingbyinductiononnandusingLemma4.2weseethatHom(X;K)=0forallXinhEin.Lemma36.5.09SLLetDbeatriangulatedcategory.LetEbeanobjectofD.IfEisaclassicalgeneratorofD,thenEisagenerator.Proof.AssumeEisaclassicalgenerator.LetKbeanobjectofDsuchthatHom(E;K[i])=0foralli2Z.ByLemma36.4Hom(E0;K)=0forallE0inhEi.However,sinceD=hEiweconcludethatidK=0,i.e.,K=0.Lemma36.6.0FXALetDbeatriangulatedcategorywhichhasastronggenerator.LetEbeanobjectofD.IfEisaclassicalgeneratorofD,thenEisastronggenerator.Proof.LetE0beanobjectofDsuchthatD=hE0in.SinceD=hEiweseethatE02hEimforsomem1byLemma36.2.ThenhE0i1hEimhenceD=hE0in=smd(hE0i1?:::?hE0i1)smd(hEim?:::?hEim)=hEinmasdesired.HereweusedLemma36.1.Remark36.7.0ATHLetDbeatriangulatedcategory.LetEbeanobjectofD.LetTbeapropertyofobjectsofD.Supposethat(1)ifKi2D(A),i=1;:::;rwithT(Ki)fori=1;:::;r,thenT(L
110 Ki),(2)ifK!L!M!K[1]isadistinguishedtrian![Ki),(2)ifK!L!M!K[1]isadistinguishedtrian](884125/ki-2-ifk-l-m-k-1-isadistinguishedtriangl.jpg "Ki),(2)ifK!L!M!K[1]isadistinguishedtrian")
Ki),(2)ifK!L!M!K[1]isadistinguishedtriangleandTholdsfortwo,thenTholdsforthethirdobject,(3)ifT(KL)thenT(K)andT(L),and(4)T(E[n])holdsforalln.ThenTholdsforallobjectsofhEi.37.Compactobjects09SMHereisthede nition.De nition37.1.07LSLetDbeanadditivecategorywitharbitrarydirectsums.AcompactobjectofDisanobjectKsuchthatthemapMi2IHomD(K;Ei)�!HomD(K;Mi2IEi)isbijectiveforanysetIandobjectsEi2Ob(D)parametrizedbyi2I. DERIVEDCATEGORIES109Thisnotionturnsouttobeveryusefulinalgebraicgeometry.Itisanintrinsicconditiononobjectsthatforcestheobjectstobe,well,compact.Lemma37.2.09QHLetDbea(pre-)triangulatedcategorywithdirectsums.ThenthecompactobjectsofDformtheobjectsofaKaroubian,saturated,strictlyfull,(pre-)triangulatedsubcategoryDcofD.Proof.Let(X;Y;Z;f;g;h)beadistinguishedtriangleofDwithXandYcompact.ThenitfollowsfromLemma4.2andthe velemma(Homology,Lemma5.20)thatZisacompactobjecttoo.ItisclearthatifXYiscompact,thenX,Yarecompactobjectstoo.HenceDcisasaturatedtriangulatedsubcategory.SinceDisKaroubianbyLemma4.14weconcludethatthesameistrueforDc.Lemma37.3.09SNLetDbeatriangulatedcategorywithdirectsums.LetEi,i2IbeafamilyofcompactobjectsofDsuchthatLEigeneratesD.TheneveryobjectXofDcanbewrittenasX=hocolimXnwhereX1isadirectsumofshiftsoftheEiandeachtransitionmorphism tsintoadistinguishedtriangleYn!Xn!Xn+1!Yn[1]whereYnisadirectsumofshiftsoftheEi.Proof.SetX1=L(i;m;')Ei[m]wherethedirectsumisoveralltriples(i;m;')suchthati2I,m2Zand':Ei[m]!X.ThenX1comesequippedwithacanonicalmorphismX1!X.GivenXn!XwesetYn=L(i;m;')Ei[m]wherethedirectsumisoveralltriples(i;m;')suchthati2I,m2Z,and':Ei[m]!XnisamorphismsuchthatEi[m]!Xn!Xiszero.ChooseadistinguishedtriangleYn!Xn!Xn+1!Yn[1]andletXn+1!XbeanymorphismsuchthatXn!Xn+1!Xisthegivenone;suchamorphismexistsbyourchoiceofYn.WeobtainamorphismhocolimXn!XbytheconstructionofourmapsXn!X.ChooseadistinguishedtriangleC!hocolimXn!X!C[1]LetEi[m]!Cbeamorphism.SinceEiiscompact,thecompositionEi[m]!C!hocolimXnfactorsthroughXnforsomen,saybyEi[m]!Xn.ThentheconstructionofYnshowsthatthecompositionEi[m]!Xn!Xn+1iszero.Inotherwords,thecompositionEi[m]!C!hocol
111 imXniszero.ThismeansthatourmorphismEi[m]![imXniszero.ThismeansthatourmorphismEi[m]](884125/imxniszero-thismeansthatourmorphismei-m-.jpg "imXniszero.ThismeansthatourmorphismEi[m]")
