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.ArtinMotivesandnoncommutativ

2、espaces93.1.Complexcoe±cients103.2.Artinmotives103.3.Endomotives114.ScalingasFrobeniusincharacteristiczero204.1.Preliminaries214.2.Thecoolingmorphism224.3.Thedistilled¤-moduleD(A;')264.4.Dualactiononcyclichomology304.5.TypeIIIfactorsandunrami¯edextensions355.Geometryofthe

3、adeleclassspace375.1.ThespectralrealizationonH137K;C5.2.TheTraceFormula405.3.Theperiodicclassicalpoints"ofXK415.4.Weilpositivity426.Higherdimensionalvirtualcorrespondences446.1.GeometriccorrespondencesandKK-theory446.2.CyclesandK-theory456.3.AlgebraicversustopologicalK-t

4、heory467.ALefschetzformulaforarchimedeanlocalfactors477.1.Weilformoflogarithmicderivativesoflocalfactors497.2.Lefschetzformulaforcomplexplaces527.3.Lefschetzformulaforrealplaces527.4.Thequestionofthespectralrealization547.5.Localfactorsforcurves56References5712CONNES,CONS

5、ANI,ANDMARCOLLI1.IntroductionAfewunexpectedencountersbetweennoncommutativegeometryandthetheoryofmo-tiveshavetakenplacerecently.A¯rstinstanceoccurredin[14],wheretheWeilexplicitformulaeacquirethegeometricmeaningofaLefschetztraceformulaoverthenoncommu-tativespaceofadeleclass

6、es.Thereasonwhytheconstructionof[14]shouldberegardedasmotivicistwofold.Ontheonehand,aswediscussinthispaper,theadeleclassspaceisobtainedfromanoncommutativespace,theBost{Connessystem,whichsitsnaturallyinacategoryofnoncommutativespacesextendingthecategoryofArtinmotives.Moreo

7、ver,aswediscussbrie°yinthispaperandinfulldetailinourforthcomingwork[15],itispossibletogiveacohomologicalinterpretationofthespectralrealizationofthezerosoftheRiemannzetafunctionof[14]onthecyclichomologyofanoncommutativemotive.Thereasonwhythisconstructiontakesplaceinacatego

8、ryof(noncommutative)motivesisthatthegeometricspaceweneedtouseisobtainedasacokernelofamorphismofa