2B.HUISGEN-ZIMMERMANN
ofhowstructuralpropertiesoftheobjectsunderdiscussionaremirroredbygeometricpropertiesofthepertinentparametrizingvariety.Theleastonehopesforis someformof“continuous”bijectivedependenceoftheequivalenceclassesofobjects ontheirclassifyingparameters(whatonemeansby“continuity”ismadeprecise throughthenotionofa“family”),preferablysatisfyingauniversalproperty.
Theprototypicalexampleofahighlysuccessfulclassificationofthisilkgoes backtoRiemann:In1857,heclassifiedtheisomorphismclassesofnonsingular complexprojectivecurvesoffixedgenus g ≥ 2intermsofwhathecalled“moduli”. AccordingtoRiemann,theisomorphismclassofanonsingularcurveofgenus g “haengtvon3g 3stetigveraenderlichenGroessenab,welchedieModulndieser Klassegenanntwerdensollen”.Ittookaboutacenturyfortheterm“moduli”to begivenaprecisemeaning,namely,aselementsofafineorcoarsemodulispace. SuchmodulispaceswereaxiomaticallyintroducedbyMumfordinthe1960s.At thebeginningofSection2,wewillroughlysketchtheunderlyingidea,adaptedto therepresentationtheoryofafinitedimensionalalgebraΛ,inordertomotivatea firstroundoftool-building.PrecisedefinitionsofmodulispacesaregiveninSection 4,whichfollowsthetransparentexpositionofNewstead[26].
Todelineateourgoals:Ourpresentinterestisinanimprovedunderstanding ofselectclassesofrepresentationsofabasicfinitedimensionalalgebraΛoveran algebraicallyclosedfield K bywayofmoduli.Wemayassumewithoutlossof generalitythatΛ= KQ/I ,where Q isaquiverwithvertexset Q0 = {e1 ,...,en }, and I isanadmissibleidealinthepathalgebra KQ.Ourprimaryobjectivehereis toreviewandcomparepresentlyavailabletechniquesandresultsthatharnessfine orcoarsemodulispacesfortheclassificationoffinitedimensionalrepresentations ofsuchanalgebraΛ.Adiscussionfromaunifiedperspectiveshould,inparticular, makethesubjectmoreaccessibletonewcomerstothearea;tomeetthispurpose, wewillincludesomeelementaryobservations,tobeskippedbyexperts.Asecondaryaimistopromoteaseriesofproblemswhichappeartobe“nextinline” towardsbroadeningtheimpactofthegeneralmodulimachineryonrepresentation theory.Someoftheseproblems–thoseaimingat“genericclassification”–extend aninvestigationthatwasinitiatedbyKacintheearly1980s(see[21,22])and pickedupbySchofield[33],Derksen-Weyman[12],Reineke[29]andothersinthe case I =0,bySchr¨oer[34],Crawley-Boevey-Schr¨oer[9],Babson-Thomasandthe author[3]inthegeneralsituation.
Hereisanoutlineofthearticle:InSection1,werevisittwostartingpoints forageometricclassificationoffinitedimensionalΛ-modules.Wefirstreviewthe classicalaffinevariety Modd (Λ)parametrizingthe(left)Λ-moduleswithfixeddimensionvector d (werefertoitasParametrizationA);nextweturntotheprojectivevarietyGRASSd (Λ)parametrizingthesameisomorphismclassesofmodules (ParametrizationB).Ineithercase,theparametrizingvarietycomesequippedwith analgebraicgroupaction,theorbitsofwhichareinbijectivecorrespondencewith theisomorphismclassesofmodulesunderconsideration.However,thewidelydifferentstructuresofthesevarietiesandtheirrespectiveactinggroupsgivethetwo pointsofdeparturedistincttypesofpotential,onsomeoccasionsyieldingalternate roadstothesameconclusion.Inbothsettings,oneobservesthatthegroupaction canhardlyeverbefactoredoutoftheoriginalparametrizingvarietyinageometricallymeaningfulmanner,whichpromptsustoincludeabriefgeneraldiscussion ofquotientsofalgebraicvarietiesmoduloactionsofalgebraicgroupsinSection3.
ThissectionoverlapswithexpositoryarticlesbyBongartz[5]andGeiss[13].The modestoverlapisrequiredforaconsistentdevelopmentofthesubsequentideas.
ThenwereturntoRiemann’sclassificationprogramanddiscuss/exemplifythe conceptsofafine/coarsemodulispaceintherepresentation-theoreticcontext(Section4).Todate,therearetwodifferentstrategiestogetmileageoutoftheconceptualframework.Inlightofthefactthatfineorcoarsemodulispacesforthe fullcollectionofisomorphismclassesofΛ-moduleswithagivendimensionvector hardlyeverexist,eachmethodproposesamodeofslicingΛ-modsoastoextract portionsonwhichtheconceptualvehicleofmodulispacesacquirestraction.The strategiesofslicingtakeadvantageoftheparticularsoftheinitialparametrizing setups,andhence,ineachcase,specificmethodologyiscalledfortomatchthe target.SincethereexisttwopriorsurveyarticlesdealingwithApproachA,by Geiss[13]andReineke[29],wewillgivemoreroomtoApproachBinthepresent overview.
OneofthemethodsmimicksastrategyMumfordusedintheclassificationof vectorbundlesoncertainprojectivevarieties.ItwasadaptedtotherepresentationtheoreticsettingbyKingin[23](seeSection5).Startingwithanadditivefunction θ : K0 (Λ)= Zn → Z,KingfocusesontheΛ-moduleswithdimensionvector d whichare θ -stable ,resp. θ -semistable ;interpretsthesestabilityconditionsinterms ofthebehaviorof θ onsubmodulelattices;andshowshowtoapplytechniques fromgeometricinvarianttheorytosecureafine,resp.coarse,modulispacefor θ(semi)stablemodules.Theresultingstabilityclassesarenotapriorirepresentationtheoreticallydistinguished,whenceafundamentalchallengeliesin“good”choices ofthefunction θ andasolidgraspofthecorresponding θ -(semi)stablemodules.
Asthismethodisbasedontheaffineparametrizingvariety Modd (Λ),crucially leaningonthefeaturesofthissetup,itwillbelabeledApproachA.Sofar,itsmain applicationsaretothehereditarycaseΛ= KQ,eventhough,inprinciple,King extendedthemethodtoincludearbitrarypathalgebrasmodulorelations.
Bycontrast,thesecondapproach(labeledApproachBanddescribedinSections6-8)startswithclasses C ofmodulesoverΛ= KQ/I whicharecutoutby purelyrepresentation-theoreticfeatures,andaimsatunderstandingtheseclasses throughananalysisofthesubvarietiesofGRASSd (Λ)thatencodethem.Thename ofthegameistoexploitprojectivityoftheparametrizingvarietyandthetypically largeunipotentradicaloftheactinggrouptofindusefulnecessaryandsufficient conditionsfortheexistenceofageometricquotientofthesubvarietyencoding C , andtosubsequentlyestablishsuchaquotientasamodulispacethatclassifies therepresentationsin C uptoisomorphism.Simultaneously,oneseekstheoretical and/oralgorithmicaccesstomodulispaceswheneverexistenceisguaranteed.
Indescribingeithermethod,westatesampletheoremswitnessingviability andillustratethemwithexamples.Eachofthetwooutlineswillconcludewitha discussionofprosandconsoftheexhibitedapproach.
Acknowledgements
IwishtothanktheorganizersoftheAuslanderConferenceandDistinguished LectureSeries(WoodsHole,April2012),K.Igusa,A.Martsinkovsky,and G.Todorov,andtheorganizersF.BleherandC.ChindrisoftheConferenceon GeometricMethodsinRepresentationTheory(UniversityofMissouri-Columbia,
4B.HUISGEN-ZIMMERMANN
November2012)forhavingprovidedmewithcongenialvenuesfortheexpository lecturesthatgaverisetothesenotes.
Furtherconventions. Throughout,Λwillbeabasicfinitedimensionalalgebra overanalgebraicallyclosedfield K ,and J willdenotetheJacobsonradicalofΛ. WethusdonotloseanygeneralityinassumingthatΛ= KQ/I foraquiver Q and anadmissibleideal I ofthepathalgebra KQ.Thevertexset Q0 = {e1 ,...,en } of Q willbeidentifiedwithafullsetofprimitiveidempotentsofΛ.Moreover,we let Si =Λei /Jei bethecorrespondingrepresentativesofthesimplemodules.The absolutevalue ofadimensionvector d =(d1 ,...,dn )is |d| = i di .
Wewillsystematicallyidentifyisomorphicsemisimplemodules.The top ofa (left)Λ-module M istop(M )= M/JM .The radicallayering of M isthesequence ofsemisimplemodules S(M )= J l M/J l+1 M 0≤l≤L ,where L +1istheLoewy lengthofΛ.Inparticular,thezero-thentryof S(M )equalsthetopof M . Forourpresentpurpose,itsufficestoconsiderclassicalquasi-projectivevarieties.Bya subvariety ofasuchavarietywewillmeanalocallyclosedsubset.
2.Affineandprojectiveparametrizationsofthe Λ-modulesof dimensionvectord
Supposethat C isaclassofobjectsinsomealgebro-geometriccategory,andlet ∼ beanequivalencerelationon C .
Riemann’sclassificationphilosophyinlooseterms. (I) Identifydiscreteinvariantsoftheobjectsin C ,inordertosubdivide C intofinitelymany(or countablymany)subclasses Ci ,theobjectsofwhicharesufficientlyakintoeach othertoallowforanormalformcharacterizingthemuptothechosenequivalence. (II) Foreachindex i,findanalgebraicvariety Vi ,togetherwithabijection Vi ←→{equivalenceclassesin Ci }, whichyieldsa continuousparametrization oftheequivalenceclassesofobjectsin Ci .(Theideaof“continuity”willbeclarifiedinSection4.Typically,sucha parametrizationwill–apriorioraposteriori–beaclassificationofnormalforms.) Onceaparametrizationthatmeetstheseciriteriaisavailable,explorepotential universalproperties .Moreover,investigatetheinterplaybetweenthegeometryof Vi ononehandandstructuralpropertiesofthemodulesin Ci ontheother.