imXniszero.ThismeansthatourmorphismEi[m]!CcomesfromamorphismEi[m]!X[�1].Theconstruc-tionofX1thenshowsthatsuchmorphismliftstohocolimXnandweconcludethatourmorphismEi[m]!Ciszero.TheassumptionthatLEigeneratesDimpliesthatCiszeroandtheproofisdone.Lemma37.4.09SPWithassumptionsandnotationasinLemma37.3.IfCisacompactobjectandC!Xnisamorphism,thenthereisafactorizationC!E!XnwhereEisanobjectofhEi1:::Eitiforsomei1;:::;it2I.Proof.Weprovethisbyinductiononn.Thebasecasen=1isclear.Ifn�1considerthecompositionC!Xn!Yn�1[1].ThiscanbefactoredthroughsomeE0[1]!Yn�1[1]whereE0isa nitedirectsumofshiftsoftheEi.LetI0Ibe DERIVEDCATEGORIES110the nitesetofindicesthatoccurinthisdirectsum.ThusweobtainE0// C0// C// E0[1] Yn�1// Xn�1// Xn// Yn�1[1]ByinductionthemorphismC0!Xn�1factorsthroughE00!Xn�1withE00anobjectofhLi2I00Eiiforsome nitesubsetI00I.ChooseadistinguishedtriangleE0!E00!E!E0[1]thenEisanobjectofhLi2I0[I00Eii.Byconstructionandtheaxiomsofatri-angulatedcategorywecanchoosemorphismsC!EandamorphismE!Xn ttingintomorphismsoftriangles(E0;C0;C)!(E0;E00;E)and(E0;E00;E)!(Yn�1;Xn�1;Xn).ThecompositionC!E!Xnmaynotequalthegivenmor-phismC!Xn,butthecompositionsintoYn�1areequal.LetC!Xn�1beamorphismthatliftsthedierence.ByinductionassumptionwecanfactorthisthroughamorphismE000!Xn�1withE00anobjectofhLi2I000Eiiforsome nitesubsetI0I.ThusweseethatwegetasolutiononconsideringEE000!XnbecauseEE000isanobjectofhLi2I0[I00[I000Eii.De nition37.5.09SQLetDbeatriangulatedcategorywitharbitrarydirectsums.WesayDiscompactlygeneratedifthereexistsasetEi,i2IofcompactobjectssuchthatLEigeneratesD.Thefollowingpropositionclari estherelationshipbetweenclassicalgeneratorsandweakgenerators.Proposition37.6.09SRLetDbeatriangulatedcategorywithdirectsums.LetEbeacompactobjectofD.Thefollowingareequivalent(1)EisaclassicalgeneratorforDcandDiscompactlygenerated,and(2)EisageneratorforD.Proof.IfEisaclassicalgeneratorforDc,thenDc=hEi.ItfollowsformallyfromtheassumptionthatDiscompactl
112 ygeneratedandLemma36.4thatEisageneratorf
ygeneratedandLemma36.4thatEisageneratorforD.Theconverseismoreinteresting.AssumethatEisageneratorforD.LetXbeacompactobjectofD.ApplyLemma37.3withI=f1gandE1=EtowriteX=hocolimXnasinthelemma.SinceXiscompactwe ndthatX!hocolimXnfactorsthroughXnforsomen(Lemma33.9).ThusXisadirectsummandofXn.ByLemma37.4weseethatXisanobjectofhEiandthelemmaisproven.38.Brownrepresentability0A8EAreferenceforthematerialinthissectionis[Nee96].Lemma38.1.0A8F[Nee96,Theorem3.1].LetDbeatriangulatedcategorywithdirectsumswhichiscom-pactlygenerated.LetH:D!Abbeacontravariantcohomologicalfunctorwhichtransformsdirectsumsintoproducts.ThenHisrepresentable. DERIVEDCATEGORIES111Proof.LetEi,i2IbeasetofcompactobjectssuchthatLi2IEigeneratesD.WemayanddoassumethatthesetofobjectsfEigispreservedundershifts.Considerpairs(i;a)wherei2Ianda2H(Ei)andsetX1=M(i;a)EiSinceH(X1)=Q(i;a)H(Ei)weseethat(a)(i;a)de nesanelementa12H(X1).SetH1=HomD(�;X1).ByYoneda'slemma(Categories,Lemma3.5)theelementa1de nesanaturaltransformationH1!H.WearegoingtoinductivelyconstructXnandtransformationsan:Hn!HwhereHn=HomD(�;Xn).Namely,weapplytheprocedureabovetothefunctorKer(Hn!H)togetanobjectKn+1=M(i;k);k2Ker(Hn(Ei)!H(Ei))EiandatransformationHomD(�;Kn+1)!Ker(Hn!H).ByYoneda'slemmathecompositionHomD(�;Kn+1)!HngivesamorphismKn+1!Xn.WechooseadistinguishedtriangleKn+1!Xn!Xn+1!Kn+1[1]inD.Theelementan2H(Xn)mapstozeroinH(Kn+1)byconstruction.SinceHiscohomologicalwecanliftittoanelementan+12H(Xn+1).WeclaimthatX=hocolimXnrepresentsH.ApplyingHtothede ningdistin-guishedtriangleMXn!MXn!X!MXn[1]weobtainanexactsequenceYH(Xn) YH(Xn) H(X)Thusthereexistsanelementa2H(X)mappingto(an)inQH(Xn).HenceanaturaltransformationHomD(�;X)!HsuchthatHomD(�;X1)!HomD(�;X2)!HomD(�;X3)!:::!HomD(�;X)!Hcommutes.ForeachithemapHomD(Ei;X)!H(Ei)issurjective,byconstruc-tionofX1.Ontheotherhand,byconstructionofXn!Xn+1thekernelofHomD(Ei;Xn)!H(Ei)iskilledbythemapHomD(Ei;Xn)!HomD(Ei;Xn+1).SinceHomD(Ei;X)=colimHomD(Ei;Xn)byLemma33.9weseethatHomD(Ei;X)!H(Ei)isinjective.To nishtheproof,considerthesubcategoryD0=fY2Ob(D)jHomD(Y[n];X)!H(Y[n])i
113 sanisomorphismforallngAsHomD(�;X)!His
