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[美] 莱恩 著 / 世界图书出版公司 / 2009-08 / 平装
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Inpresentingthistreatmentofhomologicalalgebra,itisapleasuretoacknowledgethehelpandencouragementwhichIhavehadfromallsides.Homologicalalgebraarosefrommanysourcesinalgebraandtopology.Decisiveexamplescamefromthestudyofgroupextensionsandtheirfactorsets,asubjectIlearnedinjointworkwithOTTOSCHIL-LING.Afurtherdevelopmentofhomologicalideas,withaviewtotheirtopologicalapplications,cameinmylongcollaborationwithSAHUELEZLENBERG;tobothcollaborators,especialthanks.FormanyyearstheAirForceOfficeofScientificResearchsupportedmyresearchprojectsonvarioussubjectsnowsummarizedhere;itisapleasuretoacknowledgetheirlivelyunderstandingOfbasicscience.
SaundersMacLanewasbornonAugust4,1909inConnecticut.HestudiedatYaleUniversityandthenattheUniversityofChicagoandatG6ttingen,wherehereceivedtheD.Phil.in1934.HehastaughtatHarvard,Cornel1andtheUniversityofChicago.MacLanesinitialresearchwasinlogicandinalgebraicnumbertheory(valuationtheory).WithSamuelEilenberghepublishedfifteenpapersonalgebraictopology.Anumberoftheminvolvedtheinitialstepsinthecohomologyofgroupsandinotheraspectsofhomologicalalgebra-aswellasthediscoveryofcategorytheory.HisfamousundergraduatetextbookSurveyofmodernalgebra,writtenjointlywithG.Birkhoff,hasremainedinprintforover50years.MacLaneisalsotheauthorofseveralotherhighlysuccessfulbooks.
IntroductionChapterI.Modules,Diagrams,andFunctors1.TheArrowNotation2.Modules3.Diagrams4.DirectSums5.FreeandProjectiveModules6.TheFunctorHorn7.Categories8.FunctorsChapterIf.HomologyofComplexes1.DifferentialGroups2.Complexes3.Cohomology4.TheExactHomologySequence5.SomeDiagramLemmas6.AdditiveRelations7.SingularHomology8.Homotopy9.AxiomsforHomologyChapterIII.ExtensionsandResolutions1.ExtensionsofModules2.AdditionofExtensions3.ObstructionstotheExtensionofHomomorphisms4.TheUniversalCoefficientTheoremforCohomology5.CompositionofExtensions6.Resolutions7.InjectiveModules8.InjectiveResolutions9.TwoExactSequencesforExtn10.AxiomaticDescriptionofExt11.TheInjectiveEnvelopeChapterIV.Cohomol0gyofGroups1.TheGroupRing2.CrossedHomomorphisms3.GroupExtensions4.FactorSets5.TheBarResolution6.TheCharacteristicClassofaGroupExtension7.CohomologyofCyclicandFreeGroups8.ObstructionstoExtensions9.RealizationofObstructions10.SCHURSTheorem11.SpaceswithOperatorsChapterV.TensorandTorsionProducts1.TensorProducts2.ModulesoverCommutativeRings3.Bimodules4.DualModules5.RightExactnessofTensorProducts.6.TorsionProductsofGroups7.TorsionProductsofModules8.TorsionProductsbyResolutions9.TheTensorProductofComplexes10.TheKONNETHFormula11.UniversalCoefficientTheoremsChapterVI.TypesofAlgebras1.AlgebrasbyDiagrams2.GradedModules3.GradedAlgebras4.TensorProductsofAlgebras5.ModulesoverAlgebras6.CohomologyoffreeAbelianGroups7.DifferentialGradedAlgebras8.IdentitiesonHornand9.CoalgebrasandHoPFAlgebrasChapterVII.Dimension1.HomologicalDimension2.DimensionsinPolynomialRings3.ExtandTorforAlgebras4.GlobalDimensionsofPolynomialRings5.SeparableAlgebras6.GradedSyzygies7.LocalRingsChapterVIII.Products1.HomologyProducts2.TheTorsionProductofAlgebras3.ADiagramLemma4.ExternalProductsforExt5.SimplicialObjects6.Normalization7.AcyclicModels8.TheEILENBERG-ZILBERTheorem9.CupProductsChapterIX.RelativeHomologicalAlgebra1.AdditiveCategories2.AbelianCategories3.CategoriesofDiagrams4.ComparisonofAllowableResolutions5.RelativeAbelianCategories6,RelativeResolutions7.TheCategoricalBarResolution8.RelativeTorsionProducts9.DirectProductsofRingsChapterX.CohomologyofAlgebraicSystemsChapterXI.SpectralSequencesChapterXII.DerivedFunctorsBibliographyListofStandardSymbolsIndex
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