交换代数:Commutative Algebra With a View Toward Algebraic Geometry

交换代数:Commutative Algebra With a View Toward Algebraic Geometry
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作者: (David Eisenbud) ,
2004-01
版次: 1
ISBN: 9787506292450
定价: 118.00
装帧: 平装
开本: 16开
纸张: 胶版纸
页数: 797页
正文语种: 英语
分类: 自然科学
  • ThisbookprovidesanintroductiontoLiegroups,Liealgebras,andrepresentationtheory,aimedatgraduatestudentsinmathematicsandphysics.Althoughtherearealreadyseveralexcellentbooksthatcovermanyofthesametopics,thisbookhastwodistinctivefeaturesthatIhopewillmakeitausefuladditiontotheliterature.First,ittreatsLiegroups(notjustLiealgebras)inawaythatminimizestheamountofmanifoldtheoryneeded.Thus,Ineitherassumeapriorcourseondifferentiablemanifoldsnorprovideacon-densedsuchcourseinthebeginningchapters.Second,thisbookprovidesagentleintroductiontothemachineryofsemisimplegroupsandLiealgebrasbytreatingtherepresentationtheoryofSU(2)andSU(3)indetailbeforegoingtothegeneralcase.Thisallowsthereadertoseeroots,weights,andtheWeylgroup"inaction"insimplecasesbeforeconfrontingthegeneraltheory.
    ThestandardbooksonLietheorybeginimmediatelywiththegeneralcase:asmoothmanifoldthatisalsoagroup.TheLiealgebraisthendefinedasthespaceofleft-invariantvectorfieldsandtheexponentialmappingisdefinedintermsoftheflowalongsuchvectorfields.Thisapproachisundoubtedlytherightoneinthelongrun,butitisratherabstractforareaderencounteringsuchthingsforthefirsttime.Furthermore,withthisapproach,onemusteitherassumethereaderisfamiliarwiththetheoryofdifferentiablemanifolds(whichrulesoutasubstantialpartofonesaudience)oronemustspendconsiderabletimeatthebeginningofthebookexplainingthistheory(inwhichcase,ittakesalongtimetogettoLietheoryproper). Introduction
    AdvicefortheBeginner
    InformationfortheExpert
    Prerequisites
    Sources
    Courses
    Acknowledgements

    0ElementaryDefinitions
    0.1RingsandIdeals
    0.2UniqueFactorization
    0.3Modules

    ⅠBasicConstructions
    1RootsofCommutativeAlgebra
    1.1NumberTheory
    1.2AlgebraicCurvesandFhnctionTheory
    1.3InvariantTheory
    1.4TheBasisTheorem
    1.5GradedRings
    1.6AlgebraandGeometry:TheNullstellensatz
    1.7GeometricInvariantTheory
    1.8ProjectiveVarieties
    1.9HilbertFunctionsandPolynomials
    1.10FreeResolutionsandtheSyzygyTheorem
    1.11Exercises

    2Localization
    2.1Fractions
    2.2HornandTensor
    2.3TheConstructionofPrimes
    2.4RingsandModulesofFiniteLength
    2.5ProductsofDomains
    2.6Exercises

    3AssociatedPrimesandPrimaryDecomposition
    3.1AssociatedPrimes
    3.2PrimeAvoidance
    3.3PrimaryDecomposition
    3.4PrimaryDecompositionandFactoriality
    3.5PrimaryDecompositionintheGradedCase
    3.6ExtractingInformationfromPrimaryDecomposition
    3.7WhyPrimaryDecompositionIsNotUnique
    3.8GeometricInterpretationofPrimaryDecomposition
    3.9SymbolicPowersandFunctionsVanishingtoHighOrder
    3.10Exercises

    4IntegralDependenceandtheNullstellensatz
    4.1TheCayley-HamiltonTheoremandNakayamasLemma
    4.2NormalDomainsandtheNormalizationProcess
    4.3NormalizationintheAnalyticCase
    4.4PrimesinanIntegralExtension
    4.5TheNullstellensatz
    4.6Exercises

    5FiltrationsandtheArtin-ReesLemma
    5.1AssociatedGradedRingsandModules
    5.2TheBlowupAlgebra
    5.3TheKrullIntersectionTheorem
    5.4TheTangentCone
    5.5Exercises

