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

图书标准信息

• 作者 艾森巴德（David Eisenbud） 著；世界图书出版公司北京公司 编
• 出版社 世界图书出版公司
• 出版时间 2004-01
• 版次 1
• ISBN 9787506292450
• 定价 118.00元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 797页
• 正文语种 英语

【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