sanisomorphismforallngAsHomD(�;X)!Hisatransformationbetweencohomologicalfunctors,thesubcategoryD0isastrictlyfull,saturated,triangulatedsubcategoryofD(detailsomitted;seeproofofLemma6.3).Moreover,asbothHandHomD(�;X)transformdirectsumsintoproducts,weseethatdirectsumsofobjectsofD0areinD0.ThusderivedcolimitsofobjectsofD0areinD0.SincefEigispreservedundershifts,weseethatEiisanobjectofD0foralli.ItfollowsfromLemma37.3thatD0=Dandtheproofiscomplete. DERIVEDCATEGORIES112Proposition38.2.0A8G[Nee96,Theorem4.1].LetDbeatriangulatedcategorywithdirectsumswhichiscompactlygenerated.LetF:D!D0beanexactfunctoroftriangulatedcategorieswhichtransformsdirectsumsintodirectsums.ThenFhasanexactrightadjoint.Proof.ForanobjectYofD0considerthecontravariantfunctorD!Ab;W7!HomD0(F(W);Y)ThisisacohomologicalfunctorasFisexactandtransformsdirectsumsintoproductsasFtransformsdirectsumsintodirectsums.ThusbyLemma38.1we ndanobjectXofDsuchthatHomD(W;X)=HomD0(F(W);Y).TheexistenceoftheadjointfollowsfromCategories,Lemma24.2.ExactnessfollowsfromLemma7.1.39.Admissiblesubcategories0CQPAreferenceforthissectionis[BK89,Section1].De nition39.1.0FXBLetDbeanadditivecategory.LetADbeafullsubcategory.TherightorthogonalA?ofAisthefullsubcategoryconsistingoftheobjectsXofDsuchthatHom(A;X)=0forallA2Ob(A).Theleftorthogonal?AofAisthefullsubcategoryconsistingoftheobjectsXofDsuchthatHom(X;A)=0forallA2Ob(A).Lemma39.2.0CQQLetDbeatriangulatedcategory.LetADbeafullsubcategoryinvariantunderallshifts.ConsideradistinguishedtriangleX!Y!Z!X[1]ofD.Thefollowingareequivalent(1)ZisinA?,and(2)Hom(A;X)=Hom(A;Y)forallA2Ob(A).Proof.ByLemma4.2thefunctorHom(A;�)ishomologicalandhencewegetalongexactsequenceasin(3.5.1).Assume(1)andletA2Ob(A).ThenweconsidertheexactsequenceHom(A[1];Z)!Hom(A;X)!Hom(A;Y)!Hom(A;Z)SinceA[1]2Ob(A)weseethatthe rstandlastgroupsarezero.Thusweget(2).Assume(2)andletA2Ob(A).ThenweconsidertheexactsequenceHom(A;X)!Hom(A;Y)!Hom(A;Z)!Hom(A[�1];X)!Hom(A[�1];Y)andweconcludethatHom(A;Z)=0asdesired.Lemma39.3.0FXCLetDbeatriangulatedcategory.LetADbeafullsubcategoryinvariantunde
114 rallshifts.ThenboththerightorthogonalA?a
rallshifts.ThenboththerightorthogonalA?andtheleftorthogonal?AofAarestrictlyfull,saturated7,triangulatedsubcagoriesofD.Proof.Itisimmediatefromthede nitionsthattheorthogonalsarepreservedundertakingshifts,directsums,anddirectsummands.ConsideradistinguishedtriangleX!Y!Z!X[1]ofD.ByLemma4.16itsu cestoshowthatifXandYareinA?,thenZisinA?.ThisisimmediatefromLemma39.2. 7De nition6.1. DERIVEDCATEGORIES113Lemma39.4.0CQRLetDbeatriangulatedcategory.LetAbeafulltriangulatedsubcategoryofD.ForanobjectXofDconsiderthepropertyP(X):thereexistsadistinguishedtriangleA!X!B!A[1]inDwithAinAandBinA?.(1)IfX1!X2!X3!X1[1]isadistinguishedtriangleandPholdsfortwooutofthree,thenitholdsforthethird.(2)IfPholdsforX1andX2,thenitholdsforX1X2.Proof.LetX1!X2!X3!X1[1]beadistinguishedtriangleandassumePholdsforX1andX2.ChoosedistinguishedtrianglesA1!X1!B1!A1[1]andA2!X2!B2!A2[1]asinconditionP.SinceHom(A1;A2)=Hom(A1;X2)byLemma39.2thereisauniquemorphismA1!A2suchthatthediagramA1 // X1 A2// X2commutes.ChooseanextensionofthistoadiagramA1// X1// Q1// A1[1] A2// X2// Q2// A2[1] A3// X3// Q3// A3[1] A1[1]// X1[1]// Q1[1]// A1[2]asinProposition4.23.ByTR3weseethatQ1=B1andQ2=B2andhenceQ1;Q22Ob(A?).AsQ1!Q2!Q3!Q1[1]isadistinguishedtriangleweseethatQ32Ob(A?)byLemma39.3.SinceAisafulltriangulatedsubcategory,weseethatA3isisomorphictoanobjectofA.ThusX3satis esP.Theothercasesof(1)followfromthiscasebytranslation.Part(2)isaspecialcaseof(1)viaLemma4.11.Lemma39.5.0CQSLetDbeatriangulatedcategory.LetADbeafulltriangulatedsubcategory.Thefollowingareequivalent(1)theinclusionfunctorA!Dhasarightadjoint,and(2)foreveryXinDthereexistsadistinguishedtriangleA!X!B!A[1]inDwithA2Ob(A)andB2Ob(A?).Ifthisholds,thenAissaturated(De nition6.1)andifAisstrictlyfullinD,thenA=?(A?). DERIVEDCATEGORIES114Proof.Assume(1)anddenotev:D!Atherightadjoint.LetX2Ob(D).SetA=v(X).WemayextendtheadjunctionmappingA!XtoadistinguishedtriangleA!X!B!A[1].SinceHomA(A0;A)=
