4MARKHAIMANcomplexesandfisaboundaryf2imd01ifandonlyiffisnullhomotopicThusHomKAA15B1524H0Hom15A15B15Proposition18ThefollowingpropertiesofahomomorphismfABinCAareequivalentaf240bffactorsthroughthecanoni ID: 884130
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1 2MARKHAIMANonaringedspaceX,andsomeofthei
2MARKHAIMANonaringedspaceX,andsomeoftheirabeliansubcategories,suchasthecategoryofquasi-coherentOX-modules.Inthese`concrete'abeliancategories,theobjectsaresetsanditisconvenienttoformulateargumentsintermsoftheirelements(sectionsonopensetsandgermsinstalks,inthecaseofsheaves).ArgumentsintermsofelementscanalsobejustiedinabstractabeliancategoriesbyassociatingtoanobjectAtheabeliangroupsHom(T;A)forvariousobjectsT.OnethinksofarrowsT!Aas`T-valuedelements'ofA,justasinthecategoryofschemeswethinkofmorphismsT!XasT-valuedpointsofX.Denition1.2.AcomplexinanabeliancategoryAisasequenceAofobjectsandmaps(calleddi erentials)d�1!A0d0!A1d1!A2d2!suchthatdi+1di=0foralli.Ahomomorphismf:A!Bofcomplexesconsistsofmapsfi:Ai!Bicommutingwiththedi erentials.ThecomplexesinAformanabeliancategoryC(A).TheobjectHi(A)=ker(di)=im(di�1)isthei-thcohomologyofA.I'llsticktocohomologyindexing.Itisalsoconventiona
2 lsometimestouse\homology"indexingde
lsometimestouse\homology"indexingdenedbyAi=A�i,Hi(A)=H�i(A).TheshiftA[n]ofAisthecomplexwithtermsA[n]i=Ai+nanddi erentialsdiA[n]=(�1)ndi+nA.Iff:A!Bisahomomorphismofcomplexes,wedenef[n]:A[n]!B[n]byf[n]i=fi+n.ThismakestheshiftafunctorfromC(A)toitself.Thegeneralprinciplegoverningsignrulesisthatallconstructionsinvolvingcomplexesshouldbe`graded-commutative,'meaningthathomogeneousoperatorss,tofdegreesp,qshouldsatisfyts=(�1)pqst.Inthepresentcase,theshifthasdegreenandthedi erentialshavedegree1,soshiftanddi erentialsshouldcommuteuptoasign(�1)n,whichexplainsthedenitiondiA[n]=(�1)ndi+nA.Ahomomorphismf,bycontrast,hasdegreezero,sonosignappearsinthedenitionf[n]i=fi+n.AnobjectAofAcanbeidentiedwiththecomplexwhichisAindegree0andzeroinallotherdegrees.ThismakesAafullsubcategoryofC(A).TheshiftA[n]isthenthecomplexwhichisAindegree�n.Denition1.3.Ahomomorphismofcom
3 plexesf:A!Bisaquasi-isomorphis
plexesf:A!Bisaquasi-isomorphism(\qis"forshort)iffinducesisomorphismsHi(A)'!Hi(B)foralli.Acomplexisquasi-isomorphictozeroifandonlyifHi(A)=0foralli,thatis,ifAisacyclic.Denition1.4.Themappingconeofahomomorphismf:A!BisthecomplexC(f)withtermsC(f)n=An+1Bn(thesameasA[1]B),anddi erentialsdn(a;b)=(�dn+1A(a);fn+1(a)+dnB(b)). 4MARKHAIMANcomplexes,andfisaboundary(f2im(d�1))ifandonlyiffisnull-homotopic.ThusHomK(A)(A;B)=H0(Hom(A;B)).Proposition1.8.Thefollowingpropertiesofahomomorphismf:A!BinC(A)areequivalent:(a)f0.(b)ffactorsthroughthecanonicalmapi:A!C(1A)givenby1.5(ii)for1A.(c)ffactorsthroughthecanonicalmapp[�1]:C(1B)[�1]!Bgivenby1.5(ii)for1B.(d)Theexactsequence1.5(ii)forfsplits.Proof.Exercise.Corollary1.9.Homotopicmapsfginducethesamemapsoncohomology.Inparticular,thecohomologyfunctorsK(A)!A,A7!Hi(A)arewell-dened.Proof.Anull-homotopicmapf:
4 A!Binducesthezeromaponcohomologybecausei