Wewillfocusonthesituationwhere C isaclassofrepresentationsofΛ.In thissituation,themostobviousequivalencerelationisisomorphism,orgradedisomorphismifapplicable.Riemann’sphilosophythensuggeststhefollowingasafirst step:Namely,totentativelyparametrizetheisomorphismclassesofmoduleswith fixeddimensionvectorin some plausiblewaybyavariety.Wewillreviewtwosuch parametrizations,bothhighlyredundantinthesensethatlargesubvarietiesmap tosingleisomorphismclassesingeneral.Ineachcase,theconsideredparametrizing varietycarriesamorphicactionbyanalgebraicgroup G whoseorbitscapturethe redundancy;inotherwords,the G-orbitsarepreciselythesetsofpointsindexing objectsfromthesameisomorphismclassofmodules.Sinceeachofthesesettings willhaveadvantagesanddownsidescomparedwiththeother,itwillbedesirable toshiftdatabackandforthbetweenthem.Suchatransferofinformationbetween ScenariosAandBwillturnouttobeoptimallysmooth.Wewilldeferadetailed
discussionofthispointtotheendofSection3,however,sincewewishtospecificallyaddressthepassageofinformationconcerningquotientsbytherespective groupactions.
(A)Theclassicalaffineparametrizationoftheisomorphismclassesof Λ-moduleswithdimensionvectord
Thissetupiswell-knownandmuch-used.Toourknowledge,thefirstprominent applicationtotherepresentationtheoryoffinitedimensionalalgebraswasinthe proofofGabriel’sTheorempinningdownthepathalgebrasoffiniterepresentation type.
Theaffineparametervarietyanditsgroupaction. (1)Let Q1 be thesetofarrowsof Q,andlet
Mod d (Λ)= {x =(xα )α∈Q1 | the xα ∈ Mdend(α) ×dstart(α) (K )satisfytherelationsin I }. Here Mr ×s (K )denotesthespaceof r × s matricesover K
(2)Thegroupaction:SetGL(d)= 1≤i≤n GLdi (K ),andconsiderthefollowingactionofGL(d)on Modd (Λ):For g =(g1 ,...,gn ) ∈ GL(d)and x =(xα ) ∈ Modd (Λ),define g.x = gend(α) xα g 1 start(α) α∈Q1
Evidently, Modd (Λ)isaZariski-closedsubsetoftheaffine K -spaceofdimension α∈Q1 dstart(α) dend(α) ,thepointsofwhichdetermineΛ-modulestructureson thevectorspace K |d| = 1≤i≤n K di via α 1≤i≤n vi = xα vstart(α) ∈ K dend(α) foranyarrow α and vi ∈ K di .Clearly,thefibersoftheresultingmapfrom Modd (Λ)tothesetofisomorphismclassesofmoduleswithdimensionvector d arepreciselytheorbitsofthedescribedGL(d)-actionon Modd (Λ).Thus,weobtainaone-to-onecorrespondencebetweentheGL(d)-orbitsof Modd (Λ)onone handandtheisomorphismclassesofΛ-moduleswithdimensionvector d onthe other.Moreover,weobservethattheconsideredgroupactionismorphic,meaning thatthepertinentmapGL(d) × Modd (Λ) → Modd (Λ)isamorphismofvarieties. (B)Theprojectiveparametrizationofthesamesetofisomorphism classes
Analternateparametrizingvarietyforthesameisomorphismclassesofmodules wasintroducedbyBongartzandtheauthorin[6,7],togetherwithamorphic algebraicgroupactionwhoseorbits,inturn,areinone-to-onecorrespondencewith theseisomorphismclasses.
Theprojectiveparametervarietyanditsgroupaction. (1)Let P = 1≤i≤n (Λei )di (thesmallestprojectiveΛ-moduleadmittingarbitrarymoduleswith dimensionvector d asquotientsmodulosuitablesubmodules),anddefine
GRASSd (Λ)= {C ∈ Gr(d , P) | Λ C ⊆ Λ P withdim P/C = d}, where d =dim P −|d| andGr(d , P)istheGrassmannvarietyofall d -dimensional subspacesofthe K -vectorspace P.
(2)Thegroupaction:LetAutΛ (P)betheautomorphismgroupof P,and considerthecanonicalactiononGRASSd (Λ)givenby f.C = f (C ).
Thistime,wearelookingataZariski-closedsubsetoftheclassicalGrassmann varietyGr(d , P);inparticular,GRASSd (Λ)isaprojectivevariety.Again,wehave anobviousmapfromthevarietyGRASSd (Λ)tothesetofisomorphismclassesof
6B.HUISGEN-ZIMMERMANN
Λ-moduleswithdimensionvector d,namely ρ : C → [P/C ].Bythechoiceof P,everymodule M withdimensionvector d isoftheform M ∼ = P/C forsome point C ∈ GRASSd (Λ).Moreover,thefibersof ρ againcoincidewiththeorbitsof thegroupaction;indeed,twomodules P/C and P/D areisomorphiciff C and D belongtothesameorbit,thistimetheAutΛ (P)-orbitinGRASSd (Λ).Moreover, thegroupactionisinturnmorphic.
Recallthatthe unipotentradical ofalinearalgebraicgroupistheuniquelargest normalconnectedunipotentsubgroup.Thegroupiscalled reductive ifitsunipotent radicalistrivial.IncontrasttothereductivegroupGL(d)actingintheaffine case,thelineargroupAutΛ (P)hasalargeunipotentradicalinmostinteresting cases.Namely,theunipotentradical,AutΛ (P)u ,equalsthesubgroup {id+h | h ∈ HomΛ (P,J P)}.WeobservemoreoverthatAutΛ (P) ∼ = GL(P/J P) AutΛ (P)u .
3.Quotientvarietiesonthegeometricmarket—generalitiesand representation-theoreticparticulars
InSection2,wehave,inbothcases,arrivedatascenariothatisfrequently encounteredinconnectionwithclassificationproblems:Onestartswithacollectionofalgebro-geometricobjectswhichonewishestoclassifyuptoanequivalence relation–inourcasetheobjectsarerepresentationswithfixeddimensionvector andthepreferredequivalencerelationisisomorphism.Ontheroad,onearrives atasetupthatplacestheequivalenceclassesofobjectsintoanaturalone-to-one correspondencewiththeorbitsofanalgebraicgroupactiononaparametrizing variety.Suchascenario,ofcourse,triggerstheimpulsetofactorthegroupaction outoftheconsideredvariety.Tosayitindifferentwords:Theideaistoreducethe orbitsofthegroupactiontopointsinanewvarietywhichisrelatedtotheoriginal onebyauniversalpropertywhichtakesthegeometryintoaccount.
Thecruxliesinthefactthatthetopologicalquotientof Modd (Λ)modulo GL(d),(resp.ofGRASSd (Λ)moduloAutΛ (P)),relativetotheZariskitopology, hardlyevercarriesavarietystructure,atleastnotonethatmeritsthelabel“quotientvariety”.Tocopewiththisdifficultyinabroadspectrumofsituations,algebraicgeometersintroducedquotientsofvariouslevelsofstringency.Notsurprisingly,theunderlyingguidelineisthis:TheclosertheZariskitopologyofa“quotient variety”comestothatofthetopologicalquotient,thebetter.Wewilltouchthis subjectonlybrieflyandreferthereadertothesurveybyBongartz[5]andthe expositionbyPopovandVinberg[27].
Categoricalandgeometricquotients. Let X beanalgebraicvariety, andlet G bealinearalgebraicgroupactingmorphicallyon X .
(1) A categorical (oralgebraic )quotientof X by G isamorphism ψ : X → Z of varietiessuchthat ψ isconstantontheorbitsof G,andeverymorphism ψ : X → Y whichisconstantonthe G-orbitsfactorsuniquelythrough ψ .Write Z = X//G in casesuchaquotientexists.
(2) Acategoricalquotient ψ : X → X//G iscalledan orbitspace fortheactionin casethefibersof ψ coincidewiththeorbitsof G in X
(3) A geometricquotientof X by G isanopensurjectivemorphism ψ : X → Z , whosefibersequaltheorbitsof G in X ,suchthat,moreover,foreveryopensubset U of Z ,thecomorphism ψ ◦ inducesanalgebraisomorphismfromthering O (U ) ofregularfunctionson U tothering O ψ 1 (U ) G of G-invariantregularfunctions on ψ 1 (U ).
Itiseasytoseethatageometricquotientisanorbitspace,andhence,in particular,isacategoricalquotient.Thisguaranteesuniquenessincaseofexistence. Wegivetwoelementaryexamplesinordertobuildintuition:For n ≥ 2,the conjugationactionofGLn (K )onthevarietyof n × n matriceshasacategorical quotient,which,however,failstobeanorbitspace.Givenalinearalgebraicgroup G andanyclosedsubgroup H ,therighttranslationactionof H on G hasageometric quotient;inparticular,thepointsofthisquotientmaybeidentifiedwiththeleft cosetsof H in G
OnereadilyverifiesthattheZariskitopologyonageometricquotientcoincides withthequotienttopology.So,inlightoftheaboveguideline,existenceofa geometricquotientisthebestpossibleoutcomewheneverwelookforaquotientof asubvarietyof Modd (Λ)modulo GL(d)orofasubvarietyofGRASSd (Λ)modulo AutΛ (P).Ontheotherhand,anorbitspaceforasuitableaction-stablesubvariety istheleastwerequireinordertoimplementRiemann’sidea.Evidently, • theexistenceofanorbitspaceimpliesclosednessofallorbits, whichplacesastrongnecessaryconditiononpotentialscenariosofsuccess.
LetustakealookatourtwoparametrizationsoftheΛ-moduleswithdimension vector d.HereiswhatGeometricInvariantTheorygrantsusintheaffinesetting: Namely,everymorphicactionofareductivelinearalgebraicgroup G onanaffine variety X hasacategoricalquotient(see,e.g.,[26,Chapter3]).Thepivotalasset ofthissetupliesinthefactthatthering K [X ]G of G-invariantregularfunctions (i.e.,ofregularfunctions f : X → K suchthatf(gx)=f(x),forall g ∈ G and x ∈ X )isfinitelygeneratedover K .Wewillrepeatedlyrefertothisresult.
Theorem 3.1(Haboush,Hilbert,Mumford,Nagata,Weyl,etal.) Suppose that X isanaffinevarietywithcoordinatering K [X ]and G areductivegroupacting morphicallyon X .Thenthecanonicalmap
ψ :Spec K [X ] → Spec K [X ]G isacategoricalquotient X//G.Moreover,thepointsof X//G areinone-to-one correspondencewiththeclosed G-orbitsof X .
Inparticular,Theorem3.1guaranteesacategoricalquotient Modd (Λ)// GL(d). Atfirstglance,thisconclusionmaylookbetterthanitis,sincetheonlyclosedorbitin Modd (Λ)isthatofthesemisimplemoduleofdimensionvector d.Indeed, givenanymodule M andanysubmodule N ⊆ M ,theGL(d)-orbitcorresponding tothedirectsum N ⊕ M/N in Mod d (Λ)iscontainedintheclosureoftheGL(d)orbitcorrespondingto M .So,bythetheorem, Modd (Λ)// GL(d)isasingleton. Expresseddifferently:ThecatchliesinthefactthattheringofGL(d)-invariant regularfunctionson Modd (Λ)equalsthefield K ofconstants,andhencehasonly asingleprimeideal.TheresponseofGeometricInvariantTheorytosuchasparsity ofclosedorbitsistoparedowntheparametrizingvarietyand,intandem,torelax theinvariancerequirementsplacedontheregularfunctionsthatareexpectedto separatetheorbits,soastoobtainalargeralgebraoffunctionsthatmaybeused toconstructausefulquotient.
