非交换几何与动机（孔涅）

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1、NONCOMMUTATIVEGEOMETRYANDMOTIVES:THETHERMODYNAMICSOFENDOMOTIVESALAINCONNES,CATERINACONSANI,ANDMATILDEMARCOLLIContents1.Introduction22.Correspondencesfornoncommutativespaces52.1.KK-theory62.2.Theabeliancategoryof¤-modulesandcyclic(co)homology73.ArtinMotivesandnoncommutativespaces93.1.Complexcoe±c

2、ients103.2.Artinmotives103.3.Endomotives114.ScalingasFrobeniusincharacteristiczero204.1.Preliminaries214.2.Thecoolingmorphism224.3.Thedistilled¤-moduleD(A;')264.4.Dualactiononcyclichomology304.5.TypeIIIfactorsandunrami¯edextensions355.Geometryoftheadeleclassspace375.1.ThespectralrealizationonH13

3、7K;C5.2.TheTraceFormula405.3.Theperiodicclassicalpoints"ofXK415.4.Weilpositivity426.Higherdimensionalvirtualcorrespondences446.1.GeometriccorrespondencesandKK-theory446.2.CyclesandK-theory456.3.AlgebraicversustopologicalK-theory467.ALefschetzformulaforarchimedeanlocalfactors477.1.Weilformofloga

4、rithmicderivativesoflocalfactors497.2.Lefschetzformulaforcomplexplaces527.3.Lefschetzformulaforrealplaces527.4.Thequestionofthespectralrealization547.5.Localfactorsforcurves56References5712CONNES,CONSANI,ANDMARCOLLI1.IntroductionAfewunexpectedencountersbetweennoncommutativegeometryandthetheoryof

5、mo-tiveshavetakenplacerecently.A¯rstinstanceoccurredin[14],wheretheWeilexplicitformulaeacquirethegeometricmeaningofaLefschetztraceformulaoverthenoncommu-tativespaceofadeleclasses.Thereasonwhytheconstructionof[14]shouldberegardedasmotivicistwofold.Ontheonehand,aswediscussinthispaper,theadeleclass

6、spaceisobtainedfromanoncommutativespace,theBost{Connessystem,whichsitsnaturallyinacategoryofnoncommutativespacesextendingthecategoryofArtinmotives.Moreover,aswediscussbrie°yinthispaperandinfulldetailinourforthcomingwork[15],itispossibletogiveacohomologicalinterpretationofthespectralrealizationof

7、thezerosoftheRiemannzetafunctionof[14]onthecyclichomologyofanoncommutativemotive.Thereasonwhythisconstructiontakesplaceinacategoryof(noncommutative)motivesisthatthegeometricspaceweneedtouseisobtainedasacokernelofamorphismofa