    6FlatFamilies
    6.1ElementaryExamples
    6.2IntroductiontoTor
    6.3CriteriaforFlatness
    6.4TheLocalCriterionforFlatness
    6.5TheReesAlgebra
    6.6Exercises

    7CompletionsandHenselsLemma
    7.1ExamplesandDefinitions
    7.2TheUtilityofCompletions
    7.3LiftingIdempotents
    7.4CohenStructureTheoryandCoefficientFields
    7.5BasicPropertiesofCompletion
    7.6MapsfromPowerSeriesRings
    7.7Exercises

    ⅡDimensionTheory
    8IntroductiontoDimensionTheory
    8.1AxiomsforDimension
    8.2OtherCharacterizationsofDimensionFundamentalDefinitionsofDimensionTheory
    9.1DimensionZero
    9.2Exercises

    10ThePrincipalIdealTheoremandSystemsofParameters
    10.1SystemsofParametersandIdealsofFiniteColength
    10.2DimensionofBaseandFiber
    10.3RegularLocalRings
    10.4Exercises

    11DimensionandCodimensionOne
    11.1DiscreteValuationRings
    11.2NormalRingsandSerresCriterion
    11.3InvertibleModules
    11.4UniqueFactorizationofCodimension-OneIdeals
    11.5DivisorsandMultiplicities
    11.6MultiplicityofPrincipalIdeals
    11.7Exercises

    12DimensionandHilbert-SamuelPolynomials
    12.1Hilbert-SamuelFunctions
    12.2Exercises

    13TheDimensionofAffineRings
    13.1NoetherNormalization
    13.2TheNullstellensatz
    13.3FinitenessoftheIntegralClosure
    13.4Exercises

    14EliminationTheory,GenericFreeness,andtheDimensionofFibers
    14.1EliminationTheory
    14.2GenericPreeness
    14.3TheDimensionofFibers
    14.4Exercises

    15GrSbnerBases
    15.1MonomialsandTerms
    15.2MonomialOrders
    15.3TheDivisionAlgorithm
    15.4Gr5bnerBases
    15.5Syzygies
    15.6HistoryofGr5bnerBases
    15.7APropertyofReverseLexicographicOrder
    15.8Gr5bnerBasesandFlatFamilies
    15.9GenericInitialIdeals
    15.10Applications
    15.11Exercises
    15.12Appendix:SomeComputerAlgebraProjects

    16ModulesofDifferentials
    16.1ComputationofDifferentials
    16.2DifferentialsandtheCotangentBundle
    16.3ColimitsandLocalization
    16.4TangentVectorFieldsandInfinitesimalMorphisms
    16.5DifferentialsandFieldExtensions
    16.6JacobianCriterionforRegularity
    16.7SmoothnessandGenericSmoothness
    16.8Appendix:AnotherConstructionofKahlerDifferentials
    16.9Exercises

    ⅢHomologicalMethods
    17RegularSequencesandtheKoszulComplex
    17.1KoszulComplexesofLengthsIand2
    17.2KoszulComplexesinGeneral
    17.3BuildingtheKoszulComplexfromParts
    17.4DualityandHomotopies
    17.5TheKoszulComplexandtheCotangentBundleofProjectiveSpace
    17.6Exercises

    18Depth,Codimension,andCohen-MacaulayRings
    18.1Depth
    18.2Cohen-MacaulayRings
    18.3ProvingPrimenesswithSerresCriterion
    18.4FlatnessandDepth
    18.5SomeExamples
    18.6Exercises

    19HomologicalTheoryofRegularLocalRings
    19.1ProjectiveDimensionandMinimalResolutions
    19.2GlobalDimensionandtheSyzygyTheorem
    19.3DepthandProjectiveDimension:TheAuslander-BuchsbaumFormula
    19.4StablyFreeModulesandFactorialityofRegularLocalRings
    19.5Exercises

    20FreeResolutionsandFittingInvariants
    20.1TheUniquenessofFreeResolutions
    20.2FittingIdeals
    20.3WhatMakesaComplexExact?
    20.4TheHilbert-BurchTheorem
    20.5Castelnuovo-MumfordRegularity
    20.6Exercises