Inordertobenefitfromthefactthatdifferentarsenalsoftechniquesapplyto ourtwoparametrizations,wefirstexplainhowtomovebackandforthbetween them.
Proposition 3.2(InformationtransferbetweenParametrizationsAandB). (see[7],PropositionC)
8B.HUISGEN-ZIMMERMANN
Considertheone-to-onecorrespondencebetweentheorbitsof GRASSd (Λ) on onehandand Modd (Λ) ontheother,whichassignstoanyorbit AutΛ (P).C ⊆ GRASSd (Λ) theorbit GL(d).x ⊆ Mod d (Λ) representingthesame Λ-moduleupto isomorphism.Thiscorrespondenceextendstoaninclusion-preservingbijection
Φ: {AutΛ (P)-stablesubsetsof GRASSd (Λ)}→{GL(d)-stablesubsetsof Modd (Λ)} whichpreservesandreflectsopenness,closures,connectedness,irreducibility,and typesofsingularities.
Moreover,let X bea GL(d)-stablesubvarietyof Modd (Λ),withcorresponding AutΛ (P)-stablesubvariety Φ(X ) of GRASSd (Λ).Then X hasanalgebraicquotient (resp.,orbitspace/geometricquotient ) by GL(d) ifandonlyif Φ(X ) hasan algebraicquotient (resp.,orbitspace/geometricquotient ) by AutΛ (P).Incaseof existence,thequotientsareisomorphicandhavethesameseparationproperties relativetoactionstablesubvarietiesof X and Φ(X ),respectively.
Thetransferresultthusallowsustosymmetrizetheunhelpfulconclusionwe drewfromTheorem3.1.TheprojectivevarietyGRASSd (Λ)hasacategorical quotientbyAutΛ (P),andthisquotientisisomorphicto Modd (Λ)// GL(d),asingleton.
Whereshouldwegofromhere?Weareontheoutlookforinterestingsubvarietiesof Modd (Λ),resp.GRASSd (Λ)whicharestableunderthepertinentgroup actionsandhavethepropertythatallorbitsarerelativelyclosed.Proposition3.2 tellsusthatwemayinterchangeablyusethetwosettings,AandB,inthisquest.
InSections5and6,7wewillreviewandillustratetwodifferentmethodsto identifysubvarietiesofthisilk.Butfirstwewillfleshoutthevagueclassification philosophypresentedinSection1.
4.RenderingRiemann’sclassificationphilosophymoreconcrete
ThecurrentunderstandingofRiemann’s“moduli”viewsthemas“elementsof afineorcoarsemodulispace”.Thetwonotionsofmodulispace,onesignificantly strongerthantheother,wereintroducedandputtousebyMumfordinthe1960’s (seethestandardGITtext[25]).WewillfollowNewstead’sexposition[26].
Bothtypesofmodulispacesbuildontheconceptofa familyofobjects parametrizedbyanalgebraicvariety .Theupcomingdefinitionclarifiestheidea ofa continuousparametrization ,asopposedtoarandomindexingofobjectsbythe pointsofavariety.The(only)plausibledefinitionofa family intherepresentationtheoreticcontextwasputforthbyKingin[23].
Definition:Familiesofrepresentations. Set d = |d|,andlet ∼ bean equivalencerelationontheclassof d-dimensionalΛ-modules.
(1) A familyof d-dimensional Λ-modulesparametrizedbyavariety X isapair (Δ,δ ),whereΔisavectorbundleofrank d over X ,and δ a K -algebrahomomorphismΛ → End(Δ).
(2) Extending ∼ tofamilies:Twosuchfamilies(Δ1 ,δ1 )and(Δ2 ,δ2 )parametrized bythesamevariety X willbecalled similar incase,foreach x ∈ X ,thefibersof Δ1 andΔ2 over x are ∼ equivalentasΛ-modulesunderthestructuresinducedby δ1 and δ2 ,respectively.Wewrite(Δ1 ,δ1 ) ∼ (Δ2 ,δ2 )inthissituation.
(3) Inducedfamilies:Givenafamily(Δ,δ )parametrizedby X asabove,together withamorphism τ : Y → X ofvarieties,thepull-backbundleofΔalong τ isa
familyofΛ-modulesparametrizedby Y (seetheremarkbelow).Itiscalledthe familyinducedfrom (Δ,δ ) by τ andisdenotedby τ ∗ (Δ,δ ).
Here,thevectorbundlesconsideredarewhatHartshorne[14]calls geometric vectorbundles :ThismeansthatΔcarriesthestructureofavariety,andallof theoccurringmaps–thebundleprojection,thelocalsectionsresponsibleforlocal triviality,andthecompatibilitymapsforthetrivializedpatches–aremorphisms ofvarieties.Therequirementthat δ (λ),for λ ∈ Λ,beanendomorphismofΔjust meansthat δ (λ):Δ → Δisamorphismofvarietiesthatrespectsthefibersof thebundleundertheprojectionmap;sowefindthateachfiberisindeedendowed withaΛ-modulestructure.Sinceeach δ (λ)isaglobalmorphismfromΔtoΔ,this meansthattheΛ-modulestructuresontheindividualfibersarecompatibleina stronggeometricsense,thusjustifyingtheinterpretationasacontinuousarrayof modules.
Remarkconcerningthepull-backconstruction :Usingthecorrespondingtrivializations,wereadilycheckthat,for y ∈ Y ,thepullbackdiagram
π X
(Δ) π
Y
permitsustopullbacktheΛ-modulestructure(stemmingfrom δ )onthefibre π 1 (τ (y ))ofΔtoaΛ-modulestructureonthefiber(π ∗ ) 1 (y )of τ ∗ (Δ);oneverifies thatthesemodulestructuresontheindividualfibersof τ ∗ (Δ)arecompatible,so astoyielda K -algebrahomomorphism δ ∗ :Λ → End τ ∗ (Δ) thatinducesthem.
Set τ ∗ (Δ,δ )=(τ ∗ (Δ),δ ∗ ).
Itiseasilyverifiedthatthedefinitionsof“family”and“inducedfamily”satisfy thefunctorialconditionsspelledoutasprerequisitesforawell-defined“moduli problem”in([26,Conditions1.4,p.19]).Namely: • Theequivalencerelationon familiesboilsdowntotheinitialequivalencerelation ∼ onthetargetclass C ,ifone identifiesafamilyparametrizedbyasinglepointwiththecorrespondingmodule; infact,theequivalencerelationweintroducedunder(2)aboveisthecoarsestwith thisproperty.(Itisnotthemostnaturaloption,buttheeasiesttoworkwithinour context.) • If τ : Y → X and σ : Z → Y aremorphismsofvarietiesand(Δ,δ )isa familyofmodulesover X ,then(τ ◦ σ )∗ (Δ,δ )= σ ∗ τ ∗ (Δ,δ ) ;moreover(idX )∗ isthe identityonfamiliesparametrizedby X • Similarityoffamiliesiscompatiblewith thepullbackoperation,thatis:If(Δ1 ,δ1 )and(Δ2 ,δ2 )arefamiliesparametrized by X with(Δ1 ,δ1 ) ∼ (Δ2 ,δ2 )and τ isasabove,then τ ∗ (Δ1 ,δ1 ) ∼ τ ∗ (Δ2 ,δ2 ).
Example 4.1. LetΛbetheKroneckeralgebra,i.e.,Λ= KQ,where Q isthe quiver1 α1 α2 2,andtake d =(1, 1).Thenon-semisimple2-dimensional Λ-modulesformafamilyindexedbytheprojectivelineover K .Itcaninformallybe presentedas M[c1 :c2 ] [c1 :c2 ]∈P1 with M[c1 :c2 ] =Λe1 /Λ(c1 α1 c2 α2 ).Foraformal renderinginthesenseoftheabovedefinition,considerthetwostandardaffine patches, Uj = {[c1 : c2 ] ∈ P1 | cj =0},andletΔj = Uj × K 2 for j =1, 2 bethecorrespondingtrivialbundles.TomakeΔ1 intoafamilyofΛ-modules,let δ1 :Λ → End(Δ1 )besuchthat δ1 (α1 )actsonthefibreabove[c1 : c2 ]viathematrix
10B.HUISGEN-ZIMMERMANN
00 c2 /c1 0 and δ1 (α2 )actsvia 00 10 .Define δ2 :Λ → End(Δ2 )symmetrically, andgluethetwotrivialbundlestoabundleΔover P1 viathemorphism
Observethatthe δj arecompatiblewiththegluing,thatis,theyyielda K -algebra homomorphism δ :Λ → End(Δ),andthusafamily(Δ,δ ).
Definitionoffineandcoarsemodulispaces. Wefixadimensionvector d,set d = |d|,andlet C beaclassofΛ-moduleswithdimensionvector d.Denoting by C (Modd (Λ)),resp. C (GRASSd (Λ)),theunionofallorbitsin Modd (Λ),resp. inGRASSd (Λ),whichcorrespondtotheisomorphismclassesin C ,weassumethat C (Modd (Λ))isasubvarietyof Modd (Λ)(or,equivalently,that C (GRASSd (Λ))is asubvarietyofGRASSd (Λ)).Again,welet ∼ beanequivalencerelationon C and extendtherelation ∼ tofamiliesasspelledoutintheprecedingdefinition. (1)A finemodulispace classifying C upto ∼ isavariety X withthepropertythat thereexistsafamily(Δ,δ )ofmodulesfrom C whichisparametrizedby X and hasthefollowinguniversalproperty:Whenever(Γ,γ )isafamilyofmodulesfrom C indexedbyavariety Y ,thereexistsauniquemorphism τ : Y → X suchthat (Γ,γ ) ∼ τ ∗ (Δ,δ ).
Inthissituation,wecall(Δ,δ )a universalfamily forourclassificationproblem. (Clearly,suchauniversalfamilyisuniqueupto ∼ wheneveritexists.)
(2)Specializingtothecasewhere ∼ is“isomorphism”(forthemoment),wesaythat avariety X isa coarsemodulispace fortheclassificationof C uptoisomorphismin case X isanorbitspacefor C (Modd (Λ))undertheGL(d)-action(or,equivalently, for C (GRASSd (Λ))undertheAutΛ (P)-action).