A!BinducesthezeromaponcohomologybecauseitfactorsthroughC(1A),whichisacyclicby1.5(vi).Corollary1.10.Everyhomotopyequivalence(i.e.,everyhomomorphisminvertibleinK(A))isaquasi-isomorphism.Remark1.11.Proposition1.5(vi)canbestrengthened(exercise)tosaythatiffisbijective,thenC(f)ishomotopy-equivalenttozero.However,itisnottruethateveryacycliccomplexishomotopy-equivalenttozero,nordo1.5(iv,v)holdforhomotopy-equivalence.2.TrianglesThehomotopycategoryK(A)andthederivedcategoryD(A),tobeintroducedinx3,areadditivebutnotabeliancategories.Instead,theyshareanextrastructuredescribedbyadistinguishedcollectionofexacttriangles.Althoughwearemainlyinterestedinthederivedcategory,werstconsidertrianglesinthehomotopycategory.Itwillbeeasiertodeducethemainpropertiesofthederivedcategoryafterthisintermediatestep.Lemma2.1.Givenahomomorphismofcomplexesf:A!B,eachcompositeoftwosuc-cessivemapsinthesequence!Af!Bi!C(f)p!A[1]f[1]�!B[1]!inducedby1.
5 5(ii)iszeroinK(A).Proof.ThecompositeB!C(
5(ii)iszeroinK(A).Proof.ThecompositeB!C(f)!A[1]isalreadyzeroinC(A).ForAf!Bi!C(f),thediagramA1A���!A1A??yf??yAf���!B 6MARKHAIMAN(iii)GivenexacttrianglesAf!B!C!A[1]andA0g!B0!C0!A0[1],everycommutativediagraminK(A)Af���!Bh??yh0??yA0g���!B0extendstoamorphismoftrianglesAf���!B���!C���!A[1]??y??y??y??yA0g���!B0���!C0���!A0[1]:(iv)Adirectsumofexacttrianglesisexact.Proof.Axioms(o),(iv)andtherstpartof(i)areobvious.Forthesecondpartof(i),wehaveC(1A)=0inK(A)byRemark1.11.For(ii),usingshiftinvariance(seeremarkbelow),itsucestoverifytherstrotation.Wecanassumethegiventriangleisstandard,Af!Bi!C(f)p!A[1].ThenwemustshowthatA[1]=C(i)inK(A),viaanisomorphismsuchthatthecompositeA[1]!C(i)p(i)!B[1]is�f[1]andC(f)i(i)!C(i)!A[1]isp.Now,C(i)isidenticaltothemappingconeofthemaph:A!C(1B)obtainedbycomposingi(1B):B!C(1B)withf.Thisgivesacanonicalmap=p(h):C(
6 i)!A[1],by1.5(ii).ButC(i)isalsoidentical![i)!A[1],by1.5(ii).ButC(i)isalsoidentical](884130/i-a-1-by1-5-ii-butc-i-isalsoidenticaltot.jpg "i)!A[1],by1.5(ii).ButC(i)isalsoidentical")
i)!A[1],by1.5(ii).ButC(i)isalsoidenticaltothemappingconeof(f;1B):AB!B,whosekernelisisomorphictoA.Thisgivesamap=k[1]:A[1]!C(i),by1.5(iii).Onechecksthat=1A[1],p(i)=�f[1],i(i)=p,and1C(i).For(iii),wecanassumebothtrianglesarestandard.Ifthegivendiagramcommutesuptoahomotopys:ghh0f,youcancheckthat(ai+1;bi)7!(h(ai+1);h0(bi)+s(ai+1))isahomomorphismC(f)!C(g)thatyieldsthedesiredmorphismoftrianglesinK(A).Remarks2.4.(a)Warning:Themorphismoftrianglesin(iii)isnotcanonical,butdependsonthechoiceofanisomorphismbetweeneachofthegiventrianglesandastandardtriangle.Thuswedonothaveafunctorial\mapping-cone"constructionassigningtoeacharrowf:A!BinK(A)anexacttrianglewithfasitsbase.(b)TrianglesAf!Bg!Ch!A[1]andA�f!B�g!Ch!A[1]areisomorphicvia�1B,andlikewiseifwechangeanytwosigns.IfAf!Bg!Ch!A[1]isexact,atrianglesuchasA�f!B�g!C�h!A[1]withoneorthreesignschangedisanti-exact.NotethatC(f
7 )[1]=C(�f[1]),sotheshiftA[1]f[1]�!![)[1]=C(�f[1]),sotheshiftA[1]f[1]�!](884130/1-c-0-f-1-sotheshifta-1-f-1-0-b-1-g-1-0-.jpg ")[1]=C(�f[1]),sotheshiftA[1]f[1]�!")