    21Duality,CanonicalModules,andGorensteinRings
    21.1DualityforModulesofFiniteLength
    21.2Zero-DimensionalGorensteinRings
    21.3CanonicalModulesandGorensteinRingsinHigherDimension
    21.4MaximalCohen-MacaulayModules
    21.5ModulesofFiniteInjectiveDimension
    21.6Uniquenessand(Often)Existence
    21.7LocalizationandCompletionoftheCanonicalModule
    21.8CompleteIntersectionsandOtherGorensteinRings
    21.9DualityforMaximalCohen-MacaulayModules
    21.10Linkage
    21.11DualityintheGradedCase
    21.12Exercises

    Appendix1FieldTheory
    A1.1TranscendenceDegree
    A1.2Separability
    A1.3p-Bases

    Appendix2MultilinearAlgebra
    A2.1Introduction
    A2.2TensorProducts
    A2.3SymmetricandExteriorAlgebras
    A2.4CoalgebraStructuresandDividedPowers
    A2.5SchurFunctors
    A2.6ComplexesConstructedbyMultilinearAlgebra

    Appendix3HomologicalAlgebra
    A3.1Introduction
    PartI:ResolutionsandDerivedFunctors
    A3.2FreeandProjectiveModules
    A3.3FreeandProjectiveResolutions
    A3.4InjectiveModulesandResolutions
    A3.5BasicConstructionswithComplexes
    A3.6MapsandHomotopiesofComplexes
    A3.7ExactSequencesofComplexes
    A3.8TheLongExactSequenceinHomology
    A3.9DerivedFunctors
    A3.10Tor
    A3.11Ext

    PartⅡI:FromMappingConestoSpectralSequences
    A3.12TheMappingConeandDoubleComplexes
    A3.13SpectralSequences
    A3.14DerivedCategories

    Appendix4ASketchofLocalCohomology
    A4.1LocalCohomologyandGlobalCohomology
    A4.2LocalDuality
    A4.3DepthandDimension

    Appendix5CategoryTheory
    A5.1Categories,Functors,andNaturalTransformations
    A5.2AdjointFunctors
    A5.3RepresentableFunctorsandYonedasLemma

    Appendix6LimitsandColimits
    A6.1ColimitsintheCategoryofModules
    A6.2FlatModulesasColimitsofFreeModules
    A6.3ColimitsintheCategoryofCommutativeAlgebras
    A6.4Exercises

    Appendix7WhereNext
    HintsandSolutionsforSelectedExercises
    References
    IndexofNotation
    Index
  • 内容简介:
    ThisbookprovidesanintroductiontoLiegroups,Liealgebras,andrepresentationtheory,aimedatgraduatestudentsinmathematicsandphysics.Althoughtherearealreadyseveralexcellentbooksthatcovermanyofthesametopics,thisbookhastwodistinctivefeaturesthatIhopewillmakeitausefuladditiontotheliterature.First,ittreatsLiegroups(notjustLiealgebras)inawaythatminimizestheamountofmanifoldtheoryneeded.Thus,Ineitherassumeapriorcourseondifferentiablemanifoldsnorprovideacon-densedsuchcourseinthebeginningchapters.Second,thisbookprovidesagentleintroductiontothemachineryofsemisimplegroupsandLiealgebrasbytreatingtherepresentationtheoryofSU(2)andSU(3)indetailbeforegoingtothegeneralcase.Thisallowsthereadertoseeroots,weights,andtheWeylgroup"inaction"insimplecasesbeforeconfrontingthegeneraltheory.
    ThestandardbooksonLietheorybeginimmediatelywiththegeneralcase:asmoothmanifoldthatisalsoagroup.TheLiealgebraisthendefinedasthespaceofleft-invariantvectorfieldsandtheexponentialmappingisdefinedintermsoftheflowalongsuchvectorfields.Thisapproachisundoubtedlytherightoneinthelongrun,butitisratherabstractforareaderencounteringsuchthingsforthefirsttime.Furthermore,withthisapproach,onemusteitherassumethereaderisfamiliarwiththetheoryofdifferentiablemanifolds(whichrulesoutasubstantialpartofonesaudience)oronemustspendconsiderabletimeatthebeginningofthebookexplainingthistheory(inwhichcase,ittakesalongtimetogettoLietheoryproper).
  • 目录:
    Introduction
    AdvicefortheBeginner
    InformationfortheExpert
    Prerequisites
    Sources
    Courses
    Acknowledgements