InSection6,wewillalsolookformodulispacesclassifyingclasses C ofgraded modulesuptogradedisomorphism.Bythiswewillmeananorbitspaceof C (GRASSd (Λ))relativetotheactionofthegroupofgradedautomorphismsin AutΛ (P).
Comments 4.2. Ratherthangivingtheoriginalfunctorialdefinitionsof fine/coarsemodulispaces,wehaveintroducedtheseconceptsviaequivalentcharacterizationsofhigherintuitiveappeal.
(i) Thestandardfunctorialdefinitionsofafine/coarsemodulispaceareasfollows (cf.[26]):
Considerthecontravariantfunctor
F :Var=categoryofvarietiesover K −→ categoryofsets, Y →{equivalenceclassesoffamiliesofobjectsfrom C parametrizedby Y }. Thisfunctorisrepresentableintheform F ∼ = HomVar ( ,X )preciselywhen X isa finemodulispaceforourproblem.
Thatavariety X beacoarsemodulispaceforourproblemamountstothe followingcondition:ThereexistsanaturaltransformationΦ: F → HomVar ( ,X ) suchthatanynaturaltransformation F → HomVar ( ,Y )forsomevariety Y factors uniquelythroughΦ.
(ii) Ourdefinitionofacoarsemodulispace X isequivalenttoMumford’sinthe situationsonwhichwearefocusing,butnotingeneral.Wearechippinginthe
factthatthemodulesfrom C belongtoafamily(Δ,δ )thatenjoysthe localuniversalproperty inthesenseof[26,Proposition2.13];indeed,weonlyneedto restrictthetautologicalbundleon Mod d (Λ)to C (Modd (Λ)).If X denotesthe parametrizingvarietyofΔ,thisconditionpostulatesthefollowing:Foranyfamily (Γ,γ )ofmodulesfrom C ,parametrizedbyavariety Y say,andany y ∈ Y ,there isaneighborhood N (y )of y suchthattherestrictedfamilyΓ|N (y ) isinducedfrom Δbyamorphism N (y ) → X .Notethatlocaluniversalitycarriesnouniqueness requirement.
InclassifyinggradedrepresentationsofagradedalgebraΛuptogradedisomorphism,analogousconsiderationsensurethatourconceptofacoarsemoduli spacecoincideswiththeoriginalone.Inthissituation,gradedisomorphismtakes ontheroleoftheequivalencerelation ∼ . (iii) Clearly,anyfinemodulispacefor C isacoarsemodulispace.Inparticular,by Proposition3.2,eithertypeofmodulispaceforourproblemisanorbitspacebased onourchoiceofparametrizingvariety(thesubvariety C (Modd (Λ))of Mod d (Λ)or C (GRASSd (Λ))ofGRASSd (Λ)correspondingto C )modulotheappropriategroup action.Fromthedefinitionofanorbitspace,wethusgleanthatclassificationbya coarsemodulispace X alsoyieldsaone-to-onecorrespondencebetweenthepoints of X andtheisomorphismclassesofmodulesfrom C .Concerningfineclassification, wemoreoverobserve:If X isafinemodulispacefor C ,theneachisomorphismclass from C isrepresentedbypreciselyonefibreofthecorrespondinguniversalfamily parametrizedby X .
Inessence,theroleofafineorcoarsemodulispacethusistonotonlyrecord parameterspinningdownnormalformsfortheobjectsintheclass C underdiscussion,buttodosoinanoptimallyinteractiveformat.Consequently,underthe presentangle,the“effectiveness”ofanormalformismeasuredbythelevelof universalityitcarries.Letussubjectsome familiarinstancestothisqualitytest, recruitingschoolbookknowledgefromtherepresentationtheoryofthepolynomial algebra K [t].
Firstexamples4.3. (1) Itisnotdifficulttocheckthatthefamilypresented in4.1isuniversalfortheclass C ofnon-semisimplemoduleswithdimensionvector (1, 1)overtheKroneckeralgebra.Thisfactwillbere-encounteredasaspecialcase ofCorollaries5.2and6.7below.
(2) (cf.[26,Chapter2])Let d beaninteger ≥ 2.Supposethat V isa ddimensional K -space,and C aclassofendomorphismsof V .Inotherwords,weare consideringaclassof d-dimensionalmodulesover K [t].Rephrasingtheabovedefinitionofafamilyofmodules,weobtain:Afamilyfromthisclass,parametrizedby avariety X say,isavectorbundleofrank d,togetherwithabundleendomorphism δ (t)thatinducesendomorphismsfromtheclass C onthefibers.Theequivalence relationtobeconsideredissimilarityintheusualsenseoflinearalgebra.
Animmediatequestionarises:Doesthefullclass C ofendomorphismsof V haveacoarseorfinemodulispace?
Giventhatourbasefield K isalgebraicallyclosed,wehaveJordannormal formswhichareinone-to-onecorrespondencewiththesimilarityclasses.Sothe firstquestionbecomes:Cantheinvariantsthatpindownthenormalformsbe assembledtoanalgebraicvariety?ThefactthattheblocksizesinJNFsarepositive integers–thatis,arediscreteinvariants–while Md×d (K )isanirreduciblevariety, doesnotbodewell.Indeed,onereadilyfindsthatallconjugacyclassesinEndK (V )
12B.HUISGEN-ZIMMERMANN
encodingnon-diagonalizableendomorphisms(=non-semismple K [t]-modules)fail tobeclosed;indeed,theZariski-closureofanysuchclasscontainsthediagonalizable endomorphismwiththesameeigenvaluesandmultiplicities.Consequently,thefull collectionofendomorphismsof V doesnotevenhaveacoarsemodulispace.
Ifonerestrictstotheclass C ofdiagonalizableendomorphismsof V ,thereisa coarsemodulispace;thisorbitspace C // GL(V )isisomorphicto K d andrecordsthe coefficientsofthecharacteristicpolynomial(disregardingtheleadingcoefficient). Butthereisnouniversalfamilyfortheproblem,sothecoarsemodulispacefailsto befineinthiscase.Foraproof,seee.g.[26,Corollary2.6.1].Ontheotherhand, ifonefurtherspecializestothecyclicendomorphisms,i.e.,
C = {f ∈ EndK (V ) | f correspondstoacyclic K [t]-module}, onefinallydoesobtainafinemodulispace,namely K d ;auniversalfamilyforthe endomorphismsin C tracestheirrationalcanonicalforms.
(3) Riemann’scelebratedclassificationofsmoothprojectivecurvesoffixed genusover C isimplementedbya coarse modulispace,whichfailstobefine.This appearstobethesituationprevalentinsweepingclassificationresultsinalgebraic geometry.
Toreturntotherepresentationtheoryofa finite dimensionalalgebraΛ:Two strategieshaveemergedtodrawprofitfromtheconceptsofcoarseorfinemoduli spacesinthiscontext.InlinewiththeconclusionofSection3,eachofthemreduces thefocustosuitablesubclassesofthefullclassofmoduleswithfixeddimension vector.However,theyarebasedondifferentexpectations,andthedichotomyis paralleledbydifferenttechniques.Inthefollowing,wesketchbothofthesemethods andprovidesampleresults.
5.ApproachA:King’sadaptationofMumfordstability:Focusingon theobjectswhichare(semi-)stablerelativetoaweightfunction
Asthecaptionindicates,thisapproachbuildsontheaffineparametrization AofSection2.Giventhattherearealreadytwosurveyarticlesrecordingit,by Geiss[13]andReineke[29],wewillbecomparativelybriefandrefertotheexisting overviewsfortechnicaldetailandfurtherapplications.
Thestrategyunderdiscussionwasoriginallydevelopedforthepurposeofclassifyingcertaingeometricobjects(vectorbundlesovercertainprojectivevarieties,in particular)subjecttothefollowing,aprioriunfavorable,startingconditions:The equivalenceclassesoftheobjectsareinbijectivecorrespondencewiththeorbitsof areductivegroupactiononanaffineparametrizingvariety,butclosedorbitsare inshortsupply.ThisispreciselytheobstacleweencounteredattheendofSection 3relativeto Mod d (Λ)withitsGL(d)-action.Asaconsequence,theattemptto constructanorbitspacefrominvariantregularfunctionsontheconsideredvariety, onthemodelofTheorem3.1,isdoomed.Theideanowistousemoreregular functions,ratherthanjusttheclassicalinvariants(constantontheorbits),looseningtheirtietothegroupactiontoacontrollableextent:namely,touseallregular functionswhichare semi-invariantrelativetoacharacteroftheactinggroup .In tandem,oneparesdowntheoriginalvarietytoanaction-stablesubvarietywitha richersupplyof(relatively)closedorbits.Inanutshell:Oneallowsforalargersupplyofregularfunctionstopalpateacurtailedcollectionoforbits.Wefollowwith
asomewhatmoreconcreteoutline.FirstwesketchtheoriginalGIT-scenariowithoutincludingthegeneraldefinitionsof(semi)stabilityand S -equivalence.Thenwe specializetothevariety Modd (Λ)withitsGL(d)-actionandfillintheconceptual blanks,usingKing’sequivalentcharacterizationsofstabilityandsemistabilityfor thiscase.(Formoreprecisiononthegeneralcase,seealso[10].)
Thetypicalscenariotowhichthisstrategyappliesisasfollows:Namely,a finitedimensional K -vectorspace V (forexample, V = Modd (Λ),whereΛ= KQ isahereditaryalgebra),togetherwithareductivealgebraicgroup G whichoperates linearlyon V .Thenaregularfunction V → K iscalleda semi-invariant forthe actionincasethereexistsacharacter χ : G → K ∗ suchthat f (g.x)= χ(g )f (x)for all g ∈ G and x ∈ V .Next,onesinglesoutasubvariety V st of V whose G-orbitsare separatedby χ-semi-invariants;thepointsof V st arecalled“χ-stable”.Inaddition, oneconsidersalargersubvariety V sst whosepointsareseparatedbysemi-invariants moduloasomewhatcoarser(butoftenstilluseful)equivalencerelation,labeled Sequivalence(“S ”for“Seshadri”);thepointsof V sst aredubbed“χ-semistable”. Moreaccurately,the S -equivalenceclassesof χ-semistablepointsareseparatedby semi-invariantsoftheform χm forsome m ≥ 0.Themotivationforthissetup liesinthefollowingconsequences:Thecollection V sst ofsemistablepointsisan open(possiblyempty)subvarietyof V whichallowsforacategoricalquotientthat classifiestheorbitsin V sst upto S -equivalence.Thesubset V st ofstablepoints in V isinturnopenin V andfarbetterbehavedfromourpresentviewpoint:It frequentlypermitsevenageometricquotientmodulo G.Asistobeexpected,the quotientof V sst modulothe G-actionisconstructedfromsemi-invariants,namely asProjofthefollowinggradedringofsemi-invariantfunctions: m≥0 K [V ]χ m , where K [V ]χ m isthe K -subspaceofthecoordinatering K [V ]consistingofthe polynomialfunctionswhicharesemi-invariantrelativeto χm
Forthemodule-theoreticscenariothatresemblestheGIT-templatethemost closely,King’sadaptationoftheoutlinedstrategyhasbeenthemostsuccessful.It isthecaseofahereditaryalgebraΛ= KQ.Inthissituation, V = Modd (Λ)is afinite-dimensional K -vectorspace,andthereductivegroupactionistheGL(d)conjugationaction.Inparticular,Kingshowedthat χ-(semi)stability,foracharacter χ ofGL(d),translatesintoamanageableconditionforthemodulesrepresented bythe χ-(semi)stablepoints;seebelow.HethenproceededtocarryoverthetechniquetoarbitraryfinitedimensionalalgebrasΛ= KQ/I .