)[1]=C(�f[1]),sotheshiftA[1]f[1]�!B[1]g[1]�!C[1]h[1]�!A[1]isanti-exact,whileA[1]�f[1]�!B[1]�g[1]�!C[1]�h[1]�!A[1],gottenbyrotatingouroriginaltrianglethreetimes,isexact. 8MARKHAIMANInanyevent,thegenuinemathematicalissuesinvolveddonotconcernset-theoretictech-nicalities,butratherconcretequestionsofhowtodescribeanarrowinD(A),andhowtorecognizewhentwoarrowsareequal.Bythedenitionofquasi-isomorphism,thecohomologyfunctorsHi:C(A)!AfactorthroughuniquefunctorsHi:D(A)!A.IfAisanobjectofA,letA[0]denotethecomplexwhichisAindegreezero,and0inotherdegrees.ThenH0(A[0])=A,soA7!A[0]isafullyfaithfulembeddingofAintoD(A),withleftinversegivenbythefunctorH0.UsuallywejustidentifyAwithA[0]andregardAasafullsubcategoryofD(A).Proposition3.3.ThecanonicalfunctorC(A)!D(A)factors(uniquely)throughK(A).Proof.Itsucestoprovethatf0impliesf=0inD(A).Thisisimmediatefrom1.8(b),sinceC(1A)=0inD(A),by1.5(vi).Corollary3.4.Thederivedcateg
8 oryD(A)canalsobeidentiedwithK(A)[Q&
oryD(A)canalsobeidentiedwithK(A)[Q�1].Remarks3.5.(a)Traditionally,D(A)isoftendenedasK(A)[Q�1].Thistendstooverem-phasizetheroleofthehomotopycategory,whichisnotessentialtothedenition,althoughitisausefulauxiliarydeviceforunderstandingmanypropertiesofD(A).(b)EqualityinD(A)ofhomomorphismsf;g2HomC(A)(A;B)doesnotimplythatfandgarehomotopic.Acriterionforequalityofarrowsinthederivedcategoryisgivenby3.22(ii),below.Denition3.6.AnexacttriangleinD(A)isatriangleisomorphicinD(A)toastandardtriangle,asin2.2.Equivalently(by1.10),atriangleinD(A)isexacti itisisomorphicinD(A)toanexacttriangleofK(A).AnadvantageofthederivedcategoryisthateveryexactsequenceinC(A)givesrisetoanexacttriangleinD(A),whichisnotthecaseinK(A).Proposition3.7.Let0!Af!Bg!C!0beanexactsequenceinC(A).ThenthediagramC(f)'!qisq(f)Cp(f)??yi(g)??yA[1]'!qisk(g)[1]C(g)anti-commutesinK(A),andhenceinD(A).Proof.Theformulafork(g)[1]p(f)is(ai+1;bi)7!(f(ai+1);0),andfori(g)
9 4;q(f)itis(ai+1;bi)7!(0;g(bi)).Thensi(ai
4;q(f)itis(ai+1;bi)7!(0;g(bi)).Thensi(ai+1;bi)=(bi;0)isahomotopybetweenk(g)[1]p(f)and�i(g)q(f).Denition3.8.Themaph:C!A[1]inD(A),givenintermsofthediagramin3.7byh=p(f)q(f)�1=�k(g)[1]�1i(g),istheconnectinghomomorphismoftheexactsequence0!Af!Bg!C!0. 