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.ArtinMotivesandnoncomm

2、utativespaces93.1.Complexcoe±cients103.2.Artinmotives103.3.Endomotives114.ScalingasFrobeniusincharacteristiczero204.1.Preliminaries214.2.Thecoolingmorphism224.3.Thedistilled¤-moduleD(A;')264.4.Dualactiononcyclichomology304.5.TypeIIIfactorsandunrami¯edextensions355.G

3、eometryoftheadeleclassspace375.1.ThespectralrealizationonH137K;C5.2.TheTraceFormula405.3.Theperiodicclassicalpoints"ofXK415.4.Weilpositivity426.Higherdimensionalvirtualcorrespondences446.1.GeometriccorrespondencesandKK-theory446.2.CyclesandK-theory456.3.Algebraicve

4、rsustopologicalK-theory467.ALefschetzformulaforarchimedeanlocalfactors477.1.Weilformoflogarithmicderivativesoflocalfactors497.2.Lefschetzformulaforcomplexplaces527.3.Lefschetzformulaforrealplaces527.4.Thequestionofthespectralrealization547.5.Localfactorsforcurves56R

5、eferences5712CONNES,CONSANI,ANDMARCOLLI1.IntroductionAfewunexpectedencountersbetweennoncommutativegeometryandthetheoryofmo-tiveshavetakenplacerecently.A¯rstinstanceoccurredin[14],wheretheWeilexplicitformulaeacquirethegeometricmeaningofaLefschetztraceformulaovertheno

6、ncommu-tativespaceofadeleclasses.Thereasonwhytheconstructionof[14]shouldberegardedasmotivicistwofold.Ontheonehand,aswediscussinthispaper,theadeleclassspaceisobtainedfromanoncommutativespace,theBost{Connessystem,whichsitsnaturallyinacategoryofnoncommutativespacesexte

7、ndingthecategoryofArtinmotives.Moreover,aswediscussbrie°yinthispaperandinfulldetailinourforthcomingwork[15],itispossibletogiveacohomologicalinterpretationofthespectralrealizationofthezerosoftheRiemannzetafunctionof[14]onthecyclichomologyofanoncommutativemotive.There

8、asonwhythisconstructiontakesplaceinacategoryof(noncommutative)motivesisthatthegeometricspaceweneedtouseisobtainedasacokernelofamorphismofa