Table Of ContentCAMBRIDGE 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
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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