115 HomA(A0;v(X))=HomD(A0;X)forA02Ob(A),weco
HomA(A0;v(X))=HomD(A0;X)forA02Ob(A),weconcludethatB2Ob(A?)byLemma39.2.Assume(2).Wewillcontructtheadjointvexplictly.LetX2Ob(D).ChooseA!X!B!A[1]asin(2).Setv(X)=A.Letf:X!YbeamorphisminD.ChooseA0!Y!B0!A0[1]asin(2).SinceHom(A;A0)=Hom(A;Y)byLemma39.2thereisauniquemorphismf0:A!A0suchthatthediagramAf0 // Xf A0// Ycommutes.Hencewecansetv(f)=f0togetafunctor.ToseethatvisadjointtotheinclusionmorphismuseLemma39.2again.Proofofthe nalstatement.InothertoprovethatAissaturatedwemayreplaceAbythestrictlyfullsubcategoryhavingthesameisomorphismclassesasA;detailsomitted.AssumeAisstrictlyfull.IfweshowthatA=?(A?),thenAwillbesaturatedbyLemma39.3.SincetheincusionA?(A?)isclearitsu cestoprovetheotherinclusion.LetXbeanobjectof?(A?).ChooseadistinguishedtriangleA!X!B!A[1]asin(2).AsHom(X;B)=0byassumptionweseethatA=XB[�1]byLemma4.11.SinceHom(A;B[�1])=0asB2A?thisimpliesB[�1]=0andA=Xasdesired.Lemma39.6.0CQTLetDbeatriangulatedcategory.LetADbeafulltriangulatedsubcategory.Thefollowingareequivalent(1)theinclusionfunctorA!Dhasaleftadjoint,and(2)foreveryXinDthereexistsadistinguishedtriangleB!X!A!B[1]inDwithA2Ob(A)andB2Ob(?A).Ifthisholds,thenAissaturated(De nition6.1)andifAisstrictlyfullinD,thenA=(?A)?.Proof.Omitted.DualtoLemma39.5.De nition39.7.0FXDLetDbeatriangulatedcategory.ArightadmissiblesubcategoryofDisastrictlyfulltriangulatedsubcategorysatisfyingtheequivalentconditionsofLemma39.5.AleftadmissiblesubcategoryofDisastrictlyfulltriangulatedsubcategorysatisfyingtheequivalentconditionsofLemma39.6.Atwo-sidedad-missiblesubcategoryisonewhichisbothrightandleftadmissible.40.Postnikovsystems0D7YAreferenceforthissectionis[Orl97].LetDbeatriangulatedcategory.LetXn!Xn�1!:::!X0beacomplexinD.Inthissectionweconsidertheproblemofconstructingatotalizationofthiscomplex. DERIVEDCATEGORIES115De nition40.1.0D7ZLetDbeatriangulatedcategory.LetXn!Xn�1!:::!X0beacomplexinD.APostnikovsystemisde nedinductivelyasfollows.(1)Ifn=0,thenitisanisomorphismY0!X0.(2)Ifn=1,thenitisachoiceofadistinguishedtriangleY1!X1!Y0!Y1[1]wh
116 ereX1!Y0composedwithY0!X0isthegivenmorph
ereX1!Y0composedwithY0!X0isthegivenmorphismX1!X0.(3)Ifn�1,thenitisachoiceofaPostnikovsystemforXn�1!:::!X0andachoiceofadistinguishedtriangleYn!Xn!Yn�1!Yn[1]wherethemorphismXn!Yn�1composedwithYn�1!Xn�1isthegivenmorphismXn!Xn�1.Givenamorphism(40.1.1)0D80Xn// Xn�1// :::// X0 X0n// X0n�1// :::// X00betweencomplexesofthesamelengthinDthereisanobviousnotionofamorphismofPostnikovsystems.Hereisakeyexample.Example40.2.0D8ZLetAbeanabeliancategory.Let:::!A2!A1!A0beachaincomplexinA.ThenwecanconsidertheobjectsXn=AnandYn=(An!An�1!:::!A0)[�n]ofD(A).WiththeevidentcanonicalmapsYn!XnandY0!Y1[1]!Y2[2]!:::thedistinguishedtrianglesYn!Xn!Yn�1!Yn[1]de neaPostnikovsystemasinDe nition40.1for:::!X2!X1!X0.HereweareusingtheobviousextensionofPostnikovsystemsforanin nitecomplexofD(A).Finally,ifcolimitsoverNexistandareexactinAthenhocolimYn[n]=(:::!A2!A1!A0!0!:::)inD(A).ThisfollowsimmediatelyfromLemma33.7.GivenacomplexXn!Xn�1!:::!X0andaPostnikovsystemasinDe nition40.1wecanconsiderthemapsY0!Y1[1]!:::!Yn[n]Thesemaps ttogetherincertaindistinguishedtrianglesand twiththegivenmapsbetweentheXi.Hereisapictureforn=3:Y0// Y1[1]{{ // Y2[2]{{ // Y3[3]{{ X1[1] +1aa X2[2]+1oo +1cc X3[3]+1oo +1cc DERIVEDCATEGORIES116WeencouragethereadertothinkofYn[n]asobtainedfromX0;X1[1];:::;Xn[n];forexampleifthemapsXi!Xi�1arezero,thenwecantakeYn[n]=Li=0;:::;nXi[i].Postnikovsystemsdonotalwaysexist.Hereisasimplelemmaforlown.Lemma40.3.0D81LetDbeatriangulatedcategory.ConsiderPostnikovsystemsforcomplexesoflengthn.