Finite Schur filtration dimension for modules over

arXiv:0706.0604v2[math.RT]15May2008FiniteSchurfiltrationdimensionformodulesoveranalgebrawithSchurfiltrationVasudevanSrinivasandWilberdvanderKallenMay15,2008AbstractLetG=GLNorSLNasreductivelinearalgebraicgroupoverafieldkofcharacteristicp0.WeproveseveralresultsthatwerepreviouslyestablishedonlywhenN≤5orp2N:LetGactra-tionallyonafinitelygeneratedcommutativek-algebraAandletgrAbetheGrosshansgradedring.WeshowthatthecohomologyalgebraH∗(G,grA)isfinitelygeneratedoverk.IfmoreoverAhasagoodfiltrationandMisanoetherianA-modulewithcompatibleGaction,thenMhasfinitegoodfiltrationdimensionandtheHi(G,M)arenoetherianAG-modules.Toobtainresultsinthisgenerality,weem-ployfunctorialresolutionoftheidealofthediagonalinaproductofGrassmannians.1IntroductionConsideraconnectedreductivelinearalgebraicgroupGdefinedoverafieldkofpositivecharacteristicp.WesaythatGhasthecohomologicalfinitegen-erationproperty(CFG)ifthefollowingholds:LetAbeafinitelygeneratedcommutativek-algebraonwhichGactsrationallybyk-algebraautomor-phisms.(SoGactsfromtherightonSpec(A).)ThenthecohomologyringH∗(G,A)isfinitelygeneratedasak-algebra.Here,asin[9,I.4],weusethecohomologyintroducedbyHochschild,alsoknownas‘rationalcohomology’.Theintentofthispaperistotakeonemoresteptowardsprovingtheconjecturethateveryreductivelinearalgebraicgrouphasproperty(CFG).1Otherinputwillhopefullycomefromthetheoryofstrictpolynomialbifunc-tors,cf.[15].Thekeypointofthepresentworkistoremoverestrictionsonthecharacteristicfrom[19].OurproofsuseresolutionofthediagonalinproductsofGrassmannians.ThustheyapplyonlytothegroupsSLN,GLN.Butrecall([17],[18],[19])thatfortheconjecturethesecasessuffice.Alsorecallthattheconjectureimpliesthemainresultsofthispaper,aswellastheiranaloguesforotherreductivegroups.Toformulatethemainresults,letN≥1andletGbetheconnectedreductivelinearalgebraicgroupGLNorSLNoveranalgebraicallyclosedfieldkofcharacteristicp0.LetAbeafinitelygeneratedcommutativek-algebraonwhichGactsrationallybyk-algebraautomorphisms.LetMbeanoetherianA-moduleonwhichGactscompatibly.ThismeansthatthestructuremapA⊗M→MisaG-modulemap.OurmaintheoremisTheorem1.1IfAhasagoodfiltration,thenMhasfinitegoodfiltrationdimensionandeachHi(G,M)isanoetherianAG-module.Onemayalsoformulatethefirstpartintermsofpolynomialrepresenta-tionsofGLN.Recallthatafinitedimensional(askvectorspace)rationalrepresentationofGLNiscalledpolynomialifitextendstothemonoidofNbyNmatriceswithoutpolesalongthelocuswherethedeterminantvanishes.UnlikeGreen[5]wecannotrestrictourselvestofinitedimensionalrepresen-tations,sowedefinearepresentationtobepolynomialifitisaunionoffinitedimensionalpolynomialrepresentations.Inotherwords,weallowinfi-nitedimensionalcomodulesfortheHopfalgebraofregularfunctionsonthemonoid.SoletAbeafinitelygeneratedcommutativek-algebraonwhichGLNactspolynomiallybyk-algebraautomorphisms.LetMbeanoetherianA-moduleonwhichGLNactscompatiblyandpolynomially.Theorem1.2IfAhasSchurfiltration,thenMhasfiniteSchurfiltrationdimension.Remark1.3TheHi(GLN,M)arelessinterestingnow,becausethepartofnonzeropolynomialdegreeinMdoesnotcontributetoHi(GLN,M).NowletAbeafinitelygeneratedcommutativek-algebraonwhichSLNactsrationallybyk-algebraautomorphisms.OnethenhasaGrosshans2gradedalgebragrAandwecanremovetherestrictionsonthecharacter-isticin[17,Theorem1.1]:Corollary1.4Thek-algebraH∗(SLN,grA)isfinitelygenerated.Themethodofproofofthemainresultisbasedonthefunctorialresolu-tion[12]ofthediagonalofZ×ZwhenZisaGrassmannianofsubspacesofkN.ThisisusedinductivelytostudyequivariantsheavesonaproductXofsuchGrassmannians.Thatleadstoaspecialcaseofthetheorems,withAequaltotheCoxringofX,multigradedbythePicardgroupPic(X),andMcompatiblymultigraded.NextonetreatscaseswhenonthesameAthemultigradingisreplacedwitha‘collapsed’gradingwithsmallervaluegroupandMisonlyrequiredtobemultigradedcompatiblywiththisnewgrading.HerethetrickisthatanassociatedgradedofMhasamultigradingthatiscollapsedalittleless.ThesuitablymultigradedCoxringsnowreplacethe‘gradedpolynomialalgebraswithgoodfiltration’of[17]andthemethodof[19]appliestofinishtheproofofTheorem1.1.ThenCorollary1.4followsinthemannerof[17].AcknowledgementsOurcollaborationgotstartedthankstothe60thbirthdayconferencesforV.B.MehtaandS.M.BhatwadekaratTIFRMum-baiin2006.MuchofthesubsequentworkwasdoneattheuniversityofBielefeld,whichwethankforitshospitality.2RecollectionsandconventionsSomeunexplainednotations,terminology,properties,...canbefoundin[9].Fromnowon,withtheexceptionofsection8,weputG=GLN,withB+itssubgroupofuppertriangularmatrices,B−theoppositeBorelsubgroup,T=B+∩B−thediagonalsubgroup,U=U+theunipotentradicalofB+.TherootsofUarepositive.ThecharactergroupX(T)hasabasisǫ1...,ǫNwithǫi(diag(t1,...,tN))=ti.Anelementλ=PiλiǫiofX(T)isoftendenoted(λ1,...,λN).Itiscalledapolynomialweightiftheλiarenonnegative.Itiscalledadominantweightifλ1≥···≥λN.Itiscalledanti-dominantifλ1≤···≤λN.Thefundamentalweights̟1,...,̟Naregivenby̟i=Pij=1ǫj.Ifλ∈X(T)isdominant,thenindGB−(λ)isthedualWeylmoduleorcostandardmodule∇G(λ),orsimply∇(λ),withhighestweightλ.TheGrosshansheightofλisht(λ)=Pi(N−2i+1)λi.3Itextendstoahomomorphismht:X(T)⊗Q→Q.Thedetermi