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1、JOURNALOFALGEBRA65,453--494(1980)........?.AJacobianCriterionforSeparabilitySTUARTSuI-SHENGWANG*tDepartmentofMathematics,TexasTechUniversity,Lubbock,Texas79409CommunicatedbyD.A.BuchsbaumReceivedFebruary23,1978Contents.1.IntroductionandNotation;2.~tJacobianCr
2、iterionforSeparability;3.AJacobianCriterionforExtensionofDerivations;4.ExplicitFormulasfortheSeparabilityIdempotent;4.1.l-variable,1-relationcaseandatheoremofEuler,4.2.1-variable,11I-relationcase,4.3.Symmetricexpansionsandn-algebras,4.4.n-variable,n-relation
3、casewherethebaseringcontainstherationalnumberfield,4.5.n-variable,IIt-relati0ncasewherethebaseringcontainstherationalnumberfield,4.6.n-variable,n-relationcasewhere2isaunit,therelationsarequadraticandtheJacobiandeterminantisinthebasering;5.ApplicationstotheJa
4、cobianProblem;5.1.Briefhistory,5.2.InterpretationsoftheJacobiancondition,5.3.Consequencesofflatness,5.4.Equivalentconditionsunderseparability,5.5.Pseudo-inversepolynomials,5.6.Counter-examples,5.7.Quadraticease.1.INTRODUCTIONANDNOTATIONThemainpurposeofthispa
5、peristodevelopvariousJacobiancriteriaandexplicitformulasforseparabilityidempotents.Thesearethenappliedtoalong-standingunsolvedprobleminalgebraicgeometry,knownastheJacobianproblem.Inparticular,wesettlethequadraticcaseaffirmatively.InSection2,itisshownthatsepa
6、rabilityimpliesleft-invertibilityoftheJacobianmatrixandhencethatthenumberofrelationsmustbegreaterthanorequaltothenumberofgenerators[Theorem2].Itisalsoshownthatforanalgebraoffinitetype,separabilityisequivalenttoleft-invertibilityoftheJacobianmatrix[Theorem5].
7、Ofspecialinterestisthecaseinwhichthenumberofvariablesisthesameasthenumberofrelations[Theorem7].Aseparablealgebraoffinitetypeisahomomorphicimageofaseparablealgebraoffinitepresentationwithoutchangingthesetofgenerators[Corollary6].Section3dealswiththeproblemofe
8、xtendingderivations.Itisshownthatleft-invertibilityoftheJacobianmatriximpliesuniquenesswhileright-inver-tibilityimpliesextendibility[Theorem12].Ifthereisafamilyofderivationswhichbehavelikeordina