10MARKHAIMANProof.Weprovethesecondsequence;asimilarargumentappliestotherst.Supposef:B!Xsatisesfu0.From2.3(i,iii)wegetamorphismoftrianglesAu���!Bv���!Cw���!A[1]??yf??yg??y??y0���!X1X���!X���!0;whichshowsfgvforsomeg:C!X.Inotherwords,HomK(A)(A;X) HomK(A)(B;X) HomK(A)(C;X)isexact,andtherestfollowsbytherotationaxiom,2.3(ii).Corollary3.15.Corollary3.12alsoholdsinK(A),andindeedinany\weakly"triangulatedcategory,satisfyingaxioms2.3(o-iv).Proof.Intheproofof3.12wecanuseeitherlongexactsequencein3.14inplaceoftheonein3.11,togetherwiththefactthatthefunctorHom(�;C)(resp.Hom(C;�))determinesCupto
10 canonicalisomorphism.Remark3.16.Thep
canonicalisomorphism.Remark3.16.Theproofsshowthat3.12and3.14holdinanyweaklytriangulatedcategory.OurnextgoalistodescribethearrowsinD(A).Denition3.17.AclassofQofarrowsinacategoryCisa(left)Oresystemifitsatisesthefollowingconditions:(a)Qismultiplicative,i.e.QQQand1X2QforeveryobjectXofC.(b)EverypairofarrowsA0q Af!Bwithq2QcanbecompletedtoacommutativediagramAf���!Bq??yr??yA0g���!B0withr2Q.(c)Iffq=0withq2Q,thereexistsr2Qsuchthatrf=0.ArightOresystemisdeneddually.Remark3.18.Acategoryislteredifeverypairf:A!B,f0:A!B0ofmorphismsfromthesameobjectAcanbecompletedtoacommutativesquare.Aninductivesystemofsets(Xi)i2IindexedbyalteredcategoryIisalsosaidtobeltered.Alteredinductivesystemhasthepropertythatelementsx2Xi,x02Xi0representthesameelementofthedirectlimitlim�!(Xi)ifandonlyifthereexistarrows :Xi!Xj, 0:Xi0!XjinIsuchthat (x)= 0(x0).Toseethis,onechecksthatinalter
11 edinductivesystem,theprecedingconditiond
edinductivesystem,theprecedingconditiondenesanequivalencerelationxx0,andthenlim�!(Xi)=(Fi2IXi)=. 12MARKHAIMANCorollary3.22.(i)EveryarrowinD(A)factorsasq�1fandasgr�1,wheref,g,q,rarehomomorphismsinC(A),withq,rquasi-isomorphisms.(ii)Ahomomorphismf:A!BinC(A)iszeroinD(A)ifandonlyiftheequivalentconditionshold:(a)thereexistsaquasi-isomorphismq:B!B0suchthatqf0;(b)thereexistsaquasi-isomorphismr:A0!Asuchthatfr0.Corollary3.23.TheexacttrianglesinD(A)satisfyaxioms2.3(o{iv).Proof.Axiom(o)holdsbydenition,(iv)isclear,andwehavealreadyseen(ii).Axioms(i)and(iii)followeasilyfromthecorrespondingaxiomsinK(A),using3.22(i).Corollary3.24.Proposition3.14alsoholdsinD(A).Proof.See3.16.Corollary3.25.Theexacttrianglebasedonanarrowf:A!BinD(A)isuniqueupto(non-canonical)isomorphism.Proof.Followsfromaxiom(iii)and3.12.Remarks3.26.(a)Theoctahedralaxiomfollowssimilarly,soD(A)isatriangulatedcat-egoryinthesenseofVerdier.(b)Therea
12 soningemployedaboveappliesmoregenerally.