Asiswell-known,thecharactersofGL(d)areinnaturalcorrespondencewith themaps Q0 → Z (see[10],forinstance).Namely,everycharacter χ isoftheform χ(g )= i∈Q0 det(gi )θ (i) forasuitablemap θ : Q0 → Z;conversely,allmapsof thisilkareobviouslycharacters.Startingwiththeadditiveextension ZQ0 → Z of suchamap—calledbythesamename—onelets χθ bethecorrespondingcharacterofGL(d).AswasprovedbyKing[23,Theorem4.1],apointin Modd (Λ)is χθ -semistableintheGIT-senseifandonlyifthecorrespondingmodule M satisfies θ (dim M )=0and θ (dim M ) ≥ 0forallsubmodules M of M ;stabilityrequires thatdim M belongtothekernelof θ and θ (dim M ) > 0forallpropernonzerosubmodules M of M .Forconvenience,onealsoreferstoamodule M as θ -(semi)stable ifitisrepresentedbya θ -(semi)stablepointin Modd (Λ).(Note:Thefunction θ iscalleda weight byDerksen,a stability byReineke.)
14B.HUISGEN-ZIMMERMANN
Sincethesetsof θ -semistable,resp.of θ -stable,pointsin Modd (Λ)areopenin Modd (Λ),theclasses C of θ -semistable,resp. θ -stablemodulessatisfytheblanket hypothesisweimposedinourdefinitionsofafineorcoarsemodulispacefor C Thissetupyieldsthefollowing:
Theorem 5.1(see[23,Propositions4.3,5.2and5.3]). Let Λ= KQ/I .
• The θ -semistableobjectsin Λ-mod forma (full ) abeliansubcategoryof Λ-mod inwhichallobjectshaveJordan-H¨olderseries.Thesimpleobjectsinthiscategory arepreciselythe θ -stablemodules.Twosemistableobjectsare S -equivalentprecisely whentheyhavethesamestablecompositionfactors.
• The θ -semistablemodulesofafixeddimensionvector d =(d1 ,...,dn ) have acoarsemodulispace Msst Λ (d,θ ) whichclassifiesthemupto S -equivalence.This coarsemodulispaceisprojectiveandcontains,asanopensubvariety,acoarsemodulispace Mst Λ (d,θ ) classifyingthe θ -stablemodulesuptoisomorphism. Mst Λ (d,θ ) isafinemodulispaceprovidedthatgcd(d1 ,...,dn )=1
Evidently,conditionsguaranteeingthat Msst Λ (d,θ ) resp., Mst Λ (d,θ ) be nonemptyareamongthemostpressingpointstobeaddressed.Wepinpointone oftheluckysituations,whereaneffectiveweightfunction θ iseasytocomeby.It concernstheclassificationoflocalmodules(=moduleswithasimpletop),when thequiverofΛhasnoorientedcycles.
Corollary 5.2(Crawley-Boevey,oralcommunication). Suppose Λ= KQ/I , where Q isaquiverwithoutorientedcycles, T isasimple Λ-moduleand d adimensionvector.Thenthe (local ) moduleswithtop T anddimensionvector d havea finemodulispace, Mst Λ (d,θ ) forasuitableweightfunction θ ,whichclassifiesthem uptoisomorphism.
Proof. Supposethat T = S1 .Let θ : Q0 → Z bedefinedby θ (ej )=1 for j> 1and θ (e1 )= 2≤j ≤n dj .Thenclearlythemodulesaddressedbythe corollaryarethe θ -stableones,andKing’stheoremapplies.
ProsandconsofApproachA
Pros:
• Thistacticalwaysleadstoamodulispaceifoneextendsthenotiontothe emptyset.Indeed,foranychoiceofweightfunction,existenceofcoarse,resp.fine modulispaces,forthecorrespondingsemistable,resp.stablemodules,isguaranteed byGIT.
• Sincethismethodhasprovedveryeffectiveforvectorbundlesonnon-singular projectivecurves,alargearsenalofmethodsforanalyzingtheresultingmoduli spaceshasbeendeveloped.ThisincludescohomologygroupsandtheirBettinumbers,aswellascelldecompositions.(InterestingadaptationstotherepresentationtheoreticsettingoftechniquesdevelopedtowardstheunderstandingofvectorbundlemodulicanbefoundintheworkofReineke,e.g.,in[28].)
• Thespotlightplacedonsemi-invariantfunctionson Modd (Λ)bythismethod appearstohavereinforcedresearchintoringsofsemi-invariants,asubjectofgreat interestinitsownright.
Cons:
• Howtojudiciouslychooseweightfunctionsisatoughproblem.Inthiscontext,aweightfunction θ : Q0 → Z meritstheattribute“good”ifoneisableto securearichsupplyof θ -stablerepresentations,nexttoasolidgraspof“whothey
are”.(Mst Λ (d,θ )maybeempty.)Therearenot(yet)anysystematicresponsesto thisproblem,beyondsomepartialinsightsinthehereditarycase.
• Ingeneral,the θ -(semi)stablemodulesdonothavedescriptionsinstructural termsthatturnthemintorepresentation-theoreticallydistinguishedclasses.
• Thestablemodulestypicallyhavelargeorbits,whichmeansthatthemoduli space Mst Λ (d,θ )isunlikelytocaptureboundaryphenomenainthegeometryof Modd (Λ).
• Thisreferstoaweightfunction θ suchthat Msst Λ (d,θ )isnonempty:Thefact thatitistypicallydifficulttointerpret S -equivalenceinrepresentation-theoretic termsdetracts–atleastforthemoment–fromthevalueoftheexistenceofcoarse modulispacesthatclassifythesemistablemodulesuptothisequivalence.
Exploringandaddressingtheseproblems: Hereisaselectionofinsightsfor thespecialcasewhereΛ= KQ:
• Existenceofaweightfunction θ withthepropertythat Mst Λ (d,θ ) = ∅ is equivalentto d beingaSchurroot(see[23,Proposition4.4]).Infact,stabilityof amodule M relativetosomeweightfunctionforces M tobeaSchurianrepresentation,thatis,tohaveendomorphismring K .SincetheSchurianrepresentations withdimensionvector d clearlyhavemaximalorbitdimensionin Modd (Λ),the unionofthe Mst Λ (d,θ ),where θ tracesdifferentweightfunctions,iscontainedin theopensheetof Modd (Λ)(forsheets,see,e.g.,[24]).Thefactthatthevariety Mst Λ (d,θ )isalwayssmooth(see[23,Remark5.4])oncemorepointstoabsenceof boundaryphenomena.
• GivenaSchurroot d,thereisingeneralnochoiceof θ suchthatallSchur representations M ofΛwithdimensionvector d are θ -stable.Infact,foragiven Schurianrepresentation,thereneednotbeanyweightfunction θ makingit θ -stable (see[29,Section5.2],wherethe5-arrowKroneckerquiverisusedtodemonstrate this).
• Onthepositiveside:Given Q, θ and d,Schofield’salgorithmin[33]permitstodecidewhether d isaSchurrootof KQ and,ifso,whetherthereis a θ -stableSchurianrepresentationofdimensionvector d;cf.[23,Remarks4.5, 4.6].Furthermore,Reinekedevelopedarecursiveprocedurefordecidingwhether Msst Λ (d,θ ) = ∅;see[29,Section5.3]foranoutline.(Theargumentisbasedon anadaptationtothequiverscenarioofresultsduetoHarderandNarasimhanand providesaspecificinstanceofoneoftheplusseslistedabove.)Seealsotheworkby AdriaenssensandLeBruyn[1]onassessingthesupplyof θ -(semi)stablemodules withdimensionvector d
Pointersforfurtherreading :Overthepast20years,thisangleonmoduli ofrepresentationshasinspiredanenormousamountofresearch,withinteresting resultsnotonlydirectlytargetingmodulispaces,butalsoringsofsemi-invariants ofthevarieties Modd (Λ)intheirownright,nexttosurprisingapplications,for instancetoHorn’sProblem.Letusjustmentiona(necessarilyincomplete)listof furthercontributors:Chindris,Crawley-Boevey,delaPe˜na,Derksen,Geiss,Hille, LeBruyn,Nakajima,Procesi,Reineke,Schofield,VandenBergh,Weyman.
6.ApproachB.Slicing Λ-mod intostratawithfixedtop Inthefollowing,werelyontheprojectiveparametrizationintroducedinSection 2.
16B.HUISGEN-ZIMMERMANN
Insteadofusingstabilityfunctionstosingleoutclassifiablesubvarietiesof Modd (Λ),westartbypartitioningGRASSd (Λ)intofinitelymanylocallyclosed subvarieties,basedonmodule-theoreticinvariants.Theprimaryslicingisinterms oftops.Let T ∈ Λ-modbesemisimple.Therestrictiontomoduleswithfixedtop T hasanimmediatepayoff.Namely,thelocallyclosedsubvariety
GRASST d = {C ∈ GRASSd (Λ) | top(P/C )= T }
ofGRASSd (Λ)maybereplacedbyaprojectiveparametrizingvariety, GrassT d , whichhasfarsmallerdimensioninallinterestingcases.Infact,thepared-down variety GrassT d appearstogopartofthewaytowardsaquotientofGRASST d byits AutΛ (P)-action.Inmanyinstances,modulispacesforsubstantialclassesofrepresentationswithfixedtop T will,infact,beidentifiedassuitableclosedsubvarieties of GrassT d .ThisisforinstancetrueinthelocalcaseaddressedinCorollary5.2: Thefinemodulispace Mst Λ (d,θ ),guaranteedbyApproachAinthatcase,equals GrassT d ;seeCorollary6.7belowforjustification.
