Coherent sheaves and cohesive sheaves

arXiv:0808.1717v1[math.CV]12Aug2008COHERENTSHEAVESANDCOHESIVESHEAVESL´aszl´oLempertAbstract.WeconsidercoherentandcohesivesheavesofO–modulesoveropensetsΩ⊂Cn.Weprovethatcoherentsheaves,andcertainothersheavesderivedfromthem,arecohesive;andconversely,certainsheavesderivedfromcohesivesheavesarecoherent.Animportanttoolinallthis,alsoprovedhere,isthatthesheafofBanachspacevaluedholomorphicgermsisflat.ToLindaRothschildonherbirthday1.IntroductionThetheoryofcoherentsheaveshasbeencentraltoalgebraicandanalyticgeom-etryinthepastfiftyyears.Bycontrast,ininfinitedimensionalanalyticgeometrycoherenceisirrelevant,asmostsheavesassociatedwithinfinitedimensionalcom-plexmanifoldsarenotevenfinitelygeneratedoverthestructuresheaf,letalonecoherent.InarecentpaperwithPatyi,[LP],weintroducedtheclassofsocalledcohesivesheavesinBanachspaces,thatseemstobethecorrectreplacementofcoherentsheaves—wewerecertainlyabletoshowthatmanysheavesthatoccurinthesubjectarecohesive,andforcohesivesheavesCartan’sTheoremsAandBhold.WewillgooverthedefinitionofcohesivesheavesinSection2,butforapreciseformulationoftheresultsabovethereaderisadvisedtoconsult[LP].Whilecohesivesheavesweredesignedtodealwithinfinitedimensionalproblems,theymakesenseinfinitedimensionalspacesaswell,andtherearereasonstostudytheminthiscontext,too.First,somenaturalsheavesevenoverfinitedimensionalmanifoldsarenotfinitelygenerated:forexamplethesheafOEofgermsofholomor-phicfunctionstakingvaluesinafixedinfinitedimensionalBanachspaceEisnot.Itisnotquasicoherent,either(forthisnotion,see[Ha]),butitiscohesive.Sec-ond,anaturalapproachtostudycohesivesheavesininfinitedimensionalmanifoldswouldbetorestrictthemtovariousfinitedimensionalsubmanifolds.Theissuetobeaddressedinthispaperistherelationshipbetweencoherenceandcohesioninfinitedimensionalspaces.OurmainresultsareTheorems4.3,4.4,and4.1.Looselyspeaking,thefirstsaysthatcoherentsheavesarecohesive,andthesecondthattheyremaincohesiveevenaftertensoringwiththesheafOFofResearchpartiallysupportedbyNSFgrantDMS0700281,theMittag–LefflerInstitute,andtheClayInstitute.1991MathematicsSubjectClassification.32C35,32B05,14F05,13C.TypesetbyAMS-TEX12L´ASZL´OLEMPERTholomorphicgermsvaluedinaBanachspaceF.AkeyelementoftheproofisthatOFisflat,Theorem4.1.Thislatterisalsorelevantforthestudyofsubvarieties.Ontheotherhand,MasagutovshowedthatOFisnotfreeingeneral,see[Ms,Corollary1.4].Theresultsabovesuggesttwoproblems,whoseresolutionhaseludedus.First,isthetensorproductofacoherentsheafwithacohesivesheafitselfcohesive?Ofcourse,onecanalsoaskthemoreambitiousquestionwhetherthetensorproductoftwocohesivesheavesiscohesive,buthereoneshoulddefinitelyconsidersomekindof“completed”tensorproduct,anditispartoftheproblemtofindwhichone.Thesecondproblemiswhetherafinitelygeneratedcohesivesheafiscoherent.Ifso,thencoherentsheavescouldbedefinedascohesivesheavesoffinitetype.Wecouldonlysolvesomerelatedproblems:accordingtoCorollary4.2,anyfinitelygeneratedsubsheafofOFiscoherent;andcohesivesubsheavesofcoherentsheavesarealsocoherent,Theorem5.4.2.Cohesivesheaves,anoverviewInthisSectionwewillreviewnotionsandtheoremsrelatedtothetheoryofcohesivesheaves,following[LP].Weassumethereaderisfamiliarwithverybasicsheaftheory.Onegoodreferencetowhatweneedhere—andmuchmore—is[S].LetΩ⊂CnbeanopensetandEacomplexBanachspace.Afunctionf:Ω→Eisholomorphicifforeacha∈ΩthereisalinearmapL:Cn→Esuchthatf(z)=f(a)+L(z−a)+okz−ak,z→a.ThisisequivalenttorequiringthatineachballB⊂Ωcenteredatanya∈Ωourfcanberepresentedasalocallyuniformlyconvergentpowerseriesf(z)=Pjcj(z−a)j,withj=(j1,...,jn)anonnegativemultiindexandcj∈E.WedenotebyOEΩorjustOEthesheafofholomorphicE–valuedgermsoverΩ.Inparticular,O=OCisasheafofrings,andOEisasheafofO–modules.Typically,insteadofasheafofO–moduleswewilljusttalkaboutO–modules.Definition2.1.ThesheavesOE=OEΩ→Ωarecalledplainsheaves.Theorem2.2.[Bi,Theorem4],[Bu,p.331]or[L,Theorem2.3].IfΩ⊂Cnispseudoconvexandq=1,2,...,thenHq(Ω,OE)=0.GivenanotherBanachspaceF,wewriteHom(E,F)fortheBanachspaceofcontinuouslinearmapsE→F.IfU⊂Ωisopen,thenanyholomorphicΦ:U→Hom(E,F)inducesahomomorphismϕ:OE|U→OF|U,byassociatingwiththegermofaholomorphice:V→Eatζ∈V⊂Uthegermofthefunctionz7→Φ(z)e(z),againatζ.Suchhomomorphismsandtheirgermsarecalledplain.ThesheafofplainhomomorphismsbetweenOEandOFisdenotedHomplain(OE,OF).IfHomO(A,B)denotesthesheafofO–homomorphismsbetweenO–modulesAandB,then(2.1)Homplain(OE,OF)⊂HomO(OE,OF)isanO–submodule.Infact,Masagutovshowedthatthetwosidesin(2.1)areequalunlessn=0,see[Ms,Theorem1.1],butforthemomentwedonotneedthis.TheCOHERENTSHEAVESANDCOHESIVESHEAVES3O(U)–moduleofsectionsΓ(U,Homplain(OE,OF))isinone–to–onecorrespondencewiththeO(U)–moduleHomplain(OE|U,OF|U)ofplainhomomorphisms.Further,anygermΦ∈OHom(E,F)zinducesagermϕ∈Homplain(OE,OF)z.Aspointedoutin[LP,Section2],theresultingmapisanisomorphism(2.2)OHom(E,F)≈→Homplain(OE,OF)ofO–modules.Definition2.3.AnanalyticstructureonanO–moduleAisthechoice,foreachplainsheafE,ofasubmoduleHom(E,A)⊂HomO(E,A),subjectto(i)ifE,Fareplainsheavesandϕ∈Homplain(E,F)zforsomez∈Ω,thenϕ∗Hom(F,A)z⊂Hom(E,A)z;and(ii)Hom(O,A)=HomO(O,A).IfAisendowedwithananalyticst