(1)Forn=0Postnikovsystemsalwaysexistandanymorphism(40.1.1)ofcomplexesextendstoauniquemorphismofPostnikovsystems.(2)Forn=1Postnikovsystemsalwaysexistandanymorphism(40.1.1)ofcomplexesextendstoa(nonunique)morphismofPostnikovsystems.(3)Forn=2Postnikovsystemsalwaysexistbutmorphisms(40.1.1)ofcom-plexesingeneraldonotextendtomorphismsofPostnikovsystems.(4)Forn�2Postnikovsystemsdonotalwaysexist.Proof.Thecasen=0isimmediateasisomorphismsareinvertible.Thecasen=1followsimmediatelyfromTR1(existenceoftriangles)andTR
117 3(extendingmorphismstotriangles).Forthec
3(extendingmorphismstotriangles).Forthecasen=2weargueasfollows.SetY0=X0.Bythecasen=1wecanchooseaPostnikovsystemY1!X1!Y0!Y1[1]SincethecompositionX2!X1!X0iszero,wecanfactorX2!X1(nonuniquely)asX2!Y1!X1byLemma4.2.Thenwesimply tthemorphismX2!Y1intoadistinguishedtriangleY2!X2!Y1!Y2[1]togetthePostnikovsystemforn=2.Forn�2wecannotarguesimilarly,aswedonotknowwhetherthecompositionXn!Xn�1!Yn�1iszeroinD.Lemma40.4.0D82LetDbeatriangulatedcategory.Givenamap(40.1.1)considerthecondition(40.4.1)0DW1Hom(Xi[i�j�1];X0j)=0fori�j+1Then(1)IfwehaveaPostnikovsystemforX0n!X0n�1!:::!X00thenproperty(40.4.1)impliesthatHom(Xi[i�j�1];Y0j)=0fori�j+1(2)IfwearegivenPostnikovsystemsforbothcomplexesandwehave(40.4.1),thenthemapextendstoa(nonunique)mapofPostnikovsystems.Proof.We rstprove(1)byinductiononj.Forthebasecasej=0thereisnothingtoproveasY00!X00isanisomorphism.Saytheresultholdsforj�1.WeconsiderthedistinguishedtriangleY0j!X0j!Y0j�1!Y0j[1]ThelongexactsequenceofLemma4.2givesanexactsequenceHom(Xi[i�j�1];Y0j�1[�1])!Hom(Xi[i�j�1];Y0j)!Hom(Xi[i�j�1];X0j)Fromtheinductionhypothesisand(40.4.1)weconcludetheoutergroupsarezeroandwewin. DERIVEDCATEGORIES117Proofof(2).Forn=1theexistenceofmorphismshasbeenestablishedinLemma40.3.Forn�1byinduction,wemayassumegiventhemapofPostnikovsystemsoflengthn�1.TheproblemisthatwedonotknowwhetherthediagramXn// Yn�1 X0n// Y0n�1iscommutative.Denote :Xn!Y0n�1thedierence.Thenwedoknowthatthecompositionof withY0n�1!X0n�1iszero(becauseofwhatitmeanstobeamapofPostnikovsystemsoflengthn�1).BythedistinguishedtriangleY0n�1!X0n�1!Y0n�2!Y0n�1[1],thismeansthat isthecompositionofY0n�2[�1]!Y0n�1withamap 0:Xn!Y0n�2[�1].Then(40.4.1)guarantees 0iszerobypart(1)ofthelemma.Thus iszero.To nishtheproofofexistence,thecommutativityguaranteeswecanchoosethedottedarrow ttingintothediagramYn�1[�1] // Yn// Xn// Yn�1 Y0n�1[�1]// Y0n// X0n// Y0n�1
118 byTR3.Lemma40.5.0FXELetDbeatriangula
byTR3.Lemma40.5.0FXELetDbeatriangulatedcategory.Givenamap(40.1.1)assumewearegivenPostnikovsystemsforbothcomplexes.If(1)Hom(Xi[i];Y0n[n])=0fori=1;:::;n,or(2)Hom(Yn[n];X0n�i[n�i])=0fori=1;:::;n,or(3)Hom(Xj�i[�i+1];X0j)=0andHom(Xj;X0j�i[�i])=0forji�0,thenthereexistsatmostonemorphismbetweenthesePostnikovsystems.Proof.Proofof(1).LookatthefollowingdiagramY0// Y1[1]// {{ Y2[2]// uu :::// Yn[n] rr Y0n[n]ThearrowsarethecompositionofthemorphismYn[n]!Y0n[n]andthemorphismYi[i]!Yn[n].ThearrowY0!Y0n[n]isdeterminedasitisthecompositionY0=X0!X00=Y00!Y0n[n].SincewehavethedistinguishedtriangleY0!Y1[1]!X1[1]weseethatHom(X1[1];Y0n[n])=0guaranteesthatthesecondverticalarrowisunique.SincewehavethedistinguishedtriangleY1[1]!Y2[2]!X2[2]weseethatHom(X2[2];Y0n[n])=0guaranteesthatthethirdverticalarrowisunique.Andsoon.Proofof(2).ThecompositionYn[n]!Y0n[n]!Xn[n]isthesameasthecom-positionYn[n]!Xn[n]!X0n[n]andhenceisunique.Thenusingthedistin-guishedtriangleY0n�1[n�1]!Y0n[n]!X0n[n]weseethatitsu