Graded annihilators of modules over the Frobenius

arXiv:math/0605329v1[math.AC]12May2006GRADEDANNIHILATORSOFMODULESOVERTHEFROBENIUSSKEWPOLYNOMIALRING,ANDTIGHTCLOSURERODNEYY.SHARPAbstract.ThispaperisconcernedwiththetightclosureofanidealainacommutativeNoetherianlocalringRofprimecharacteristicp.Severalauthors,includingR.Fedder,K.-i.Watanabe,K.E.Smith,N.HaraandF.Enescu,haveusedthenaturalFrobeniusactiononthetoplocalcohomologymoduleofsuchanRtogoodeffectinthestudyoftightclosure,andthispaperusesthatdevice.Themainpartofthepaperdevelopsatheoryofwhatareherecalled‘specialannihilatorsubmodules’ofaleftmoduleovertheFrobeniusskewpolynomialringassociatedtoR;thistheoryisthenappliedinthelatersectionsofthepapertothetoplocalcohomologymoduleofRandusedtoshowthat,ifRisCohen–Macaulay,thenitmusthaveaweakparametertestelement,evenifitisnotexcellent.0.IntroductionThroughoutthepaper,RwilldenoteacommutativeNoetherianringofprimecharacteristicp.Weshallalwaysdenotebyf:R−→RtheFrobeniushomomorphism,forwhichf(r)=rpforallr∈R.LetabeanidealofR.Then-thFrobeniuspowera[pn]ofaistheidealofRgeneratedbyallpn-thpowersofelementsofa.WeuseR◦todenotethecomplementinRoftheunionoftheminimalprimeidealsofR.Anelementr∈Rbelongstothetightclosurea∗ofaifandonlyifthereexistsc∈R◦suchthatcrpn∈a[pn]foralln≫0.Wesaythataistightlyclosedpreciselywhena∗=a.ThetheoryoftightclosurewasinventedbyM.HochsterandC.Huneke[8],andmanyapplicationshavebeenfoundforthetheory:see[10]and[11],forexample.InthecasewhenRislocal,severalauthorshaveused,asanaidtothestudyoftightclosure,thenaturalFrobeniusactiononthetoplocalcohomologymoduleofR:see,forexample,R.Fedder[4],FedderandK.-i.Watanabe[5],K.E.Smith[17],N.HaraandWatanabe[6]andF.Enescu[3].Thisdeviceisemployedinthispaper.ThenaturalFrobeniusactionprovidesthetoplocalcohomologymoduleofRwithanaturalstructureasaleftmoduleovertheskewpolynomialringR[x,f]associatedtoRandf.Sections1and3developatheoryofwhatareherecalled‘specialannihilatorsubmodules’ofaleftR[x,f]-moduleH.Toexplainthisconcept,weneedthedefinitionofthegradedannihilatorgr-annR[x,f]HofH.NowR[x,f]hasanaturalstructureasagradedring,andgr-annR[x,f]Hisdefinedtobethelargestgradedtwo-sidedidealofR[x,f]thatannihilatesH.Ontheotherhand,foragradedtwo-sidedidealBofR[x,f],theannihilatorofBinHisdefinedasannHB:={h∈H:θh=0forallθ∈B}.IsaythatanR[x,f]-submoduleofHisaspecialannihilatorsubmoduleofHifithastheformannHBforsomegradedtwo-sidedidealBofR[x,f].Thereisanaturalbijectiveinclusion-reversingcorrespondencebetweenthesetofallspecialannihi-latorsubmodulesofHandthesetofallgradedannihilatorsofsubmodulesofH.Alargepartofthispaperisconcernedwithexplorationandexploitationofthiscorrespondence.Itisparticularlysatisfac-toryinthecasewheretheleftR[x,f]-moduleHisx-torsion-free,forthenitturnsoutthatthesetofallgradedannihilatorsofsubmodulesofHisinbijectivecorrespondencewithacertainsetofradicalDate:February2,2008.2000MathematicsSubjectClassification.Primary13A35,16S36,13D45,13E05,13E10;Secondary13H10.Keywordsandphrases.CommutativeNoetherianring,primecharacteristic,Frobeniushomomorphism,tightclosure,(weak)testelement,(weak)parametertestelement,skewpolynomialring;localcohomology;Cohen–Macaulaylocalring.TheauthorwaspartiallysupportedbytheEngineeringandPhysicalSciencesResearchCounciloftheUnitedKingdom(OverseasTravelGrantNumberEP/C538803/1).12RODNEYY.SHARPidealsofR,andoneofthemainresultsof§3isthatthissetisfiniteinthecasewhereHisArtinianasanR-module.Thetheorythatemergeshassomeuncannysimilaritiestotightclosuretheory.UseismadeoftheHartshorne–Speiser–LyubeznikTheorem(seeR.HartshorneandR.Speiser[7,Proposition1.11],G.Lyubeznik[13,Proposition4.4],andM.KatzmanandR.Y.Sharp[12,1.4and1.5])topassbetweenageneralleftR[x,f]-modulethatisArtinianoverRandonethatisx-torsion-free.In§4,thistheoryofspecialannihilatorsubmodulesisappliedtoproveanexistencetheoremforweakparametertestelementsinaCohen–Macaulaylocalringofcharacteristicp.Toexplainthis,Inowreviewsomedefinitionsconcerningweaktestelements.Apw0-weaktestelementforR(wherew0isanon-negativeinteger)isanelementc′∈R◦suchthat,foreveryidealbofRandforr∈R,itisthecasethatr∈b∗ifandonlyifc′rpn∈b[pn]foralln≥w0.Ap0-weaktestelementiscalledatestelement.AproperidealainRissaidtobeaparameteridealpreciselywhenitcanbegeneratedbyhtaelements.Parameteridealsplayanimportantrˆoleintightclosuretheory,andHochsterandHunekeintroducedtheconceptofparametertestelementforR.Apw0-weakparametertestelementforRisanelementc′∈R◦suchthat,foreveryparameteridealbofRandforr∈R,itisthecasethatr∈b∗ifandonlyifc′rpn∈b[pn]foralln≥w0.Ap0-weakparametertestelementiscalledaparametertestelement.ItisaresultofHochsterandHuneke[9,Theorem(6.1)(b)]thatanalgebraoffinitetypeoveranexcellentlocalringofcharacteristicphasapw0-weaktestelementforsomenon-negativeintegerw0;furthermore,suchanalgebrawhichisreducedactuallyhasatestelement.Ofcourse,a(weak)testelementisa(weak)parametertestelement.OneofthemainresultsofthispaperisTheorem4.5,whichshowsthateveryCohen–Macaulaylocalringofcharacteristicp,evenifitisnotexcellent,hasapw0-weakparametertestelementforsomenon-negativeintegerw0.Lastly,thefinal§5establishessomeconnectionsbetweenthetheoryde