    0ElementaryDefinitions
    0.1RingsandIdeals
    0.2UniqueFactorization
    0.3Modules

    ⅠBasicConstructions
    1RootsofCommutativeAlgebra
    1.1NumberTheory
    1.2AlgebraicCurvesandFhnctionTheory
    1.3InvariantTheory
    1.4TheBasisTheorem
    1.5GradedRings
    1.6AlgebraandGeometry:TheNullstellensatz
    1.7GeometricInvariantTheory
    1.8ProjectiveVarieties
    1.9HilbertFunctionsandPolynomials
    1.10FreeResolutionsandtheSyzygyTheorem
    1.11Exercises

    2Localization
    2.1Fractions
    2.2HornandTensor
    2.3TheConstructionofPrimes
    2.4RingsandModulesofFiniteLength
    2.5ProductsofDomains
    2.6Exercises

    3AssociatedPrimesandPrimaryDecomposition
    3.1AssociatedPrimes
    3.2PrimeAvoidance
    3.3PrimaryDecomposition
    3.4PrimaryDecompositionandFactoriality
    3.5PrimaryDecompositionintheGradedCase
    3.6ExtractingInformationfromPrimaryDecomposition
    3.7WhyPrimaryDecompositionIsNotUnique
    3.8GeometricInterpretationofPrimaryDecomposition
    3.9SymbolicPowersandFunctionsVanishingtoHighOrder
    3.10Exercises

    4IntegralDependenceandtheNullstellensatz
    4.1TheCayley-HamiltonTheoremandNakayamasLemma
    4.2NormalDomainsandtheNormalizationProcess
    4.3NormalizationintheAnalyticCase
    4.4PrimesinanIntegralExtension
    4.5TheNullstellensatz
    4.6Exercises

    5FiltrationsandtheArtin-ReesLemma
    5.1AssociatedGradedRingsandModules
    5.2TheBlowupAlgebra
    5.3TheKrullIntersectionTheorem
    5.4TheTangentCone
    5.5Exercises

    6FlatFamilies
    6.1ElementaryExamples
    6.2IntroductiontoTor
    6.3CriteriaforFlatness
    6.4TheLocalCriterionforFlatness
    6.5TheReesAlgebra
    6.6Exercises

    7CompletionsandHenselsLemma
    7.1ExamplesandDefinitions
    7.2TheUtilityofCompletions
    7.3LiftingIdempotents
    7.4CohenStructureTheoryandCoefficientFields
    7.5BasicPropertiesofCompletion
    7.6MapsfromPowerSeriesRings
    7.7Exercises

    ⅡDimensionTheory
    8IntroductiontoDimensionTheory
    8.1AxiomsforDimension
    8.2OtherCharacterizationsofDimensionFundamentalDefinitionsofDimensionTheory
    9.1DimensionZero
    9.2Exercises

    10ThePrincipalIdealTheoremandSystemsofParameters
    10.1SystemsofParametersandIdealsofFiniteColength
    10.2DimensionofBaseandFiber
    10.3RegularLocalRings
    10.4Exercises

    11DimensionandCodimensionOne
    11.1DiscreteValuationRings
    11.2NormalRingsandSerresCriterion
    11.3InvertibleModules
    11.4UniqueFactorizationofCodimension-OneIdeals
    11.5DivisorsandMultiplicities
    11.6MultiplicityofPrincipalIdeals
    11.7Exercises

    12DimensionandHilbert-SamuelPolynomials
    12.1Hilbert-SamuelFunctions
    12.2Exercises

    13TheDimensionofAffineRings
    13.1NoetherNormalization
    13.2TheNullstellensatz
    13.3FinitenessoftheIntegralClosure
    13.4Exercises

    14EliminationTheory,GenericFreeness,andtheDimensionofFibers
    14.1EliminationTheory
    14.2GenericPreeness
    14.3TheDimensionofFibers
    14.4Exercises

    15GrSbnerBases
    15.1MonomialsandTerms
    15.2MonomialOrders
    15.3TheDivisionAlgorithm
    15.4Gr5bnerBases
    15.5Syzygies
    15.6HistoryofGr5bnerBases
    15.7APropertyofReverseLexicographicOrder
    15.8Gr5bnerBasesandFlatFamilies
    15.9GenericInitialIdeals
    15.10Applications
    15.11Exercises
    15.12Appendix:SomeComputerAlgebraProjects