Followingthetenet“smallerisbetter”,wefixaprojectivecover P of T ,to replacetheprojectivecover P of 1≤i≤n S di i .Sincewearerestrictingourfocus tomoduleswithtop T ,thisprojectivecoversuffices.Accordingly,weconsiderthe subset
GrassT d = {C ∈ Gr(dim P −|d| ,P ) | Λ C isasubmoduleof Λ JP anddim P/C = d} oftheclassicalGrassmannianconsistingofthe(dim P −|d|)-dimensional K -subspacesofthe K -vectorspace JP .Clearly, GrassT d isinturnaclosedsubvarietyofthesubspaceGrassmannianGr(dim P −|d| ,P ),andhenceisprojective. Moreover,thenaturalactionoftheautomorphismgroupAutΛ (P )on GrassT d once moreprovidesuswithaone-to-onecorrespondencebetweenthesetoforbitson onehandandtheisomorphismclassesofΛ-moduleswithtop T anddimension vector d ontheother.Evidently,wehavethesamesemi-directproductdecompositionoftheactinggroupasbefore:AutΛ (P ) ∼ = AutΛ (T ) AutΛ (P ) u ,where AutΛ (P ) u = {idP +h | h ∈ HomΛ (P,JP )} istheunipotentradicalofAutΛ (P ).
Themainreasonforexpectationsofagainfromthisdownsizingisasfollows: Thesemi-directproductdecompositionoftheactingautomorphismgroup,inboth thebigandsmallscenarios,invitesustosubdividethestudyoforbitclosuresinto twoparts.Itdoes,indeed,turnouttobehelpfultoseparatelyfocusonorbitsunder theactionsofthesemidirectfactors,anditisforemostthesizeofthereductive factorgroup,AutΛ (T )=AutΛ (P/JP )resp.AutΛ (P/J P),whichdeterminesthe complexityofthistask.(InSection7,itwillbecomeapparentwhytheactionof theunipotentradicaliseasiertoanalyze.)Asaconsequence,itisadvantageousto passfromthebigautomorphismgroupAutΛ (P)toonewithreductivefactorgroup assmallaspossible.Corollaries6.6,6.7andProposition7.2,inparticular,attest tothebenefitsthatcomewithasimple,oratleastsquarefree,top.
6.1.Preliminaryexamples. (1) LetΛ= KQ,where Q isthegeneralized Kroneckerquiverwith m ≥ 2arrowsfromavertex e1 toavertex e2 .Moreover, choose T = S1 and d =(1, 1).Then P =Λe1 , GrassT d ∼ = Pm 1 ,andtheAutΛ (P )orbitsaresingletons.Thus GrassT d isanorbitspace.Corollary5.2guaranteesa finemodulispaceclassifyingthemoduleswithtop T anddimensionvector d upto isomorphism,andhence GrassT d coincideswiththismodulispace.
(2) Next,letΛ= KQ/I ,where Q isthequiver 1 α1 . α5 2 β1 . β5 and I istheidealgeneratedbythe βi αj for i = j andallpathsoflength
3.Againchoose T = S1 .For d =(d1 +1,d2 )with d1 ,d2 ≤ 5,weobtainthe followingdistinctoutcomesconcerning GrassT d :If d1 >d2 ,then GrassT d isempty. If d1 = d2 ,then GrassT d ∼ = Gr(5 d1 ,K 5 ) ∼ = Gr(d1 ,K 5 ).If d1 <d2 ,then GrassT d ∼ =
Flag(5 d1 , 5 d2 ,K 5 ),wherethelatterdenotesthevarietyofpartialsubspace
flags K 5 ⊇ U1 ⊇ U2 withdim U1 =5 d1 anddim U2 =5 d2 .Asaconsequenceof Corollary6.7below,wewillfindthat,ineithercase, GrassT d isafinemodulispace classifyingthemoduleswithtop T anddimensionvector d uptoisomorphism.
(3) Finally,letΛ= KQ/I ,where Q isthequiver1 α β 2,and I = α2
For T = S1 (hence P =Λe1 )and d =(2, 1),weobtain GrassT d ∼ = P1 .FromSection 3,wegleanthatthemoduleswithtop T anddimensionvector d donotevenhave acoarsemodulispaceclassifyingthemuptoisomorphism.Indeed,theAutΛ (P )orbitofthepoint C =Λβ ∈ GrassT d isacopyof A1 ,andconsequentlyfailstobe closedin GrassT d .Ontheotherhand,themodulesin GrassT d areclassifiablein naiveterms–uptoisomorphism,thereareonlytwoofthemafterall.Inorderto obtainthebenefitsofafineclassificationinthestrictsense,however,oneneedsto stratify GrassT d furtherintosegmentswithfixedradicallayerings.Inthepresent example,thisisatrivialstratificationinto A1 andasingleton.
Inthepresentsmallersetting,thetransferofinformationbetweentheprojectiveandtheaffineparametrizingvarietiesfollowsthesamepatternasinthebig (describedinProposition3.2).Clearly,thecounterpartof GrassT d intheaffinesettingisthesubvariety ModT d of Modd (Λ)whichconsistsofthepointsthatrepresent moduleswithtop T .Observethat GrassT d recordsthesamegeometricinformation asGRASST d ,justinalessredundantformat.
Proposition 6.2(Informationtransferrevisited) Let Ψ bethebijection
{AutΛ (P )-stablesubsetsof GrassT d }→{GL(d)-stablesubsetsof ModT d } extendedfromtheone-to-onecorrespondencebetweensetsoforbitswhichassigns toanorbit AutΛ (P ).C of GrassT d the GL(d)-orbitof ModT d thatrepresentsthe isomorphismclassof P/C .Onceagain, Ψ isaninclusion-preservingbijection whichpreservesandreflectsopenness,closures,connectedness,irreducibility,and typesofsingularities.Moreover,itpreservescategoricalandgeometricquotients of AutΛ (P )-stablesubvarietiesof GrassT d ,aswellasorbitspacesforthe AutΛ (P )action.Theinverse Ψ 1 hasanalogouspreservationproperties.
Nextwepresentaselectionofresultsaddressingexistenceand,ifpertinent, propertiesoffineorcoarsemodulispacesfor:(I)Themodulesthatdonotadmit
18B.HUISGEN-ZIMMERMANN
anypropertop-stabledegenerations.(II)Thegradedmoduleswithfixedtopand dimensionvectoroveranalgebraΛ= KQ/I ,where I isahomogeneousideal.
(I)Themoduleswhicharedegeneration-maximalamongthosewith fixedtop. Whatarethey?
Let M and M beΛ-moduleswithdimensionvector d.Recallthat M isa degeneration of M incasetheGL(d)-orbitin Modd (Λ)thatcorrespondsto M is containedintheclosureoftheGL(d)-orbitcorrespondingto M .ByProposition 3.2,thisamountstothesameaspostulatingthattheAutΛ (P)-orbitrepresenting M inGRASSd (Λ)becontainedintheclosureoftheAutΛ (P)-orbitrepresenting M .Wewrite M ≤deg M tocommunicatethisconnectionbetweentheorbits, andobservethat ≤deg definesapartialorderonisomorphismclassesofmodules. Intuitively,onemaythinkofthedegenerationsof M asacollectionofmodules thatdocumentasuccessiveunravelingofthestructureof M ,followingageometryguidedinstructionset;thisviewpointisbuttressedbyexamples.
Note:Whereasinthepresentcontext–thepursuitofmodulispaces–the typicallyenormoussizesoforbitclosuresinmodulevarietiesisapriorianobstacle, ashiftedviewpointmakesavirtueoutofnecessity.Onewayoforganizingthe categoryΛ-modistobreakitupintoposetsof(isomorphismclassesof)degenerationsofindividualmodules,andtoanalyzetheseposetsintheirownright;this directionhas,infact,movedtothemainstreamofresearch.Alongarelatedline,it isprofitabletotakeaimatthosemodulesinaspecifiedsubvariety X of Modd (Λ) whicharedistinguishedbyhavingthesame“height”(or“depth”)relativetothe degenerationorderwithin X .(Observethat,forgiven d,thelengthsofchains ofdegenerationsofmoduleswithdimensionvector d areboundedfromaboveby |d|− 1;wefollowtheRomansandstartwith0incountingchainlengths.)Thisis, infact,thetackwearetakinginthissubsection.Forbackgroundontheextensive theoryofdegenerationswereferthereadertoworkofBobinski,Bongartz,Riedtmann,Schofield,Skowronski,Zwara,andtheauthor,forinstance.Threeseminal articlesprovideagoodpointofdeparture:[30],[4],[35].
Therepresentationswhicharemaximalunder ≤deg inΛ-moddonotholdmuch interest.Itiseasytoseethat,givenanysubmodule U ofamodule M ,thedirect sum U ⊕ M/U isadegenerationof M .Hence,foranydimensionvector d,thereis, uptoisomorphism,exactlyonemodulewhichisdegeneration-maximalamongthe moduleswiththatdimensionvector,namelythesemisimplemodule 1≤i≤n S di i . Bycontrast,thereisusuallyaplethoraofdegenerationsof M whicharemaximal amongthedegenerationsthathavethesametopas M (see6.4below).Onthe otherhand,Theorem6.3belowguaranteesthattheynonethelessalwayshaveafine modulispaceclassifyingthemuptoisomorphism.
Sincewearefocusingonmoduleswithfixedtop T ,itistheorbitclosureofa module M in GrassT d (resp., ModT d )thatisrelevantforthemoment.Accordingly, wereferto M asa top-stabledegeneration of M incase M ≤deg M andtop(M )= top(M ).Clearly, M = P/C with C ∈ GrassT d isdegeneration-maximalamong themoduleswithtop T (meaningthat M hasnopropertop-stabledegeneration) preciselywhentheAutΛ (P )-orbitof C isclosedin GrassT d .
Theorem 6.3(see[11],Theorem4.4andCorollary4.5). Foranysemisimple T ∈ Λ-mod,themodulesofdimensionvector d whicharedegeneration-maximal
amongthosewithtop T haveafinemodulispace, ModuliMaxT d ,thatclassifies themuptoisomorphism.
Themodulispace ModuliMaxT d isaclosedsubvarietyof GrassT d ,andhenceis projective.
Inparticular,givenanymodule M withdimensionvector d whosetopiscontainedin T ,theclosedsubvarietyof ModuliMaxT d consistingofthepointsthat correspondtodegenerationsof M isafinemodulispaceforthemaximaltop-T degenerationsof M .
Observethattop(M ) ⊆ top(M )whenever M ≤deg M .Bythetheorem,we hitnewclassifiablestratainthehierarchyofdegenerationsof M aswesuccessively enlargetheallowabletop.
