Symplectic Reduction of Sheaves of $mathcal{A}$-mo

arXiv:0802.4224v1[math.SG]28Feb2008SymplecticReductionofSheavesofA-modulesAnastasiosMallios,PatriceP.Ntumba∗AbstractGivenanarbitrarysheafEofA-modules(orA-moduleinshort)onatopologicalspaceX,wedefineannihilatorsheavesofsub-A-modulesofEinawaysimilartotheclassicalcase,andobtainthere-aftertheanalogofthemaintheorem,regardingclassicalannihilatorsinmoduletheory,seeCurtis[[5],pp.240-242].Thefamiliarclassicalproperties,satisfiedbyannihilatorsheaves,allowustosetclearlythesheaf-theoreticversionofsymplecticreduction,whichisthemaingoalinthispaper.SubjectClassification(2000):55P05.KeyWords:A-module,annihilatorsheaves,orderedR-algebraizedspace,,symplecticA-module,affineDarbouxtheorem.IntroductionThispaperispartofourongoingprojectofalgebraizingclassicalsymplecticgeometryusingthetoolsofabstractdifferentialgeometry(`alaMallios).Ourmainreferenceasfarasabstractdifferentialgeometryisconcernedisthefirstauthor’sbook[10].Forthesakeofconvenience,werecallheresomeoftheobjectsofabstractdifferentialgeometrythatrecurallthroughout.∗Isthecorrespondingauthorforthepaper12AnastasiosMallios,PatriceP.NtumbaLetXbeatopologicalspace.AsheafofC-algebrasoraC-algebrasheaf,onX,isatripleA≡(A,τ,X)satisfyingthefollowingconditions:(i)Aisasheafofrings.(ii)FibersAx≡τ−1(x),x∈X,areC-algebras.(iii)ThescalarmultiplicationinA,viz.themapC×A−→A:(c,a)7−→c·a∈Ax⊆Awithτ(a)=x∈X,iscontinuous;inthismapping,Cisassumedtocarrythediscretetopology.Thetriple(A,τ,X)iscalledaunitalC-algebrasheafiftheindividualfibersofA,Ax,x∈X,areunitalC-algebras.Apair(X,A),withAassumedtobeunitalandcommutative,iscalledaC-algebraizedspace.Next,supposethatA≡(A,τ,X)isaunitalC-algebrasheafonX.AsheafofA-modules(oranA-module),onX,isasheaf,E≡(E,ρ,X),onXsuchthatthefollowingpropertieshold:(iv)EisasheafofabeliangroupsonX.(v)FibersEx,x∈X,ofEareAx-modules.(vi)TheleftactionA◦E−→E,describedby(a,z)7−→a·z∈Ex⊆E,withτ(a)=ρ(z)=x∈X,iscontinuous.Thesheaf-theoreticversionoftheclassicalnotionofadualmoduleisdefinedinthismanner:GivenaC-algebraizedspace(X,A)andanA-moduleEonX,theA-module(onX)E∗:=HomA(E,A)iscalledthedualA-moduleofE.FortowgivenA-modulesonatopologicalspaceX,HomA(E,F)istheA-modulegeneratedonXbythe(complete)SymplecticReductionofSheavesofA-modules3presheaf,givenbyU7−→HomA|U(E|U,F|U),whereUrunsovertheopensubsetsofX;therestrictionmapsofthispresheafarequiteobvious.AmostfamiliarconsequenceregardingdualA-modulesisthatgivenafreeA-moduleEoffiniterankonX,onehasE=E∗,withinanA-isomorphism.Section1isconcernedwithannihilatorsheavesofsub-A-modulesofarbitraryA-modulesononehand,andϕ-annihilatorsheaves,i.e.annihilatorsheaves(ofsub-A-modules)withrespecttoanon-degeneratebilinearA-morphismϕ:E⊕F−→A,whereEandFarefreeA-modulesoffiniterank.Thesheaf-theoreticversionofthemaintheoremonclassicalannihilatorsisexamined.Thesectionendswiththeinterestingresultthatgivenasub-A-moduleFofanA-moduleE,thedualA-module(E/F)∗isA-isomorphictotheannihilatorF⊥ofF.Section2dealswithpropertiesofexteriorrankwiseA-2-forms.WeprovideanotherprooffortheaffineDarbouxtheorem.TheproofisderivedfromE.Cartan[4].Section3,whichisthelastsection,outlinesthesymplecticreductionofanA-moduleEbyaco-isotropicsub-A-moduleFofE;theA-moduleEcarriesasymplectic(A−)structure,givenbytheA-morphismω:E⊕E−→A.1AnnihilatorSheavesDefinition1.1Let(S,π,X)beasheaf.ByasubsheafofS,wemeanasheafEonX,generatedbyapresheaf(E(U),σUV)whichissuchthat,forallopenU⊆XandopenV⊆U,•E(U)⊆S(U),•σUV=ρUV|E(U),where(S(U)≡Γ(U,S),ρUV)≡Γ(S)isthe(complete)presheafofsectionsofthesheafS,cf.Mallios[[10],Lemma11.1,p.48].4AnastasiosMallios,PatriceP.NtumbaWecannowdefinethenotionofsub-A-moduleofagivenA-module,whichwillbeofuseinthesequel.Definition1.2AsubsheafEofanA-moduleS,definedonatopologicalspaceX,iscalledasub-A-moduleofSifEisanA-moduleandtheinclusioni:E⊆SisanA-morphism.Lemma1.1Subsheavesareopensubsets,andconversely.Proof.LetSbeasheafon(X,T),Easubsheaf,ofS,generatedbythepresheaf(E(U),σUV),andletusdenotebyℜthesetS{E(U):U∈T}.AccordingtoMallios[[10],Theorem3.1,p.14],thefamilyB={s(U):s∈ℜandU∈TwithU=Dom(s)}isabasisforthetopologyofE,withrespecttowhichEisasheafonX.ButE(U)⊆S(U)foreveryopenU⊆X,therefore,foralls∈ℜ,s(U)isopeninS,andthusSB=EisopeninS,asdesired.Fortheconverse,seeMallios[[10],p.5].ItfollowsfromLemma1.1thatDefinition1.1andMallios’definitionofsubsheaf,seeMallios[[10],p.5],areequivalent.Definition1.3LetEbeanA-moduleonatopologicalspaceX,andFasub-A-moduleofE.Assumethat(E(U),σUV)isthe(complete)presheafofsectionsofE.BytheA-annihilatorsheaf(orsheafofA-annihilators,orjustA-annihilator)ofF,wemeanthesheafgeneratedbythepresheaf,givenbythecorrespondenceU7−→F(U)⊥,whereUisanopensubsetofXandF(U)⊥={t∈E∗(U):t(s)=0foralls∈E(U)},SymplecticReductionofSheavesofA-modules5alongwithrestrictionmaps(ρ⊥)UV:F(U)⊥−→F(V)⊥suchthat(ρ⊥)UV:=(σ∗)UV|F(U)⊥,withthe(σ∗)UV:E∗(U)−→E∗(V)beingtherestrictionmapsforthedualpresheaf(E∗(U),(σ∗)UV).WedenotebyF⊥theannihilatorsheafofF.ItfollowsfromDefinition1.3,thattheannihilatorF⊥ofasub-A-moduleFofanA-moduleEisasubsheafofthedualA-moduleE∗.Lemma1.2LetEbeanA-moduleonatopologicalspaceX,andFasub-A-moduleofE.Then,thecorrespondenceU7−→F(U)⊥alongwithm