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1、AhomologyandcohomologytheoryforrealprojectivevarietiesJyh-HaurTehNationalCenterforTheoreticalSciencesMathematicsDivisionThirdGeneralBuilding,TsingHuaUniversity,Hsinchu,30043,TaiwanAbstractInthispaperwedevelophomologyandcohomologytheorieswhichplaythesameroleforrealprojectivevarietiesthatLawsonh
2、omologyandmorphiccohomologyplayforprojectivevarietiesrespectively.Theyhavenicepropertiessuchastheexistenceoflongexactsequences,thehomotopyinvariance,theLawsonsuspensionproperty,thehomotopypropertyforbundleprojection,thesplittingprinciple,thecupproduct,theslantproductandthenaturaltransformation
3、stosingulartheories.TheFriedlander-LawsonMovingLemmaisusedtoproveadualitytheorembetweenthesetwotheories.MSC:Primary14C25Secondary14P25Keywords:Lawsonhomology;Morphiccohomology;Realprojectivevariety;Realcyclegroup;Movinglemma.1IntroductionThestudyofsolvingpolynomialequationsdatesbacktotheverybe
4、ginningofmathematics.arXiv:math.AG/0508238v114Aug2005Generalalgebraicsolutionsofagivenequationwastheoriginalgoal.Thisgoalwasachievedforequationsofdegree2,3and4.ButitwasprovedbyAbelandGaloisthatitwasimpossibleforequationsofdegree5.Galoistheorywascreatedseveraldecadeslatertostudysomepropertiesof
5、therootsofequations.Atthesametime,peoplestartedtoconsiderthemorecomplicatedproblemofsolvingpolynomialequationsofmorethanonevariable.Thezerolociofpolynomialequations,whicharecalledalgebraicvarieties,arebasicsourceofgeometryandexemplifymanyimportantgeometricphenomena.Inthispaper,westudytheproper
6、tiesofprojectivevarietieswhicharethezerolociofhomogeneouspolynomialsinprojectivespaces.Algebraiccyclesaresomefiniteformalsumofirreduciblesubvarietieswithintegralcoef-ficients.ThegroupZp(X)ofp-cyclesofaprojectivevarietyXencodesmanypropertiesof1Tel:+88635745248;Fax:+88635728161E-mailaddress:jyhhau
7、r@math.cts.nthu.edu.tw1X.Formanydecades,sinceZp(X)isaverylargegroupingeneral,quotientsofZp(X)werestudiedinstead.Forexample,thequotientofZp(X)byrationalequivalenceistheChowgroupCHp(X)onwhichtheintersectiontheoryofalgebraicvarietiescanbeb