Themodulispace ModuliMaxT d islocatedin GrassT d asfollows:Firstonezeroesinonthesubvariety M of GrassT d consistingoftheclosedorbits(thatis, ontheorbitsofthetargetclassofmodules).On M,the AutΛ (P ) u -actionis trivial,butAutΛ (T )willstilloperatewithorbitsofarbitrarilyhighdimensionin general.However,ifwepickaBorelsubgroup H ofAutΛ (T )andcut M backto theclosedsubvarietyofallpointsthathaveastabilizercontaining H ,wearriveat anincarnationof ModuliMaxT d
ThefollowingconcomitantresultprovidesevidencefortherepresentationtheoreticrichnessoftheclassesofrepresentationsaddressedbyTheorem6.3.The constructionusedhaspredecessorsin[16,Theorem6]and[15,Example].
Satelliteresult6.4 (see[11,Example5.4]). Everyprojectivevarietyis isomorphicto ModuliMaxT d forsomechoiceof Λ, T ,and d
AcrucialingredientoftheproofofTheorem6.3consistsofnormalformsofthe moduleswithoutpropertop-stabledegenerations.Infact,theshapeoftheirnormalizedprojectivepresentationsisbothofindependentinterestandguidestheexplicitconstructionofuniversalfamilies.Thereformulationofabsenceofpropertopstabledegenerationsunder(1)belowisduetoprojectivityofthevariety GrassT d . (Bydefinition,aclosedsubgroup H ofalinearalgebraicgroup G isparabolic preciselywhenthegeometricquotient G/H isaprojectivevariety.)
Theorem 6.5(see[11,Theorem3.5]). Let M beamodulewithdimension vector d andtop T = 1≤i≤n S ti i .Moreover,let C beapointin GrassT d suchthat M ∼ = P/C .Thenthefollowingstatementsareequivalent:
(1) M hasnopropertop-stabledegenerations,i.e.,thestabilizersubgroup StabAutΛ (P ) (C ) isaparabolicsubgroupof AutΛ (P )
(2) M satisfiesthesetwoconditions:
• M isadirectsumoflocalmodules,say M = 1≤i≤n 1
≤ti Mij , where Mij ∼ = Λei /Cij withthefollowingadditionalproperty:Foreach i ≤ n,the Cij arelinearlyorderedunderinclusion.
• dimK HomΛ (P,JM )=dimK HomΛ (M,JM )
Ifconditions (1),(2) aresatisfied,then AutΛ (P ) u stabilizes C ,and AutΛ (P ).C =
AutΛ (T ).C isisomorphictoadirectproductofpartialflagvarieties Fi ,where Fi dependsonlyonthenumberofdistinctleftidealsinthefamily (Cij )j ≤ti andtheir multiplicities.
20B.HUISGEN-ZIMMERMANN
Thedimensionconditioninstatement(2)ofTheorem6.5hasthefollowing interpretation:Itmeansthatthefirstsyzygyof M isinvariantunderallhomomorphisms P → JP
TheupcomingcorollariesrestonthefollowingcombinationofTheorem6.3 withSection3.ItshowsthatoneoftheconswelistedinconnectionwithMethod AarisesinMethodBaswell:Namely,forlargetops T ,few closed subvarieties X of ModT d correspondtoclassesofmodulespermittingafinemoduliclassification. Inotherwords,“most”classificationsofthisilkareexpectedtotargetonlygeneric classesofmodules,thuscircumventing“boundaryphenomena”.
ConsequenceconcerningtheclassifiabilityofclosedsubsetsofModT d .
Let X beaclosedAutΛ (P )-stablesubvarietyof GrassT d and C theclassofmodules representedbytheorbitsof X .Thenthereisafine(equivalently,acoarse)moduli spaceclassifyingthemodulesin C uptoisomorphismifandonlyif C consistsof modulesthataredegeneration-maximalamongthosewithtop T .
InCorollaries6.6and6.7,itdoesnotaffecttheoutcomeofthemoduliproblem whetherwefixadimensionvectororelsefixonlythetotaldimensionofthemodules considered.Weoptforthelatter,sincethisleadstosmootherstatements.Tothat end,weslightlyupgradeournotation:Foranypositiveinteger d,wedenoteby GrassT d theunionofthevarieties GrassT d where d rangesoverthedimensionvectors with |d| = d;thatis,
GrassT d = {C ∈ Gr(dim P d,P ) | Λ C isasubmoduleof Λ JP }.
Corollary 6.6(see[18]) Supposethat T isasquarefreesemisimplemodule and d ∈ N.Thenthefollowingstatementsareequivalent:
(a) Themoduleswithtop T anddimension d haveafinemodulispaceclassifyingthemuptoisomorphism.
(b) Themoduleswithtop T anddimension d haveacoarsemodulispace classifyingthemuptoisomorphism.
(c) Thesubmodulesof JP ofdimension dim P d areinvariantunderall endomorphismsof P
(d) GrassT d isafinemodulispaceclassifyingthe d-dimensionalmoduleswith top T uptoisomorphism.
DeducingthecorollaryfromTheorem6.5. Theimplications(d)=⇒ (a)=⇒ (b)areclear.Re“(b)=⇒ (c)”:From(b)weinferthat,foreach C ∈ GrassT d ,themodule M = P/C iswithoutpropertop-stabledegenerations.Hence, Theorem6.5yieldstheinvarianceof C underhomomorphisms P → JP ,asnoted above.For“(c)=⇒ (d)”,observethat(c)forcesallAutΛ (P )-orbitsof GrassT d to besingletonsandthusmakesallmodules P/C with C ∈ GrassT d degenerationmaximalamongthemoduleswithtop T .Hence GrassT d isanorbitspacewhose fibersaresingletons.OnemaynoweitherinvokeTheorem6.3orverifythatthe tautologicalfamilyparametrizedby GrassT d isuniversal.
Weaddanoffshoottothepreviouscorollary.ItisamildextensionofCorollary 5.2.
Corollary 6.7 Supposethat T issimpleandthattheonlyoccurrencesof T in JP areinthesocle.Thentheequivalentconditionsabovearesatisfiedforall d.Consequently,each GrassT d isafinemodulispaceclassifyingthe d-dimensional moduleswithtop T uptoisomorphism.
LetusreturntoExample(2)in6.1.That GrassT d isafinemodulispaceinthis instanceaswellisaspecialcaseofCorollary6.7.
(II)Aimingatthegradedmoduleswithfixedtop. SupposeΛ= KQ/I , where I ⊆ KQ isahomogenousidealrelativetothepath-lengthgradingof KQ. Wheneverwespeakofgraded(left)Λ-moduleswerefertothepath-lengthgrading ofΛ.Itishardlysurprisingthattheadditionalrigidityencounteredinthecategoryofgradedmoduleswithhomogeneoushomomorphisms(ofdegree0)promotes classifiabilty.Weexploretowhatextent.
Let T ∈ Λ-modbeasemisimplemoduleendowedwiththegradingthatmakes ithomogeneousofdegree0.Itisamatterofcoursethat,inaddressinggradedrepresentationswithfixedtop T anddimension d,weshouldreplacetheparametrizing variety GrassT d byagradedincarnation,thatis,by
grad-GrassT d = {C ∈ GrassT d | C isahomogeneoussubmoduleof JP }; where P standsforthegradedprojectivecoverof T .Intandem,wereplacetheactinggroupAutΛ (P )bythesubgroupgrad-AutΛ (P )consistingofthehomogeneous automorphismsof P .Onereadilyconfirmsthatthenatural(morphic)actionof grad-AutΛ (P )ongrad-GrassT d placesthegrad-AutΛ (P )-orbitsofgrad-GrassT d into acanonicalone-to-onecorrespondencewiththegradedΛ-moduleswithtop T and dimension d.ThissetupyieldsasignificantimprovementofCorollary6.7inthe gradedsituation.
Theorem 6.8 [2,Theorem4.1] Here“graded”includes“generatedindegree 0”.
Foranysimplemodule T and d ∈ N,the d-dimensionalgraded Λ-moduleswith top T possessafinemodulispaceclassifyingthemuptogradedisomorphism.This modulispaceequals grad-GrassT d and,inparticular,isaprojectivevariety.
Itisnowclearthatthefinitedirectsumsoflocalgradedmodulesareclassifiable byfinemodulispacesinsegments,namelyaftertheobvioussubdivisionaccording totopsandsequencesofdimensionsofthelocalsummandswithfixedtop.This isasfarasthiskindof“global”classificationcanbepushedinthegradedcase. Thefollowingresultatteststoaroadblock.Westillinclude“generatedindegree 0”whenwerefertogradedmodules.
Theorem 6.9 [3,Theorem4.2] Let T beanysemisimplemoduleendowedwith theobviousgradingand d ∈ N.
Ifthegradedmoduleswithtop T anddimension d haveacoarsemodulispace classifyingthemuptogradedisomorphism,thentheyarealldirectsumsoflocal modules.
(III)Threeeasypieces. Thefinemodulispacesweencounteredunder(I) and(II)andthecorrespondinguniversalfamiliesareaccessibletoalgorithmiccomputation,totheextentthatthereisanalgorithmfordeterminingthedistinguished affinecoverofthesemodulispacesintermsofpolynomialequations;itisinduced bythedistinguishedaffinecover Grass(σ ) σ oftheambient GrassT d ;seeSection 7.Therestrictionsofthetargeteduniversalfamilytothechartsofthiscovercan inturnbecalculated.
Thefirsttwopieces,theeasiest,illustrateCorollary6.7andTheorem6.8.
Another random document with no related content on Scribd:
Just as he was ready to go, Costes and Rignot, the two French aviators who were leaving on their eastward trip in an effort to beat the non-stop record he had established, came over to say good-by and he wished them Godspeed.
On the way to Cherbourg Lindbergh ran into wind, rain, hail and fog. He landed there at 11:35 amid what seemed to be the entire population of the port. He was cordially welcomed by the full staff of city officials. After lunch at the Mayor’s château he was motored into the city proper, and at the Gare Maritime a plaque was unveiled commemorating the spot where he had first flown over France on his way to Le Bourget.
To avoid pressure of the crowd he jumped upon a Cunard tender at the dock and reached the fast launch of Admiral Burrage which carried him to the U.S.S. Memphis, ordered by President Coolidge to bring the flier home.
IV
WASHINGTON
IT is probable that when Lindbergh reached America he got the greatest welcome any man in history has ever received; certainly the greatest when judged by numbers; and by far the greatest in its freedom from that unkind emotion which in such cases usually springs from one people’s triumph over another.
Lindbergh’s victory was all victory; for it was not internecine, but that of our human species over the elements against which for thousands of centuries man’s weakness has been pitted.
The striking part of it all was that a composite picture of past homecoming heroes wouldn’t look any more like Charles Lindbergh did that day of his arrival in Washington than a hitching post looks like a green bay tree.