soningemployedaboveappliesmoregenerally.LetKbeatriangulatedcategoryandNKafulltriangulatedsubcategory,closedunderisomorphismsinK.LetQconsistofthearrowsinKsuchthatthethirdvertexofanyexacttrianglebasedonq2QisanobjectofN.ThenQisaleftandrightOresysteminK,andD=K[Q�1]isatriangulatedcategory,alsodenotedD=K=N.Inourcase,K=K(A),withNconsistingoftheacycliccomplexes.By3.13,thisisequivalenttoQconsistingofthequasi-isomorphisms.4.DerivedFunctorsWewilluseDeligne'smethod[5]ofdeningandconstructingderivedfunctors.Denition4.1.GivenacomplexA,letqisnAbethecategoryofquasi-isomorphismsA'!qisA0inK(A),withmorphismsthecommutativetrianglesasin3.18.By3.18,qisnAisalteredcategory.GivenafunctorF:K(A)!CandanobjectYofC,wehavealteredinductivelimitofsets,functorialinY,lim�!A'!qisA0�HomC(Y;F(A0)):Proposition4.2.LetF:K(A)!Cbeanyfunctor.Toeacharrowf:A!BinK(A)thereiscanonicallyassociatedanaturaltransformation(1)rF(f):lim�!A'!qisA0�HomC(&#
13 0;;F(A0))!lim�!B'!qisB0�HomC(&
0;;F(A0))!lim�!B'!qisB0�HomC(�;F(B0))betweenfunctorsfromCoptoSets .ThisgivesafunctorrF:K(A)!Fun(Cop;Sets ). 14MARKHAIMANfunctorF:A!BinducesanexactfunctorF:K(A)!K(B)(alsodenotedFbyabuseofnotation).Inpractice,weonlydealwithfunctorsofthislastform.ComposingthecanonicalfunctorjB:K(B)!D(B)withF,andapplying4.2{4.5(withC=D(B)),wedeneafunctorRF=rjBF:D(A)!Ind(D(B)).Denition4.6.TherightderivedfunctorofFisdenedatAinD(A),withvalueXinD(B),ifRF(A)=XbelongstoD(B)Ind(D(B)).ItisanexerciseforthereadertoworkoutthedualdenitionofleftderivedfunctorLF.4.7.Tobemoreprecise,thedenitionmeansthatRF(A)andX2D(B)representisomor-phicfunctors(3)HomD(B)(�;X)=lim�!A'!qisA0�HomD(B)(�;F(A0))fromD(B)optoSets .ThenXisuniqueuptocanonicalisomorphism,whichjustieswritingRF(A)=X.Letusmakethisexplicit.ForeachAq!A0inqisnA,wegetanarrowA0:F(A0)!XinD(B),correspondingvia(3)tothetheelementrepresentedby1F(A0)ontherig
14 ht-handside.ThesystemofarrowsA0isco
ht-handside.ThesystemofarrowsA0iscompatiblewithF(qisnA),i.e.,foreachA0r!A00inqisnA,wehaveA0=A00F(r).Bynaturality,thearrowsA0inducethemaplimA'!qisA0�HomD(B)(�;F(A0))!HomD(B)(�;X).Intheoppositedirection,totheelement1Xontheleft-handsideof(3)therecorre-spondsanequivalenceclassofarrowsA0:X!F(A0),forsomeA02qisnA.AnytworepresentativesA0factorintoathird,andbynaturality,themapHomD(B)(�;X)!limA'!qisA0�HomD(B)(�;F(A0))isinducedbyanyrepresentativeA0.Thetwomapsbeinginversemeansthat(i)anyrepresentativeA0isasectionofA0,(ii)themapsA0:F(A0)!XmakeXtheinductivelimitX=lim�!F(qisnA)inD(B),and(iii)foranyrepresentativeA0,thearrowX!Ycorrespondingtoanysystemofmaps A0:F(A0)!Y,A02qisnAbytheuniversalpropertyoflim�!F(qisnA)isgivenby A0A0.Inthiscase,thesystemF(qisnA)issaidtobeessentiallyconstant.Thepracticalmeaningof4.6willbecomeclearerinx5,wherewewillgivecriteriathatone
15 usesinpracticetoshowthatRF(A)isdene
