CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 183 EditorialBoard B. BOLLOBA´S, W. FULTON, F. KIRWAN, P. SARNAK, B SIMON, B. TOTARO DERIVEDCATEGORIES There have been remarkably few systematic expositions of the theory of derived categories since its inception in the work of Grothendieck and Verdier in the 1960s. Thisbookisthefirstin-depthtreatmentofthisimportantcomponentofhomological algebra. It carefully explains the foundations in detail before moving on to key applicationsincommutativeandnoncommutativealgebra,manyotherwiseunavailable outsideofresearcharticles.Theseincludecommutativeandnoncommutativedualizing complexes,perfectDGmodulesandtiltingDGbimodules. Writtenwithgraduatestudentsinmind,theemphasishereisonexplicitconstructions (with many examples and exercises) as opposed to axiomatics, with the goal of demystifying this difficult subject. Beyond serving as a thorough introduction for students, it will serve as an important reference for researchers in algebra, geometry andmathematicalphysics. AmnonYekutieli isProfessorofMathematicsatBen-GurionUniversityoftheNegev, Israel.Hisresearchinterestsareinalgebraicgeometry,ringtheory,derivedcategories anddeformationquantization.Hehastaughtseveralgraduate-levelcoursesonderived categories.Thisishisfourthbook. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EditorialBoard: B.Bolloba´s,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora completeserieslisting,visitwww.cambridge.org/mathematics. AlreadyPublished 145M.VianaLecturesonLyapunovExponents 146J.-H.Evertse&K.Gyo¨ryUnitEquationsinDiophantineNumberTheory 147A.PrasadRepresentationTheory 148S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontoModelSpacesandTheirOperators 149C.Godsil&K.MeagherErdo¨s–Ko–RadoTheorems:AlgebraicApproaches 150P.MattilaFourierAnalysisandHausdorffDimension 151M.Viana&K.OliveiraFoundationsofErgodicTheory 152V.I.Paulsen&M.RaghupathiAnIntroductiontotheTheoryofReproducingKernelHilbertSpaces 153R.Beals&R.WongSpecialFunctionsandOrthogonalPolynomials 154V.JurdjevicOptimalControlandGeometry:IntegrableSystems 155G.PisierMartingalesinBanachSpaces 156C.T.C.WallDifferentialTopology 157J.C.Robinson,J.L.Rodrigo&W.SadowskiTheThree-DimensionalNavierStokesEquations 158D.HuybrechtsLecturesonK3Surfaces 159H.Matsumoto&S.TaniguchiStochasticAnalysis 160A.Borodin&G.OlshanskiRepresentationsoftheInfiniteSymmetricGroup 161P.WebbFiniteGroupRepresentationsforthePureMathematician 162C.J.Bishop&Y.PeresFractalsinProbabilityandAnalysis 163A.BovierGaussianProcessesonTrees 164P.SchneiderGaloisRepresentationsand(ϕ,(cid:3))-Modules 165P.Gille&T.SzamuelyCentralSimpleAlgebrasandGaloisCohomology(2ndEdition) 166D.Li&H.QueffelecIntroductiontoBanachSpaces,I 167D.Li&H.QueffelecIntroductiontoBanachSpaces,II 168J.Carlson,S.Mu¨ller-Stach&C.PetersPeriodMappingsandPeriodDomains(2ndEdition) 169J.M.LandsbergGeometryandComplexityTheory 170J.S.MilneAlgebraicGroups 171J.Gough&J.KupschQuantumFieldsandProcesses 172T.Ceccherini-Silberstein,F.Scarabotti&F.TolliDiscreteHarmonicAnalysis 173P.GarrettModernAnalysisofAutomorphicFormsbyExample,I 174P.GarrettModernAnalysisofAutomorphicFormsbyExample,II 175G.NavarroCharacterTheoryandtheMcKayConjecture 176P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.PerssonEisensteinSeriesandAutomorphic Representations 177E.PetersonFormalGeometryandBordismOperators 178A.OgusLecturesonLogarithmicAlgebraicGeometry 179N.NikolskiHardySpaces 180D.-C.CisinskiHigherCategoriesandHomotopicalAlgebra 181A.Agrachev,D.Barilari&U.BoscainAComprehensiveIntroductiontoSub-RiemannianGeometry 182N.NikolskiToeplitzMatricesandOperators 183A.YekutieliDerivedCategories 184C.DemeterFourierRestriction,DecouplingandApplications Derived Categories AMNON YEKUTIELI Ben-GurionUniversityoftheNegev,Israel UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 79AnsonRoad,#06—04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108419338 DOI:10.1017/9781108292825 ©AmnonYekutieli2020 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2020 PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-41933-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. DedicatedtoAlexanderGrothendieck,inmemoriam Contents 0 Introduction 1 0.1 OntheSubject 1 0.2 AMotivatingDiscussion: Duality 4 0.3 OntheBook 9 0.4 SynopsisoftheBook 11 0.5 WhatIsNotintheBook 22 0.6 PrerequisitesandRecommendedBibliography 23 0.7 Credo,WritingStyleandGoals 24 0.8 Acknowledgments 25 1 BasicFactsonCategories 26 1.1 SetTheory 26 1.2 NotationandConventions 27 1.3 EpimorphismsandMonomorphisms 28 1.4 ProductsandCoproducts 30 1.5 EquivalenceofCategories 31 1.6 Bifunctors 32 1.7 RepresentableFunctors 32 1.8 InverseandDirectLimits 34 2 AbelianCategoriesandAdditiveFunctors 37 2.1 LinearCategories 37 2.2 AdditiveCategories 38 2.3 AbelianCategories 40 2.4 AMethodforProducingProofsinAbelianCategories 44 2.5 AdditiveFunctors 49 2.6 ProjectiveObjects 55 2.7 InjectiveObjects 57 vii viii Contents 3 DifferentialGradedAlgebra 62 3.1 GradedAlgebra 62 3.2 DG(cid:139)-Modules 72 3.3 DGRingsandModules 74 3.4 DGCategories 78 3.5 DGFunctors 80 3.6 ComplexesinAbelianCategories 82 3.7 TheLongExactCohomologySequence 84 3.8 TheDGCategoryC(A,M) 90 3.9 ContravariantDGFunctors 94 4 TranslationsandStandardTriangles 101 4.1 TheTranslationFunctor 101 4.2 TheStandardTriangleofaStrictMorphism 105 4.3 TheGaugeofaGradedFunctor 107 4.4 TheTranslationIsomorphismofaDGFunctor 108 4.5 StandardTrianglesandDGFunctors 109 4.6 ExamplesofDGFunctors 113 5 TriangulatedCategoriesandFunctors 117 5.1 TriangulatedCategories 117 5.2 TriangulatedandCohomologicalFunctors 122 5.3 SomePropertiesofTriangulatedCategories 125 5.4 TheHomotopyCategoryIsTriangulated 132 5.5 FromDGFunctorstoTriangulatedFunctors 141 5.6 TheOppositeHomotopyCategoryIsTriangulated 143 6 LocalizationofCategories 146 6.1 TheFormalismofLocalization 146 6.2 OreLocalization 148 6.3 LocalizationofLinearCategories 162 7 TheDerivedCategoryD(A,M) 165 7.1 LocalizationofTriangulatedCategories 165 7.2 DefinitionoftheDerivedCategory 172 7.3 BoundednessConditionsinK(A,M) 176 7.4 ThickSubcategoriesofM 180 7.5 TheEmbeddingofMinD(M) 182 7.6 TheOppositeDerivedCategoryIsTriangulated 183