A Construction of a Differential Graded Lie Algebr

arXiv:math/0602287v1[math.AG]14Feb2006AConstructionofaDifferentialGradedLieAlgebraintheCategoryofEffectiveHomologicalMotivesKajM.GartzAbstractThistextgivesaconstructionofadifferentialgradedLiealgebrainNori’scategoryofeffectivehomologicalmotives.InfacttheconstructionworksinmoreageneralsettingthanthatofanAbeliancategory.ThisallowsustogivetherationalhomotopyLiealgebraofa1-connectedspaceamotivicstructure.AsaconsequencetherationalhomotopyLiealgebrainheritsamixedHodgestructureandGaloismodulestructure.1.IntroductionThegoalofthispaperistoshowthattherationalhomotopyLiealgebraofavariety(definedinSection5.1)carriesamotivicstructureinthesenseofNori.Thisisformalizedasfollows.Ifk֒→CisanembeddingofafieldkintothecomplexnumbersthenNori(see[Br]and[Le])definesanAbeliantensorcategoryofeffectivehomologicalmotivesoverk,calledIndEHM.LetM=Ch(IndEHM)theAbeliancategoryofchaincomplexesofmotives.TosaythatXcarriesamotivicstructuremeansthatXisanobjectinMandthatH∗(X)isanobjectofEHM.NoriuseshisBasicLemma[No]toconstructafunctorwhichassignstoanyvarietyacomplexofmotives.Asimplercaseofthisconstructionoccurswhenoneconsidersthecategoryofaffinevarieties.WedenotethisfunctorbyX7→C∗(X).Therestrictiontoaffinevarietiesisnottooseverethanksto“Jouanolou’sTrick”(see[Jo]):Foranyquasi-projectivevarietyXoverafieldk,thereexistsanaffinevarietyX′overkandamorphismX′→XwhichisaZariskilocallytrivialfibrationwithfibersisomorphictoAn.Sincetheaffinefibersarecontractible,X′ishomotopyequivalenttoX.NorialsoprovidesrealizationfunctorsfromIndEHMtothecategoryofmixedHodgestructuresandtothecategoryofGaloisrepresentations.LetMbethecategoryofchaincomplexesinIndEHM.Thereisaforgetfulfunctor,fffromIndEHMtoAbeliangroups(andthusaforgetfulfunctorfromMtochaincomplexesofAbeliangroups),suchthatH∗(ffC∗(X))∼=H∗(X(C),Z),whereH∗(X(C),Z)denotesthesingularhomologyofthetopologicalspaceX(C).InthispaperweprovideafunctorPFwhichassociatestoanyaffinevariety,X,adifferentialgradedLiealgebra(d.g.l.),PF(X)inM,suchthatifthevarietyissimplyconnectedthenthehomologyofthed.g.l.computestherationalhomotopyLiealgebra,thatisH∗(PF)∼=π∗+1(X)⊗QasgradedLiealgebras.TheknowntopologicaltechniquesforproducingsuchaLiealgebraalluseacoalgebraanalogoustothatofsingularchainswiththeAlexander-Whitneymaporthewedgeproductofforms.2000MathematicsSubjectClassification14F35,14F42,57T99KajM.GartzThisfunctor,PF,comesfromanunderlyingcombinatorialconstructionongeometricobjects,andisperhapsthesimplestLiealgebraonecanconstructfromasimplicialobjectinanAbeliancategorywhichisrepresentable(i.e.itcanbeextendedtothecategoryoffinitesetswithallmor-phisms,notjustorderpreservingmaps.)Applyingtheconstructionwithsingularchainstoanytopologicalspaceyieldsanewmethodofproducingad.g.l.whichcomputestherationalhomotopyLiealgebra.Thisfunctorialconstructionhasseveralapplications.First,sinceNori’scategoryofmotiveshasrealizationfunctors,ourfunctorgivesrisetoaGaloisstructureonthehomotopyLiealgebraaftertensoringwithQp.PreviouslytherewasthemixedHodgestructureprovidedbyHain,butsinceheusesthegradedcommutativealgebrastructureofthedeRhamcomplex,histechniqueswerenoteasilyadaptedtothecaseof´etalecohomologyandGaloisrepresentations.Second,wepavethewayforamotivicstudyofextensionsarisingfromhigherrationalhomotopytheory–analogoustoHain’sworkwithextensionsofmixedHodgestructuresarisingfromthefundamentalgroup[Ha1].TheseriousdifficultyinshowingthattherationalhomotopyLiealgebraofavarietyhasamotivicstructurereducestothefactthatalthoughthereisamotivicanalogoftheEilenberg-ZilbermapC∗(X)⊗C∗(Y)→C∗(X×Y),whichisaquasi-isomorphism,thereisnomapintheotherdirectioncorrespondingtotheAlexander-Whitneymap.WithouttheAlexander-Whitneymapthereisnocoalgebrastructureonmotivicchains.Itisunreasonabletoexpectasplittingofthismap(liketheAlexander-Whitneymap)inacategoryofmotives(otherwisethesplittingforcesnon-zeroextensionclassestovanish).ThusC∗(X)doesnothavethestructureofacoalgebra,incontrasttothetopologicalcase,whereonehasmapsonsingularchainsSing(X)Sing(Δ)−−−−−→Sing(X×X)A−W−−−→Sing(X)⊗Sing(X).Putanotherway–itisclearhowtotaketwocyclesandproduceacycleontheproduct,butnotviceversa.Thusmuchofthemachineryofalgebraictopology(inourcaseinparticularanythingrelatedtodifferentialgradedHopfalgebras)cannotbereproducedinthemotiviccategorywithoutmuchwork.Althoughwehavenocoalgebra,westillhaveamapC∗(X)⊗n→C∗(Xn)whichisΣnequivariant.InsomewaysthekeytothispaperisthatthisΣn–equivarianceisenoughtogetaLiealgebraandthatwedon’tneed(orhave)acoalgebrastructure.OnecrucialpointisthatwhenVisagradedvectorspace,consideringthetensoralgebraT(V)asadifferentialgradedHopfalgebra,theprimitiveLiealgebrainT(V)coincideswiththefreeLiealgebraonV,L(V)–andthiscanbeobtainedastheimageofaprojectordefinedintermsoftheΣnactiononC∗(Xn).ThisiscrucialbecausedefiningthisLiealgebraasprimitivesofaco-multiplicationmapwouldrequireamapC∗(X)→C∗(X)⊗C∗(X)whichdoesnotexistsincethereisnomapC∗(X×X)→C∗(X)⊗C∗(X).TheconstructionofPFisverysimpleandcombinatorialinnature.Ithasnotbeenpreviouslydiscussedbeforeprobablybecause:1.AlgebraictopologistswouldhavenoneedtomakethisconstructionwiththepresenceoftheAl