usesinpracticetoshowthatRF(A)isdened.Thecriteriaalsohavethee ectofmakingRF(A)concretelycomputable,ofteninmorethanoneway.ButrstweneedtoremainalittlelongerintheabstractcontextinordertoestablishthebasicpropertiesofRF.Denition4.8.ThecohomologyobjectsHi(RF)aredenotedRiFandcalledtheclassicalrightderivedfunctorsofF.Corollary4.9.LetDF(A)bethefullsubcategoryofD(A)whoseobjectsarethoseAsuchthatRF(A)isdened.ThenRFisafunctorfromDF(A)toD(B). 16MARKHAIMANC00!A0[1],whosecomponentarrowsareallquasi-isomorphisms,by3.12.ThisshowsthattheverticesB0in(4)areconalinqisnB.Thecorrespondingstatementholdsfortheotherverticesby2.3(ii).ProofofTheorem4.14.LetRF(A)=X,RF(B)=Y.CompletethearrowX!YtoanexacttriangleX!Y!Z!X[1]inD(B).Asin4.7,wecanndsectionsA0:X!F(A0),B0:Y!F(B0).ChangingA0,B0ifneeded,wecanttheseintoacommutativediagraminD(B)X���!YA0??yB0??yF(A0)F(u)���!F(B0);andwecanfurtherassumethatu:A0
16 !B0ispartofanexacttriangleinthebottomrow
!B0ispartofanexacttriangleinthebottomrowof(4).SinceFisexact,thisextendstoamorphismofexacttriangles(5)X���!Y���!Z���!X[1]??y??y??y??yF(A0)���!F(B0)���!F(C0)���!F(A0)[1]:Now,4.15and3.24implythatthesequence(6)!lim�!A0HomD(B)(T;F(A0))!lim�!B0HomD(B)(T;F(B0))!lim�!C0HomD(B)(T;F(C0))!lim�!A0HomD(B)(T;F(A0)[1])!isexactforeveryobjectTofD(B).Bydenition,thisisjustthesequence!RF(A)!RF(B)!RF(C)!RF(A)[1]!jjjjjjHom(�;X)Hom(�;Y)Hom(�X[1])inInd(D(B)),evaluatedatT.Themorphismin(5)providesacommutativediagram!Hom(�;X)!Hom(�;Y)!Hom(�;Z)!Hom(�;X[1])!jjjj#jj!RF(A)!RF(B)!RF(C)!RF(A)[1]!:EvaluatedatanyTinD(B),wehavejustseenthatthebottomrowisexact,andthetoprowisexactby3.24.Hencetheverticalarrowisanisomorphism.Corollary4.16.ForanytriangleA!B!C!A[1]inD(A),andinparticular,foranyex
17 actsequence0!A!B!C!0inC(A),ifRF(A),RF(B)
actsequence0!A!B!C!0inC(A),ifRF(A),RF(B),RF(C)aredened(e.g.,ifanytwoofthemare),thereisalongexactsequenceofclassicalderivedfunctors!R0F(A)!R0F(B)!R0F(C)!R1F(A)!R1F(B)!: 18MARKHAIMANdualtruncationfunctor whichkillsthecohomologyHn(A).InD(A),thisbecomesatriangle (A)!A! n(A)! (A)[1]= (n�1)(A[1]):IfAisalreadyboundedbelowatn0,thenA! n0(A)isaquasi-isomorphism.ItfollowsimmediatelythatD+(A)isequivalenttoitsfullsubcategoryofstrictlybounded-belowobjectsA,satisfyingAi=0forallilessthansomen0.Moreover,ifA!X' qisBisamorphismbetweenobjectsA;B2D+(A),thennecessarilyX2D+(A).Truncatingallthreeobjects,wecanreplaceXbyastrictlybounded-belowcomplextoo.HenceD+(A)canbeidentiedwiththelocalizationC+(A)[Q�1]ofthecategoryC+(A)ofstrictlybounded-belowcomplexesbythequasi-isomorphismsinC+(A).Proposition5.6.AssumeF:A!Bisleftexact.LetAbeaclassofobjectsinAsuchthat(i)ForeveryAinAthereisaninjectionA!Iwi
18 thI2A;(ii)Aisclosedundernitedirects
thI2A;(ii)Aisclosedundernitedirectsums,andif0!I!J!N!0isanexactsequencewithI;J2A,thenN2A;(iii)If0!I!A!B!0isanexactsequencewithI2A,then0!F(I)!F(A)!F(B)!0isexact.TheneveryAinD+(A)hasaresolutionA!I,whereIisastrictlybounded-belowcomplexofobjectsinA,andanysuchIcomputesRF,i.e.,theresolutioninducesF(A)!RF(A)=F(I).Proof.WecanassumethatAisstrictlybounded-below,sayAi=0fori0.Supposebyinductiononkthatwehaveconstructedahomomorphismr:A!I(k)asfollows:0���!A0���!A1���!���!Ak���!Ak+1���!??y??y??y??y??y0���!I0���!I1���!���!Ik���!0���!;wherethebottomrowisacomplexofobjectsinA,andrinducesanisomorphismHi(A)!Hi(I)forikandaninjectionHk(A)!coker(dk�1I)(initially,wehavethiswithk=�1).By3.11,themappingconeC(r)hasHi(C(r))=0forik.ThenC(r)!J= k(C(r))isaquasi-isomorphism,soA!I(k)!J
