Hilbert-Chow morphism for non commutative Hilbert

arXiv:0808.3753v1[math.AG]27Aug2008NonamemanuscriptNo.(willbeinsertedbytheeditor)Hilbert-ChowmorphismfornoncommutativeHilbertschemesandmodulispacesoflinearrepresentationsFedericaGalluzzi·FrancescoVaccarinoAbstractLetkbeacommutativeringandletRbeacommutativek−algebra.Theaimofthispaperistodefineanddiscusssomeconnectionmorphismsbetweenschemesassociatedtotherepresentationtheoryofa(nonnecessarilycommutative)R−algebraA.WefocusontheschemeRepnA//GLnofthen−dimensionalrepresentationsofA,ontheHilbertschemeHilbnAparameterizingtheleftidealsofcodimensionnofAandontheaffineschemeSpecΓnR(A)aboftheabelianizationofthedividedpowersofordernoverA.WegiveageneralizationoftheGrothendieck-DelignenormmapfromHilbnAtoSpecΓnR(A)abwhichspecializestotheHilbertChowmorphismonthegeometricpointswhenAiscommutativeandkisanalgebraicallyclosedfield.DescribingtheHilbertschemeasthebaseofaprincipalbundleweshallfactorthismapthroughthemodulispaceRepnA//GLngivinganicedescriptionofthisHilbert-Chowmorphism,andconsequentlyprovingthatitisprojective.KeywordsHilbert-Chowmorphism·HilbertSchemes·LinearRepresentations·DividedPowersMathematicsSubjectClassification(2000)14A15·14C05·16G99IntroductionLetkbeanalgebraicallyclosedfieldandletAbeacommutativek−algebrawithX=SpecA.Thek−pointsoftheHilbertschemeHilbnAofn−pointsonXparameterizezero-dimensionalsubschemesY⊂Xoflengthn.ItisthesimplestcaseofHilbertThefirstauthorissupportedbyProgettodiRicercaNazionaleCOFIN2006”GeometriadelleVariet`aAlgebricheedeiloroSpazidiModuli”.F.GalluzziDipartimentodiMatematica,Universit`adiTorino,ViaCarloAlberto10,10123Torino,ITALYE-mail:[e-mail]atematica,PolitecnicodiTorino,C.soDucadegliAbruzzi24,10129Torino,ITALYE-mail:[e-mail]2schemeparameterizingclosedsubschemesofXwithfixedHilbertpolynomialP,inthiscasePistheconstantpolynomialn.see[3,13].Then−foldsymmetricproductXn/SnofXisdefinedasSpec(A⊗n)Sn,wherethesymmetricgroupSnactsnaturallyonA⊗n.Itparameterizestheeffective0−cyclesoflengthnonX,(see[3]).Thereisanaturalset-theoreticmapHilbnA→Xn/Snmappingazero-dimensionalsubschemeZinHilbnAtothe0−cycleofdegreen:[Z]=XP∈|Z|dimk(OZ,P)[P].Thisisindeedamorphismofschemes,theHilbert-Chowmorphism(seeforexample[3,8,11,12]).IfXisanonsingularcurve,thismapisanisomorphism(see[5,9]).IfXisanonsingularsurface,theHilbert-ChowmorphismisaresolutionofsingularitiesofXn/Sn(see[8]),butthisisnolongertrueinhigherdimensions.Theaimofthispaperistodefineandstudyageneralizationofthissettingaccordingtothefollowingdirection:theHilbertschemeistherepresentingschemeofafunctorfromcommutativeringstosetswhichcanbeeasilyextendedtoafunctorfrom(noncommutative)algebrastosets.ThislatterisgivenbyHilbnA(B):={leftidealsIinA⊗RBsuchthatM=A⊗RB/IisprojectiveofranknasaB-module}.whereRisacommutativering,A,BareR−algebraswithBcommutative.SincethisisaclosedsubfunctoroftheGrassmannianfunctoritisclearlyrepresentable.Ithasbeenproved(inaspecialbutfundamentalcase)byM.Nori[17]thatthisfunctorisrepresentedbyascheme,whichturnsouttobeaprincipalGLn−bundle.FollowingM.Reineke[21]wecallthisschemethenoncommutativeHilbertscheme.ItshouldbeevidentthatitcoincideswiththeusualHilbertschemeofn−pointswhenAiscommutative.ThedescriptionofHilbnAasaprincipalGLn−bundlelandsustotheGeometricInvariantTheoryoflinearrepresentationsofalgebrasasdefinedbyM.Artin[1]andstudiedbyS.Donkin[6],C.Procesi[19,20]andA.Zubkov[?].InparticularwehaveadescriptionofHilbnAasthequotientbyGLnofanopensubschemeUnAofRepnA×RAnRwhereRepnAistheschemerepresentingthen−dimensionallinearrepresentationsofA.ThisgivesusamorphismHilbnA→RepnA//GLnwhichturnsouttobeprojective.Letusanticipatehowthisworkinafundamentalcase:supposeAisthefreeasso-ciativeringonmvariables.Itsn−dimensionallinearrepresentationsareinbijectionwiththeaffinespaceofm−tuplesofn×nmatrices.Onthisspaceitactsthegenerallineargroupbysimultaneousconjugationi.e.viabaseschange.Thecoarsemodulispaceparameterizingtheorbitsforthisactionisthespectrumoftheinvariantsonm−tuplesofmatrices.UsingresultsduetoV.Balaji[2]weareabletocharacterizetheHilbertschemeasthequotientofanopensubschemeofm−tuplesofmatricestimesthestan-dardrepresentationofthegenerallineargroup.ThemorphismHilbnA→RepnA//GLnisthengivenbypassingtothequotientstheprojectionRepnA×RAnR→RepnAi.e.ongeometricpointtheorbitof(a1,...,am,v)mapstotheoneof(a1,...,am).Oneforgetsthevector.Howtoinsertthesymmetricproductintothisframework?Thistheideawepur-sued:supposeAisfreeasR−module,composingarepresentationoverBwiththedeterminantwegenerateamultiplicativepolynomialmappinghomogeneousofdegree3n.Thiscorrespondstoauniquemorphism(A⊗n)Sn→Bi.e.toaB−pointofSpec(A⊗n)Sn/{xy−yx:x,y∈(A⊗n)Sn}.ThisseemstobeagoodcandidateforthecodomainoftheHilbert-Chowmorphism.Theonlyreplacementtobedonetoovercomethefreenessrequirementistosubstitute(A⊗n)SnwithΓn(A),whereΓ(A)=LnΓn(A)isthedividedpoweralgebraonA.InfactforanR−algebraAonehasthatΓn(A)representsthefunctorB→{multiplicativepolynomialmappinghomogeneousofdegreenfromAtoB},whereBisacommutativeR−algebraandapolynomiallawisageneralizationoftheconceptofpolynomialmapping