cestoshowHom(Yn[n];Y0n�1[n�1])=0.UsingthedistinguishedtrianglesY0n�i�1[n�i�1]!Y0n�i[n�i]!X0n�i[n�i] DERIVEDCATEGORIES118wegetthisvanishingfromourassumption.Smalldetailsomitted.Proofof(3).LookingattheproofofLemma40.4andarguingbyinductiononnitsu cestoshowthatthedottedarrowinthemorphismoftrianglesYn�1[�1] // Yn// Xn// Yn�1 Y0n�1[�1]// Y0n// X0n// Y0n�1isunique.ByLemma4.8part(5)itsu cestoshowthatHom(Yn�1;X0n)=0andHom(Xn;Y0n�1[�1])=0.Toprovethe rstvanishingweusethedistinguishedtrianglesYn�i�1[�i]!Yn�i[�(i�1)]!Xn�i[�(i�1)]fori�0andinductiononitoseethattheassumedvanishingofHom(Xn�i[�i+1];X0n)isenough.ForthesecondwesimilarlyusethedistinguishedtrianglesY0n�i�1[�i�1]!Y0n�i[�i]!X0n�i[�i]toseethattheassumedvanishingofHom(Xn;X0n�i[�i])isenoughaswell.Lemma40.6.0D83LetDbeatriangulatedcategory.LetXn!Xn�1!:::!X0beacomplexinD.IfHom(Xi[i�j�2];Xj)=0fori�j+2then
119 thereexistsaPostnikovsystem.IfwehaveHom(
thereexistsaPostnikovsystem.IfwehaveHom(Xi[i�j�1];Xj)=0fori�j+1thenanytwoPostnikovsystemsareisomorphic.Proof.Wearguebyinductiononn.Thecasesn=0;1;2followfromLemma40.3.Assumen�2.SupposegivenaPostnikovsystemforthecomplexXn�1!Xn�2!:::!X0.TheonlyobstructiontoextendingthistoaPostnikovsystemoflengthnisthatwehaveto ndamorphismXn!Yn�1suchthatthecompositionXn!Yn�1!Xn�1isequaltothegivenmapXn!Xn�1.ConsideringthedistinguishedtriangleYn�1!Xn�1!Yn�2!Yn�1[1]andtheassociatedlongexactsequencecomingfromthisandthefunctorHom(Xn;�)(seeLemma4.2)we ndthatitsu cestoshowthatthecompositionXn!Xn�1!Yn�2iszero.SinceweknowthatXn!Xn�1!Xn�2iszerowecanapplythedistinguishedtriangleYn�2!Xn�2!Yn�3!Yn�2[1]toseethatitsu cesifHom(Xn;Yn�3[�1])=0.ArguingexactlyasintheproofofLemma40.4part(1)thereadereasilyseesthisfollowsfromtheconditionstatedinthelemma.ThestatementonisomorphismsfollowsfromtheexistenceofamapbetweenthePostnikovsystemsextendingtheidentityonthecomplexproveninLemma40.4part(2)andLemma4.3toshowallthemapsareisomorphisms. DERIVEDCATEGORIES11941.Essentiallyconstantsystems0G38Somepreliminarylemmasonessentiallyconstantsystemsintriangulatedcategories.Lemma41.1.0G39LetDbeatriangulatedcategory.Let(Ai)beaninversesysteminD.Then(Ai)isessentiallyconstant(seeCategories,De nition22.1)ifandonlyifthereexistsaniandforalljiadirectsumdecompositionAj=AZjsuchthat(a)themapsAj0!AjarecompatiblewiththedirectsumdecompositionsandidentityonA,(b)foralljithereexistssomej0jsuchthatZj0!Zjiszero.Proof.Assume(Ai)isessentiallyconstantwithvalueA.ThenA=limAiandthereexistsaniandamorphismAi!Asuchthat(1)thecompositionA!Ai!AistheidentityonAand(2)foralljithereexistsaj0jsuchthatAj0!AjfactorsasAj0!Ai!A!Aj.From(1)weconcludethatforjithemapsA!AjandAj!Ai!AcomposetotheidentityonA.ItfollowsthatAj!AhasakernelZjandthatthemapAZj!Ajisanisomorphism,seeLemmas4.12and4.11.Thesedirectsumdecompositionsclearlysatisfy(a).From(2)weconcludethatforalljthereisaj0jsuchthatZj0!Zjiszero,so(b)holds.Proo
120 foftheconverseisomitted.Lemma41.2.0G
foftheconverseisomitted.Lemma41.2.0G3ALetDbeatriangulatedcategory.LetAn!Bn!Cn!An[1]beaninversesystemofdistinguishedtrianglesinD.If(An)and(Cn)areessen-tiallyconstant,then(Bn)isessentiallyconstantandtheirvalues tintoadistin-guishedtriangleA!B!C!A[1]suchthatforsomen1thereisamapAn // Bn // Cn // An[1] A// B// C// A[1]ofdistinguishedtriangleswhichinducesanisomorphismlimn0nAn0!Aandsim-ilarlyforBandC.Proof.AfterrenumberingwemayassumethatAn=AA0nandCn=CC0nforinversesystems(A0n)and(C0n)whichareessentiallyzero,seeLemma41.1.Inparticular,themorphismCC0n!(AA0n)[1]mapsthesummandCintothesummandA[1]forallnbyamap:C!A[1]whichisindependentofn.ChooseadistinguishedtriangleA!B!C�!A[1]Next,chooseamorphismofdistingishedtriangles(A1!B1!C1!A1[1])!