19 !A[1]isanexacttriangle.NowJisstrictlybou![!A[1]isanexacttriangle.NowJisstrictlybou](884130/a-1-isanexacttriangle-nowjisstrictlyboun.jpg "!A[1]isanexacttriangle.NowJisstrictlybou")
!A[1]isanexacttriangle.NowJisstrictlyboundedbelowatk,hencethemappingconeB=C(I(k)!J)[�1]lookslike0!I0!I1!!Ik!Jk!Jk+1!;andthereisaquasi-isomorphismA!B,i.e.,BisaresolutionofA.Chooseaninjectioni:Jk,!Ik+1withIk+12A.ConsiderthehomomorphismB!I(k+1)givenby(7)0���!I0���!I1���!���!Ikdk���!Jk���!Jk+1���!1??y1??y1??y1??yi??y??y0���!I0���!I1���!���!Ikd0k=idk����!Ik+1���!0���!; 20MARKHAIMANThesheafbFisclearly asque,so asquesheavessatify(i)(onecanalsoverify(i)byprovingthatinjectivesheavesare asque).Then5.6impliesthat asquesheavesareacyclicforf,andthatforanybounded-belowcomplexofsheavesAonX,wehaveRf(A)=f(J),whereJisastrictlybounded-below asqueresolution.Proposition5.10.Givenmorphismsofringedspacesf:X!Y,g:Y!Z,thereisanaturalisomorphismoffunctors(
20 8)RgRf=R(gf)fromD+
8)RgRf=R(gf)fromD+(A)toD+(C),whereA(resp.C)isthecateogoryofsheavesofOXmodules(resp.OZmodules).Proof.ItisimmediatefromthedenitionthatanydirectimagefFofa asquesheafFis asque(apropertynotsharedbyinjectivesheaves).GivenacomplexAofsheavesonX,theinstanceof(8)atAfollowsbytakinga asqueresolutionA!Jandobservingthat,sincefJisthena asqueresolutionofRfA,thecomplex(gf)J=gfJrepresentsbothR(gf)AandRgRfA.IfgisthetautologicalmorphismfromYtothespaceZwithonepointandOZtheconstantsheafZ,thenthefunctorsgand�Yareessentiallythesame,andsimilarlyfor(gf)and�X.Thefollowingcorollaryisthereforeaspecialcaseof5.10.Corollary5.11.Givenamorphismofringedspacesf:X!Y,thereisanaturalisomor-phismoffunctors(9)R�YRf=R�XfromthederivedcategoryD+(A),whereAisthecategoryofsheavesofOXmodules,tothederivedcategoryofsheavesofabeliangroups(or,moregenerally,ofRmodules,iffisamorphismofringedspac
21 esovertheone-pointringedspaceZwithOZ(Z)=
esovertheone-pointringedspaceZwithOZ(Z)=R).Theanalogof(8)forclassicalderivedfunctorsisaspectralsequencerelatingthefunctorsRpfRqgandRn(gf).TheidentityRgRf=R(gf)isnotonlysimplerthantheold-fashionedspectralsequence,itisastrongerresult.Spaltenstein[8]gaveadenitionofK- asquecomplexofsheaves.Usingthis,heprovedthat5.10and5.11alsoholdforunboundedcomplexes.Proposition5.12.Letf:X!Ybeamorphismofringedspaces.LetBbeabaseofthetopologyonYintheweaksense,thatis,BisasetofopensubsetsVYsuchthateveryopenUYisaunionofmembersofB,butwedonotrequirethatBbeclosedunderniteintersections.ForasheafMofOXmodulestobeacyclicforthedirectimagefunctorf,itsucesthatMbeacylicforR�f�1(V)forallV2B.Proof.LetAbetheclassofsheavesAonXsatisfyingtheacyclicityconditionwerequireofM.Explicitly,AbelongstoAifforallV2B,thecanonicalmap�(f�1(V);A)!R�(f�1(V);A)isaquasi-isomorphism,orequivalently,Ri�(f�1(V);A)