ash r.b. a course in commutative algebra (2003)(104s)

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1、1ACourseInCommutativeAlgebraRobertB.AshPrefaceThisisatextforabasiccourseincommutativealgebra,writteninaccordancewiththefollowingobjectives.Thecourseshouldbeaccessibletothosewhohavestudiedalgebraatthebeginninggraduatelevel.Forgeneralalgebraicbackground,seemyonlinetext“AbstractAlgebra:TheBasicGraduate

2、Year”,whichcanbedownloadedfrommywebsitewww.math.uiuc.edu/∼r-ashThistextwillbereferredtoasTBGY.Theideaistohelpthestudentreachanadvancedlevelasquicklyandefficientlyaspossible.InChapter1,thetheoryofprimarydecompositionisdevelopedsoastoapplytomodulesaswellasideals.InChapter2,integralextensionsaretreatedin

3、detail,includingthelyingover,goingupandgoingdowntheorems.Theproofofthegoingdowntheoremdoesnotrequireadvancedfieldtheory.ValuationringsarestudiedinChapter3,andthecharacterizationtheoremfordiscretevaluationringsisproved.Chapter4discussescompletion,andcoverstheArtin-ReeslemmaandtheKrullintersectiontheor

4、em.Chapter5beginswithabriefdigressionintothecalculusoffinitedifferences,whichclarifiessomeofthemanipulationsinvolvingHilbertandHilbert-Samuelpolynomials.ThemainresultisthedimensiontheoremforfinitelygeneratedmodulesoverNoetherianlocalrings.AcorollaryisKrull’sprincipalidealtheorem.Someconnectionswithalgeb

5、raicgeometryareestablishedviathestudyofaffinealgebras.Chapter6introducesthefundamentalnotionsofdepth,systemsofparameters,andM-sequences.Chapter7developsenoughhomologicalalgebratoprove,underappropratehypotheses,thatallmaximalM-sequenceshavethesamelength.ThebriefChapter8developsenoughtheorytoprovethatar

6、egularlocalringisanintegraldomainaswellasaCohen-Macaulayring.Aftercompletingthecourse,thestudentshouldbeequippedtomeettheKoszulcomplex,theAuslander-Buchsbaumtheorems,andfurtherpropertiesofCohen-Macaulayringsinamoreadvancedcourse.BibliographyAtiyah,M.F.andMacdonald,I.G.,IntroductiontoCommtativeAlgebr

7、a,Addison-Wesley1969Balcerzyk,S.andJozefiak,T.,CommutativeNoetherianandKrullRings,Wiley1989Balcerzyk,S.andJozefiak,T.,CommutativeRings:Dimension,MultiplicityandHomo-logicalMethods,Wiley1989Eisenbud,D.,C