(A!B!C!A[1])whichispossiblebyTR3.ForanyobjectDofDthisinducesacommutativediagram:::// HomD(C;D)// HomD(B;D)// HomD(A;D)// ::::::// colimHomD(Cn;D)// colimHomD(Bn;D)// colimHomD(An;D)// ::: DERIVEDCATEGORIES120Theleftandrightverticalarrowsareisomorphismsandsoaretheonestotheleftandrightofthose.Thusbythe5-lemmaweconcludethatthemiddlearrowisanisomorphism.Itfollowsthat(Bn)isisomorphictotheconstantinversesystemwithvalueBbythediscussioninCategories,Remark22.7.Sincethisisequivalentto(Bn)beingessentiallyconstantwithvalueBbyCategories,Remark22.5theproofiscomplete.Lemma41.3.0G3BLetAbeanabeliancategory.LetAnbeaninversesystemofobjectsofD(A).Assume(1)thereexistintegersabsuchthatHi(An)=0fori62[a;b],and(2)theinversesystemsHi(An)ofAareessentiallyconstantforalli2Z.ThenAnisanessentiallyconstantsystemofD(A)whosevalueAsatis esthatHi(A)isthevalueoftheconstantsystemHi(An)foreachi2Z.Proof.ByRemark12.4weobtainaninversesystemofdistinguishedtriangles aAn!An! a+1An!( aAn)[1]Ofcoursewehave aAn=Ha(An)[�a]inD(A).Thusbyassumptiontheseformanessentiallyconstantsystem.Byinductiononb�awe ndthattheinversesystem a+1Anisessentiallyconstant,saywithvalueA0.ByLemma41.2we ndthatAnisanesse
121 ntiallyconstantsystem.Weomittheproofofth
ntiallyconstantsystem.Weomittheproofofthestatementoncohomologies(hint:usethe nalpartofLemma41.2).Lemma41.4.0G3CLetDbeatriangulatedcategory.LetAn!Bn!Cn!An[1]beaninversesystemofdistinguishedtriangles.IfthesystemCnispro-zero(essen-tiallyconstantwithvalue0),thenthemapsAn!Bndetermineapro-isomorphismbetweenthepro-object(An)andthepro-object(Bn).Proof.ForanyobjectXofDconsidertheexactsequencecolimHom(Cn;X)!colimHom(Bn;X)!colimHom(An;X)!colimHom(Cn[�1];X)!ExactnessfollowsfromLemma4.2combinedwithAlgebra,Lemma8.8.Byas-sumptionthe rstandlasttermarezero.HencethemapcolimHom(Bn;X)!colimHom(An;X)isanisomorphismforallX.ThelemmafollowsfromthisandCategories,Remark22.7.Lemma41.5.0G3DLetAbeanabeliancategory.An!BnbeaninversesystemofmapsofD(A).Assume(1)thereexistintegersabsuchthatHi(An)=0andHi(Bn)=0fori62[a;b],and(2)theinversesystemofmapsHi(An)!Hi(Bn)ofAde neanisomorphismofpro-objectsofAforalli2Z.ThenthemapsAn!Bndetermineapro-isomorphismbetweenthepro-object(An)andthepro-object(Bn). DERIVEDCATEGORIES121Proof.WecaninductivelyextendthemapsAn!BntoaninversesystemofdistinguishedtrianglesAn!Bn!Cn!An[1]byaxiomTR3.ByLemma41.4itsu cestoprovethatCnispro-zero.ByLemma41.3itsu cestoshowthatHp(Cn)ispro-zeroforeachp.Thisfollowsfromassumption(2)andthelongexactsequencesHp(An) n��!Hp(Bn)n��!Hp(Cn)n�!Hp+1(An)n�!Hp+1(Bn)Namely,foreverynwecan ndanm�nsuchthatIm(m)mapstozeroinHp(Cn)becausewemaychoosemsuchthatHp(Bm)!Hp(Bn)factorsthrough n:Hp(An)!Hp(Bn).Forasimilarreasonwemaythenchoosek�msuchthatIm(k)mapstozeroinHp+1(Am).ThenHp(Ck)!Hp(Cn)iszerobe-causeHp(Ck)!Hp(Cm)mapsintoKer(m)andHp(Cm)!Hp(Cn)annihilatesKer(m)=Im(m).42.OtherchaptersPreliminaries(1)Introduction(2)Conventions(3)SetTheory(4)Categories(5)Topology(6)SheavesonSpaces(7)SitesandSheaves(8)Stacks(9)Fields(10)CommutativeAlgebra(11)BrauerGroups(12)HomologicalAlgebra(13)DerivedCategories(14)SimplicialMethods(15)MoreonAlgebra(16)SmoothingRingMaps(17)SheavesofModules(18)ModulesonSites(19)Injectives(20)CohomologyofSheaves(21
122 )CohomologyonSites(22)DierentialGra