Caesar was glum when he came back from Gaul; Napoleon grim; Paul Jones defiant; Peary blunt; Roosevelt abrupt; Dewey deferential; Wilson brooding; Pershing imposing. Lindbergh was none of these. He was a plain citizen dressed in the garments of an everyday man. He looked thoroughly pleased, just a little surprised, and about as full of health and spirits as any normal man of his age should be. If there was any wild emotion or bewilderment in the occasion it lay in the welcoming crowds, and not in the air pilot they were saluting.
The cruiser Memphis, on which Lindbergh travelled, passed through the Virginia Capes on her way to Washington a few minutes after five P.M. of the afternoon of June 10. Here Lindbergh got the first taste of what was to come.
A convoy of four destroyers, two army blimps from Langley Field and forty airplanes of the Army, Navy and Marine Corps accompanied the
vessel as she steamed up Chesapeake Bay. As the night fell they wheeled toward their various bases and were soon lost to view. They gave no salute; and, for all the casual observer might have noted, they were merely investigating this newcomer to their home waters. But they left an indelible impression upon those in the Memphisthat the morrow was to be extraordinary.
Saturday June 11, 1927, dawned hot and clear in Washington. It was evident early in the day that something far out of the city’s peaceful summer routine was going to happen. Streets were being roped off. Special policemen were going to their posts. Airplanes flew about overhead. Citizens began gathering in little clumps up and down Pennsylvania Avenue, many seating themselves on fruit boxes and baskets as if for a long wait.
The din that greeted the Memphis off Alexandria, suburb of Washington, began the noisy welcome that lasted for several hours. Every roof top, window, old ship, wharf and factory floor was filled with those who simply had to see Lindbergh come home. Factory whistles, automobiles, church bells and fire sirens all joined in the pandemonium.
In the air were scores of aircraft. One large squadron of nearly fifty pursuit planes maneuvered in and out of the heavy vaporous clouds that hung over the river. Beneath them moved several flights of slower bombers. The giant dirigible airship, the U.S.S. LosAngeles, wound back and forth above the course of the oncoming Memphis.
By eleven o’clock the saluting began. Vice Admiral Burrage, also returning on the Memphis, received his customary fifteen guns from the Navy yard. The President’s salute of 21 guns was exchanged. Firing from the cruisers’ battery and from the shore stations lent a fine rhythmic punctuation to the constantly increasing noise from other quarters.
Just before noon the Memphis came alongside the Navy Yard dock and a gangplank was hoisted to her rail. On the shore were collected a notable group of cabinet officers and high officials. There were the
Secretary of the Navy, Curtis D. Wilbur; the Secretary of War, Dwight F. Davis; Postmaster General Harry S. New; and former Secretary of State, Charles Evans Hughes. There were Admiral Edward W. Eberle, Chief of Naval Operations; Major General Mason W. Patrick and Rear Admiral William A. Moffett, heads of the Army and Navy air forces. There was Commander Richard E. Byrd who flew to the North Pole, and who later followed Lindbergh’s trail to France.
When the gangplank was in place Admiral Burrage came down it and a moment later returned with a lady on his arm. This lady was Mrs. Evangeline Lindbergh, the young pilot’s mother.
Instantly a new burst of cheering went up; but many wept—they knew not just why.
For a few minutes mother and son disappeared into a cabin aboard the Memphis. It was a nice touch; something more than the brass bands and cheering. And it somehow symbolized a great deal of what was being felt and said that hot morning in our country’s great capital.
Next came brief and a somewhat informal greeting by the dignitaries. In their glistening high silk hats they surrounded Lindbergh and for a bit shut him off from the pushing perspiring crowd still held at bay ashore by the bayonets of the marines.
Suddenly the crowd could hold its patience no longer. With one frantic push it broke through the ranks of “Devil Dogs” and swarmed down upon the moored vessel. Trouble was averted by the simple expedient of getting Lindbergh quickly into one of the waiting cars and starting for the Navy Yard gate.
The parade escort had been lined up some hours ahead of time. Now it got under way toward the center of the city, leading the automobiles that carried the official party. Clattering hoofs of cavalrymen, blare of bands and a rolling cheer along the ranks of waiting thousands marked the progress of the young American flier who had so gloriously come home.
Here for the first time Lindbergh saw the spirit in which his people were to greet him. They were curious, yes; crowds always are on such occasions. And they were gay with their handclapping and flagwaving, shouting and confetti throwing. But there was a note of enthusiasm everywhere that transcended just a chorus of holiday seekers witnessing a new form of circus. There was something deeper and finer in the way people voiced their acclaim. Many of them wiped their eyes while they laughed; many stood with expressionless faces, their looks glued upon the face of the lad who had achieved so great a thing and yet seemed to take it all so calmly.
When the parade reached the natural amphitheatre of the Washington Monument the hillsides were jammed with a great gathering of men, women and children. On the high stand that had been erected, the President of the United States and Mrs. Coolidge waited to receive the man who but three weeks and a day before had been a comparatively unknown adventurer hopping off for Paris by air.
Ranged about the President were the ambassadors of many foreign countries, members of the diplomatic corps with their wives and daughters, and nearly all the high officials of the government.
When Lindbergh mounted the stand the President came forward and grasped his hand. Those closest to Mr. Coolidge say that rarely has he shown the unrestrained cordiality he put into that simple greeting.
The President now moved to the front of the stand and waited for the applause to be stilled. Presently, when the multitude again was quiet, he began slowly to speak:
“My Fellow-Countrymen:
“It was in America that the modern art of flying of heavier-than-air machines was first developed. As the experiments became successful, the airplane was devoted to practical purposes. It has
been adapted to commerce in the transportation of passengers and mail and used for national defense by our land and sea forces.
“Beginning with a limited flying radius, its length has been gradually extended. We have made many flying records. Our Army fliers have circumnavigated the globe. One of our Navy men started from California and flew far enough to have reached Hawaii, but being off his course, landed in the water. Another officer of the Navy has flown to the North Pole. Our own country has been traversed from shore to shore in a single flight.
“It had been apparent for some time that the next great feat in the air would be a continuous flight from the mainland of America to the mainland of Europe. Two courageous Frenchmen made the reverse attempt and passed to a fate that is as yet unknown.
“Others were speeding their preparations to make the trial, but it remained for an unknown youth to attempt the elements and win. It is the same story of valor and victory by a son of the people that shines through every page of American history.
“Twenty-five years ago there was born in Detroit, Michigan, a boy representing the best traditions of this country, of a stock known for its deeds of adventure and exploration.
“His father, moved with a desire for public service, was a member of Congress for several years. His mother, who dowered her son with her own modesty and charm, is with us today. Engaged in the vital profession of school-teaching, she has permitted neither money nor fame to interfere with her fidelity to her duties.
“Too young to have enlisted in the World War, her son became a student at one of the big State universities. His interest in aviation led him to an Army aviation school, and in 1925 he was graduated as an airplane pilot. In November, 1926, he had reached the rank of Captain in the Officers’ Reserve Corps.
© WideWorldPhotos
NEW YORK CITY—THE PARADE PASSING THROUGH CENTRAL PARK WHERE OVER 400,000 PEOPLE WERE GATHERED. A SOLID BANK OF HUMANITY FLANKED OUR PASSAGE
©
NEW YORK CITY—PARADE IN CENTRAL PARK AS SEEN FROM A NEARBY SKYSCRAPER
“Making his home in St. Louis, he had joined the 110th Observation Squadron of the Missouri National Guard. Some of his qualities noted by the Army officers who examined him for promotion, as shown by reports in the files of the Militia Bureau of the War Department, are as follows:
“‘Intelligent,’ ‘industrious,’ ‘energetic,’ ‘dependable,’ ‘purposeful,’ ‘alert,’ ‘quick of reaction,’ ‘serious,’ ‘deliberate,’ ‘stable,’ ‘efficient,’
‘frank,’ ‘modest,’ ‘congenial’ ‘a man of good moral habits and regular in all his business transactions.’
“One of the officers expressed his belief that the young man ‘would successfully complete everything he undertakes.’ This reads like a prophecy.
“Later he became connected with the United States Mail Service, where he exhibited marked ability, and from which he is now on leave of absence.
“On a morning just three weeks ago yesterday this wholesome, earnest, fearless, courageous product of America rose into the air from Long Island in a monoplane christened ‘The Spirit of St. Louis’ in honor of his home and that of his supporters.
“It was no haphazard adventure. After months of most careful preparation, supported by a valiant character, driven by an unconquerable will and inspired by the imagination and the spirit of his Viking ancestors, this reserve officer set wing across the dangerous stretches of the North Atlantic.
“He was alone. His destination was Paris.
“Thirty-three hours and thirty minutes later, in the evening of the second day, he landed at his destination on the French flying field at Le Bourget. He had traveled over 3,600 miles, and established a new and remarkable record. The execution of his project was a perfect exhibition of art.
“This country will always remember the way in which he was received by the people of France, by their President and by their Government. It was the more remarkable because they were mourning the disappearance of their intrepid countrymen, who had tried to span the Atlantic on a western flight.
“Our messenger of peace and good-will had broken down another barrier of time and space and brought two great peoples into closer communion. In less than a day and a half he had crossed the ocean
over which Columbus had traveled for sixty-nine days and the Pilgrim Fathers for sixty-six days on their way to the New World.
“But, above all, in showering applause and honors upon this genial, modest American youth, with the naturalness, the simplicity and the poise of true greatness, France had the opportunity to show clearly her good-will for America and our people.
“With like acclaim and evidences of cordial friendship our Ambassador without portfolio was received by the rulers, the Governments and the peoples of England and Belgium. From other nations came hearty messages of admiration for him and for his country. For these manifold evidences of friendship we are profoundly grateful.
“The absence of self-acclaim, the refusal to become commercialized, which has marked the conduct of this sincere and genuine exemplar of fine and noble virtues, has endeared him to every one. He has returned unspoiled.
“Particularly has it been delightful to have him refer to his airplane as somehow possessing a personality and being equally entitled to credit with himself, for we are proud that in every particular this silent partner represented American genius and industry. I am told that more than 100 separate companies furnished materials, parts or service in its construction.
“And now, my fellow-citizens, this young man has returned. He is here. He has brought his unsullied fame home. It is our great privilege to welcome back to his native land, on behalf of his own people, who have a deep affection for him and have been thrilled by his splendid achievement, a Colonel of the United States Officers’ Reserve Corps, an illustrious citizen of our Republic, a conqueror of the air and strength for the ties which bind us to our sister nations across the sea.
“And, as President of the United States, I bestow the Distinguished Flying Cross, as a symbol of appreciation for what he is and what he has done, upon Colonel Charles A. Lindbergh.”