    16ModulesofDifferentials
    16.1ComputationofDifferentials
    16.2DifferentialsandtheCotangentBundle
    16.3ColimitsandLocalization
    16.4TangentVectorFieldsandInfinitesimalMorphisms
    16.5DifferentialsandFieldExtensions
    16.6JacobianCriterionforRegularity
    16.7SmoothnessandGenericSmoothness
    16.8Appendix:AnotherConstructionofKahlerDifferentials
    16.9Exercises

    ⅢHomologicalMethods
    17RegularSequencesandtheKoszulComplex
    17.1KoszulComplexesofLengthsIand2
    17.2KoszulComplexesinGeneral
    17.3BuildingtheKoszulComplexfromParts
    17.4DualityandHomotopies
    17.5TheKoszulComplexandtheCotangentBundleofProjectiveSpace
    17.6Exercises

    18Depth,Codimension,andCohen-MacaulayRings
    18.1Depth
    18.2Cohen-MacaulayRings
    18.3ProvingPrimenesswithSerresCriterion
    18.4FlatnessandDepth
    18.5SomeExamples
    18.6Exercises

    19HomologicalTheoryofRegularLocalRings
    19.1ProjectiveDimensionandMinimalResolutions
    19.2GlobalDimensionandtheSyzygyTheorem
    19.3DepthandProjectiveDimension:TheAuslander-BuchsbaumFormula
    19.4StablyFreeModulesandFactorialityofRegularLocalRings
    19.5Exercises

    20FreeResolutionsandFittingInvariants
    20.1TheUniquenessofFreeResolutions
    20.2FittingIdeals
    20.3WhatMakesaComplexExact?
    20.4TheHilbert-BurchTheorem
    20.5Castelnuovo-MumfordRegularity
    20.6Exercises

    21Duality,CanonicalModules,andGorensteinRings
    21.1DualityforModulesofFiniteLength
    21.2Zero-DimensionalGorensteinRings
    21.3CanonicalModulesandGorensteinRingsinHigherDimension
    21.4MaximalCohen-MacaulayModules
    21.5ModulesofFiniteInjectiveDimension
    21.6Uniquenessand(Often)Existence
    21.7LocalizationandCompletionoftheCanonicalModule
    21.8CompleteIntersectionsandOtherGorensteinRings
    21.9DualityforMaximalCohen-MacaulayModules
    21.10Linkage
    21.11DualityintheGradedCase
    21.12Exercises

    Appendix1FieldTheory
    A1.1TranscendenceDegree
    A1.2Separability
    A1.3p-Bases

    Appendix2MultilinearAlgebra
    A2.1Introduction
    A2.2TensorProducts
    A2.3SymmetricandExteriorAlgebras
    A2.4CoalgebraStructuresandDividedPowers
    A2.5SchurFunctors
    A2.6ComplexesConstructedbyMultilinearAlgebra

    Appendix3HomologicalAlgebra
    A3.1Introduction
    PartI:ResolutionsandDerivedFunctors
    A3.2FreeandProjectiveModules
    A3.3FreeandProjectiveResolutions
    A3.4InjectiveModulesandResolutions
    A3.5BasicConstructionswithComplexes
    A3.6MapsandHomotopiesofComplexes
    A3.7ExactSequencesofComplexes
    A3.8TheLongExactSequenceinHomology
    A3.9DerivedFunctors
    A3.10Tor
    A3.11Ext

    PartⅡI:FromMappingConestoSpectralSequences
    A3.12TheMappingConeandDoubleComplexes
    A3.13SpectralSequences
    A3.14DerivedCategories

    Appendix4ASketchofLocalCohomology
    A4.1LocalCohomologyandGlobalCohomology
    A4.2LocalDuality
    A4.3DepthandDimension

    Appendix5CategoryTheory
    A5.1Categories,Functors,andNaturalTransformations
    A5.2AdjointFunctors
    A5.3RepresentableFunctorsandYonedasLemma

    Appendix6LimitsandColimits
    A6.1ColimitsintheCategoryofModules
    A6.2FlatModulesasColimitsofFreeModules
    A6.3ColimitsintheCategoryofCommutativeAlgebras
    A6.4Exercises

    Appendix7WhereNext
    HintsandSolutionsforSelectedExercises
    References
    IndexofNotation
    Index
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