Graded cofinite rings of differential operators

Gradedco¯niteringsofdi®erentialoperatorsFriedrichKnopDepartmentofMathematics,RutgersUniversity,NewBrunswickNJ08903,[e-mail]ofaringofdi®erentialoperatorswhicharebiginthefollowingsense:theextensionofassociatedgradedringsis¯nite.Weshowthatthesesubalgebrascorrespond,uptoautomorphism,touniformlyrami¯ed¯nitemorphisms.ThisgeneralizesatheoremofLevasseur-Sta®ordonthegeneratorsoftheinvariantsofaWeylalgebraundera¯nitegroup.1.IntroductionInthispaperwestudysubalgebrasAofthealgebraD(X)ofdi®erentialoperatorsonasmoothvarietyXwhicharebiginthefollowingsense:usingtheorderofadi®erentialoperator,theringD(X)isequippedwitha¯ltration.ItsassociatedgradedalgebraD(X)iscommutativeandcanberegardedasthesetofregularfunctionsonthecotangentbundleofX.ThesubalgebraAinheritsa¯ltrationfromD(X)anditsassociatedgradedalgebraAisasubalgebraofD(X).WecallAgradedco¯niteinD(X)ifD(X)isa¯nitelygeneratedA-module.Ourguidingexampleofagradedco¯nitesubalgebraisthealgebraofinvariantsD(X)WwhereWisa¯nitegroupactingonX.Otherexamplescanbeconstructedasfollows.Let':X!Ybea¯nitedominantmorphismontoanormalvarietyY.Thenweput(1:1)D(X;Y)=fD2D(X)jD(O(Y))µO(Y)g:Weshow(Corollary3.6)thatthissubalgebraisgradedco¯niteifandonlyiftherami¯ca-tionof'isuniform,i.e.,therami¯cationdegreeof'alongadivisorD½Xdependsonlyontheimage'(D).Itshouldbenotedthatthesetwoconstructionsareinfactmoreorlessequivalent.InTheorem3.1weshowthatD(X)W=D(X;X=W).Conversely,weshowinProposition3.3thatD(X;Y)=D(~X)Wwhere~X!Xisasuitable¯nitecoverofXandWisa¯nitegroupactingon~X.1Ourmainresultisthatuptoautomorphismseverygradedco¯nitesubalgebraisofformabove:1.1.Theorem.LetXbeasmoothvarietyandAagradedco¯nitesubalgebraofD(X).Thenthereisanautomorphism©ofD(X),inducingtheidentityonD(X),suchthatA=©D(X;Y)forsomeuniformlyrami¯edmorphism':X!Y.ThemainmotivationforthisnotioncamefromthefollowingresultofLevasseurandSta®ord:letWbea¯nitegroupactinglinearlyonavectorspaceV.ThenD(V)WisgeneratedbytheW-invariantfunctionsO(V)WandtheW-invariantconstantcoe±cientdi®erentialoperatorsS¤(V)W.ForgeneralvarietiesX,thereisnonotionofconstantcoe±cientdi®erentialoperators.SincethealgebrageneratedbyO(V)WandS¤(V)Wisclearlygradedco¯niteourmaintheoremcanbeseenasanon-lineargeneralizationofthetheoremofLevasseur-Sta®ord.OurmaintheoremhasseveralapplicationconcerninggeneratingelementsofringsofW-invariantdi®erentialoperatorswhichgobeyondthetheoremofLevasseur-Sta®ord.Forexample,weprovethatD(X)Wcanbegeneratedbyatmost2n+1elementswhenVisann-dimensionalrepresentationofW.Moreover,weestablishakindofGaloiscorrespondenceforgradedco¯nitesubalgebras.Finally,wedetermineallgradedco¯nitesubalgebrasofD(A1),theWeylalgebraintwogenerators.Theproofconsistsessentiallyof¯vesteps:1.WeshowtheaforementionedclaimthatD(X;Y)isgradedco¯niteifandonlyif'isuniformlyrami¯ed.2.ThenweshowthatundertheseconditionsD(X;Y)isasimplering.HerewefollowanargumentofWallach[Wa].3.WeshowthatthetheoremholdsoverthegenericpointofX.4.Thenweconstructtheautomorphism©.Thisisthemosttediouspartofthepaperandrestsonexplicitcomputationsincodimensionone.5.Finally,wepasteallthisinformationtogetherbyshowingthattwogradedco¯nitesubalgebrasAµA0whichcoincidegenericallyandforwhichA0isasimpleringareactuallyequal.Herewefollowtheargumentin[LS].Finally,itshouldbementionedthattheactualmainTheorem7.1ismoregeneralinthatitallowsforcertainsingularitiesofX.Acknowledgment:ThisworkstartedwhiletheauthorwasguestoftheCRM,Montr¶eal,inSummer1997andcontinuedduringastayattheUniversityofFreiburgin2004.Theauthorthanksbothinstitutionsfortheirhospitality.Lastnotleast,theauthorwouldliketothanktherefereeforanexcellentjob.Inparticular,theshorterproofofTheorem3.1waspointedoutbyhim/her.22.Gradedco¯nitesubalgebras:de¯nitionandbasechangeAllvarietiesandalgebraswillbede¯nedoverC.Moreover,varietiesareirreduciblebyde¯nition.RecallthataC-linearendomorphismDofacommutativealgebraBisadi®erentialoperatoroforder·dif(2:1)[b0;[b1;:::[bd;D]:::]]=0forallb0;b1;:::;bd2B:LetD(B)·dbethesetofdi®erentialoperatorsoforder·dandandD(B)=SdD(B)·d.ThenD(B)isa¯lteredalgebra,i.e.,D(B)·dD(B)·eµD(B)·d+eforallintegersdande.LetD(B)beitsassociatedgradedalgebra,i.e.,D(X):=©dD(X)dwithD(X)d=D(X)·d=D(X)·d¡1.Thisisagradedcommutativealgebra.IfXisavarietywithringoffunctionsO(X)thenwede¯neD(X)=D(O(X)).EverysubalgebraAµD(X)inheritsthe¯ltrationbyA·d=A\D(X)·d.Thisway,theassociatedgradedalgebraAisasubalgebraofD(X)andwede¯ne:De¯nition:AsubalgebraAofD(X)iscalledgradedco¯niteifD(X)isa¯nitelygeneratedA-module.Example:LetWbea¯nitegroupactingonXandassumeD(X)tobe¯nitelygenerated(e.g.Xsmooth).WeclaimthatA=D(X)Wisgradedco¯niteinD(X).Infact,sinceWislinearlyreductive,wehaveA=D(X)Wwhichiswellknowntobeco¯niteinD(X).TheringA:=A·0=A\O(X)iscalledthebaseofA.2.1.Proposition.LetAµD(X)begradedco¯nite.ThenthebaseAofAisa¯nitelygeneratedalgebrawhichisco¯niteinO(X).Inotherwords,ifY=SpecAthenX!Yisa¯nitesurjectivemorphismofa±nevarieties.Proof:SinceAisco¯niteinD(X),its0-componentAisco¯niteinthe0-componentO(X)ofD(X).Nowtheassertionfollowsfromthefollowinglemma.2.2.Lemma.LetAµBbeanintegralextens