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1、?WHATIS...aMotive?BarryMazurHowmuchofthealgebraictopologyofacon-amoreintricatesetuptodealwith:foronething,nectedfinitesimplicialcomplexXiscapturedbywedon’tevenhaveacohomologytheorywithco-itsone-dimensionalcohomology?Specifically,howefficientsinZforvarietiesoverafieldkunless
2、wemuchdoyouknowaboutXwhenyouknowprovideahomomorphismk→C,sothatwecanH1(X,Z)alone?formthetopologicalspaceofcomplexpointsonFora(nearlytautological)answer,putGX:=ourvarietyandcomputethecohomologygroupsthecompact,connectedabelianLiegroupofthattopologicalspace.Oneperplexityhereis
3、(i.e.,productofcircles)whichisthePontrjaginthatthiscohomologyconstructionmay(andindualofthefreeabeliangroupH1(X,Z).Nowgeneral,does!)dependupontheimbeddingk→C.H1(GX,Z)iscanonicallyisomorphictoH1(X,Z)=And,ofcourse,therearefieldsknotadmittingHom(GX,R/Z)andthereisacanonicalhomo
4、topyembeddingsintoC.classofmappingsIncompensation,thereisaprofusionofdiffer-X−→GXentcohomologyfunctorsbeyondtheonescomingfromclassicalalgebraictopologyviaimbeddingsthatinducestheidentitymappingonH1.k→C.SomeofthesetheoriescomedependentTheanswer:weknowwhateverinformationcanup
5、onthespecificgroundfieldk,withtheirspecificbereadofffromGXandareignorantofanythingringsofcoefficients,andwithglobalrequirementsthatgetslostintheprojectionX→GX.onthevarietiesforwhichtheyaredefined.SomeThetheoryofEilenberg-MacLanespacesofferscomewiththeirownparticularattendan
6、tstructureusasomewhatanalogousanalysisofwhatweknowandwiththeirrelationstoalltheothercohomol-anddon’tknowaboutX,whenweequipourselvesogytheories:Hodgecohomology,algebraicdeRhamwithn-dimensionalcohomology,foranyspecificcohomology,crystallinecohomology,theétale-adicn,withspecif
7、iccoefficients.cohomologytheoriesforeachprimenumber,...Ifwerepeatourrhetoricalquestioninthecon-Istheresomesystematicandnaturalwayoftextofalgebraicgeometry,wherethestructureisencapsulatingallthisinformationaboutthesomewhatricher,canwehopeforasimilardis-n-dimensionalcohomolog
8、yofprojectivesmoothcussion?varietiesV(evenjustforn=1)?(ThetraditionInalgebraictopo