)CohomologyonSites(22)DierentialGradedAlgebra(23)DividedPowerAlgebra(24)DierentialGradedSheaves(25)HypercoveringsSchemes(26)Schemes(27)ConstructionsofSchemes(28)PropertiesofSchemes(29)MorphismsofSchemes(30)CohomologyofSchemes(31)Divisors(32)LimitsofSchemes(33)Varieties(34)TopologiesonSchemes(35)Descent(36)DerivedCategoriesofSchemes(37)MoreonMorphisms(38)MoreonFlatness(39)GroupoidSchemes(40)MoreonGroupoidSchemes(41)ÉtaleMorphismsofSchemesTopicsinSchemeTheory(42)ChowHomology(43)IntersectionTheory(44)PicardSchemesofCurves(45)WeilCohomologyTheories(46)AdequateModules(47)DualizingComplexes(48)DualityforSchemes(49)DiscriminantsandDierents(50)deRhamCohomology(51)LocalCohomology(52)AlgebraicandFormalGeometry(53)AlgebraicCurves(54)ResolutionofSurfaces(55)SemistableReduction(56)DerivedCategoriesofVarieties(57)FundamentalGroupsofSchemes(58)ÉtaleCohomology(59)CrystallineCohomology(60)Pro-étaleCohomology(61)MoreÉtaleCohomology DERIVEDCATEGORIES122(62)TheTraceFormulaAlgebraicSpaces(63)AlgebraicSpaces(64)PropertiesofAlgebraicSpaces(65)MorphismsofAlgebraicSpaces(66)DecentAlgebraicSpaces(67)CohomologyofAlgebraicSpaces(68)LimitsofAlgebraicSpaces(69)DivisorsonAlgebraicSpaces(70)AlgebraicSpacesoverFields(71)TopologiesonAlgebraicSpaces(72)DescentandAlgebraicSpaces(73)DerivedCategoriesofSpaces(74)MoreonMorphismsofSpaces(75)FlatnessonAlgebraicSpaces(76)GroupoidsinAlgebraicSpaces(77)MoreonGroupoidsinSpaces(78)Bootstrap(79)PushoutsofAlgebraicSpacesTopicsinGeometry(80)ChowGroupsofSpaces(81)QuotientsofGroupoids(82)MoreonCohomologyofSpaces(83)SimplicialSpaces(84)DualityforSpaces(85)FormalAlgebraicSpaces(86)AlgebraizationofFormalSpaces(87)ResolutionofSurfacesRevisitedDeformationTheory(88)FormalDeformationTheory(89)DeformationTheory(90)TheCotangentComplex(91)DeformationProblemsAlgebraicStacks(92)AlgebraicStacks(93)ExamplesofStacks(94)SheavesonAlgebraicStacks(95)CriteriaforRepresentability(96)Artin'sAxioms(97)QuotandHilbertSpaces(98)PropertiesofAlgebraicStacks(99)MorphismsofAlgebraicStacks(100)LimitsofAlgebraicStacks(101)CohomologyofAlgebraicStacks(102)DerivedC
123 ategoriesofStacks(103)IntroducingAlgebra
ategoriesofStacks(103)IntroducingAlgebraicStacks(104)MoreonMorphismsofStacks(105)TheGeometryofStacksTopicsinModuliTheory(106)ModuliStacks(107)ModuliofCurvesMiscellany(108)Examples(109)Exercises(110)GuidetoLiterature(111)Desirables(112)CodingStyle(113)Obsolete(114)GNUFreeDocumentationLi-cense(115)AutoGeneratedIndexReferences[AGV71]MichaelArtin,AlexanderGrothendieck,andJean-LouisVerdier,TheoriedetoposetcohomologieetaledesschemasI,II,III,LectureNotesinMathematics,vol.269,270,305,Springer,1971.[BK89]Alexei.BondalandMikhailKapranov,Representablefunctors,serrefunctors,andre-constructions,Izv.Akad.NaukSSSRSer.Mat.53(1989),no.6,11831205,1337.[Büh10]TheoBühler,Exactcategories,Expo.Math.28(2010),no.1,169.[BV03]AlexeiBondalandMichelVandenBergh,Generatorsandrepresentabilityoffunctorsincommutativeandnoncommutativegeometry,Mosc.Math.J.3(2003),no.1,136.[Ill72]LucIllusie,ComplexecotangentetdéformationsIandII,LectureNotesinMathematics,Vol.239and283,Springer-Verlag,Berlin,1971/1972.[Kel90]BernhardKeller,Chaincomplexesandstablecategories,ManuscriptaMath.67(1990),no.4,379417.[KS06]MasakiKashiwaraandPierreSchapira,Categoriesandsheaves,GrundlehrenderMath-ematischenWissenschaften,vol.332,Springer-Verlag,Berlin,2006.[Nee96]AmnonNeeman,TheGrothendieckdualitytheoremviaBous eld'stechniquesandBrownrepresentability,J.Amer.Math.Soc.9(1996),no.1,205236. DERIVEDCATEGORIES123[Nee01] ,Triangulatedcategories,AnnalsofMathematicsStudies,vol.148,PrincetonUniversityPress,Princeton,NJ,2001.[Orl97]D.O.Orlov,EquivalencesofderivedcategoriesandK3surfaces,J.Math.Sci.(NewYork)84(1997),no.5,13611381,Algebraicgeometry,7.[Ric89]JeremyRickard,Derivedcategoriesandstableequivalence,J.PureAppl.Algebra61(1989),no.3,303317.[Spa88]NicolasSpaltenstein,Resolutionsofunboundedcomplexes,CompositioMath.65(1988),no.2,121154.[Ver96]Jean-LouisVerdier,Descatégoriesdérivéesdescatégoriesabéliennes,Astérisque(1996),no.239,xii+253,WithaprefacebyLucIllusie,EditedandwithanotebyGeorgesMaltsiniotis.[Yon60]NobuoYoneda,OnExtandexactsequences,J.Fac.Sci.Univ.TokyoSect.I8(19