Tight closure in graded rings

TIGHTCLOSUREINGRADEDRINGSKarenE.SmithAbstract.Thispaperfacilitatesthecomputationoftightclosurebygivinggiv-ingupperandlowerboundsonthedegreesofelementsthatneedtobecheckedforinclusioninthetightclosureofcertainhomogeneousidealsinagradedring.Dif-ferentialoperatorsareintroducedtothestudyoftightclosure,andusedtoprovethatthedegreeofanyelementinthetightclosureofahomogeneousideal(butnotintheidealitself)mustexceedtheminimaldegreeofthegeneratorsoftheideal.Briancon-Skoda-typetheoremsareusedtogiveexplicitbounds(intermsofthede-greesofthegeneratorsoftheideal)suchthatallelementsofatleastthisdegreeareinthetightclosureofthishomogeneousideal.TheseideasalsoyieldanewtestforaCohen-MacaulayringtobeF-rational(andhencearationalsingularity)intermsofitsa-invariantalone.Initsprincipalsetting,tightclosureisanoperationperformedonidealsinacom-mutative,Noetherianringofprimecharacteristic.ThisoperationwasintroducedbyHochsterandHunekein[HH1],andhashadapplicationstoseveraldisparatebutclassicalproblemsincommutativealgebrasuchastheSyzygyproblem,thelocalcohomologicalconjectures,andtheBriancon-Skodatheorems.Tightclosureappearstobegivinginformationaboutthesingularitiesofalocalring.Forexam-ple,withmildhypotheses,thepropertythatallidealsofaringaretightlyclosedimpliesthatringisnormalandCohen-Macaulay[HH1]andevenpseudo-rational[S1],whichamountstorationalsingularitiesincharacteristiczero.Tightclosurealsoshedslightonlogterminalandlogcanonicalsingularities[W][H].However,aseriousdicultyinthistheoryremains:howdoesonecomputethetightclosureofagivenidealinagivenring?Thispaperattackstheproblemofcomputingthetightclosureofhomogeneousidealsinagradedring.Becauseofthesubtleinformationtightclosureprovidesaboutboththeringandtheideal,anactualalgorithmforcomputingtightclosureseemsmuchtoomuchtohopefor.However,itisofinteresttoatleastnarrowthesearch.Inthispaper,theproblemisconfrontedfrombothends.AgenerallowerboundonthedegreesofelementsinIisproven(Theorems2.2,2.4):(withmildassumptionsonR)anyelementinIImusthavedegreestrictlylargerthanthesmallestdegreeofanyoftheminimalgeneratorsforI.Foranm-primaryidealI,twoupperboundsaregiven(Propositions3.1,3.3),suchthatelementsexceedingTheauthorissupportedbytheU.S.NationalScienceFoundation.TypesetbyAMS-TEX12KARENE.SMITHthisdegreearealwaysinI:anyelementofdegreelargerthanNisalwaysinI,whereNisthesmallerofthesumofthedegreesofaminimalsetofgeneratorsforIorofthedimensionofRtimesthelargestdegreeofanyminimalgeneratorforI.Theseboundshavebeenusefultotheauthorincomputingtightclosures.Section2dealswiththelowerbounds.Thoughtheseresultsarequiteusefulinpractice,oneofthemainpointsofthissectionistointroduceanewmethodforstudyingtightclosure.Thismethodisdierentialoperators.Inoursetting,theunionoftheendomorphismringsofRasanRpemodule,aserangesthroughallnon-negativeintegers,isaringofdierentialoperatorsonR.Theseoperatorsoperateontheequationsthatdenetightclosure(seeDenition1.1),andcanbeusedtomanipulatetheseequationstogreateect.Theauthorbelievesthatthis\dierentialoperatorpointofviewontightclosurewillhavefurtherapplications,andhopesthatdeeperconnectionswilleventuallyberevealed.ThereisanothermethodforprovingsomeoftheresultsinSection2,whichinvolvestheuseoftestelementsfortightclosure.Thetheoryoftestelementsisoneofthemostimportantanddeepestaspectsoftightclosure.Toillustratethisapproach,theproofofTheorem2.4iswrittenusingtestelements,thoughitcanalsobededucedusingdierentialoperators.Thedierentialoperatorpointofviewisself-contained;itdoesnotrequiretestelements,norindeed,anyknowledgeoftightclosurebeyondthedenition.InSection3,Briancon-Skodatypetheoremsareusedtoprovethatallformsofdegreegreaterthanacertainconstant(explicitlydescribedintermsofthedegreesofthegeneratorsoftheidealI)areinthetightclosureofI,form-primaryI.Forcomputationalpurposes,thisisquiteuseful,sinceitgivesanupperboundonthedegreesofhomogeneouselementsthatneedtobeconsideredforinclusioninthetightclosureofaparticularideal.Theresultsinthissectionactuallygivemethodsforcheckingthatallelementsofhighenoughdegreeareinthe\plusclosureofI.WhenRisadomain,thisissimplytheidealIR+\R,whereR+istheintegralclosureofRinanalgebraicclosureofitsfractioneld.Whetherornottheplusclosureisthesameasthetightclosureremainsanopenquestion,althoughthisisthecaseforidealsgeneratedbyparameters[S1](seealso[Ab],wheretheclassofidealswherethisisknowntoholdisenlarged).Theworkinthispaperwaspartiallymotivatedbythisquestion,andtheresultsofSection2and3bothoerfurtherevidencefortheequalityI=IR+\R.Asidefromtheiruseincomputingtightclosures,theresultsofSections2and3haveseveralinterestingconsequences,whicharerecordedinthenalsectionofthepaper.Forexample,wededuceasucientconditionforastandardK-algebratohavethepropertythatallparameteridealsaretightlyclosedintermsofitsa-invariantalone;seeTheorem4.1.ThisringpropertyiscalledF-rationalitybe-causeofitscloseconnectionwithrationalsingularities;indeed,bythemainresultof[S3],wededucethesametestforpseudorationalrings,andthereforeforrationalsingularitieswhenKhascharacteristiczero;see4.4.Animmediateconseque