Centers of F-purity

KARLSCHWEDE

8002 lJu 01 ]CA.htam[ 1v4561.708:0viXraAbstract.Inthispaper,westudyapositivecharacteristicanalogueofthecentersoflogcanonicityofapair(R,∆).WecalltheseanaloguescentersofF-purity.Weprovepositivecharacteristicanaloguesofsubadjunction-likeresults,provenewstrongersubadjunction-likeresults,andinsomecases,liftthesenewresultstocharacteristiczero.UsingageneralizationofcentersofF-puritywhichwecalluniformlyF-compatibleideals,wegiveacharacterizationofthetestideal(whichunifiesseveralpreviouscharacterizations).Finally,inthecasethat∆=0,weshowthatuniformlyF-compatibleidealscoincidewiththeannihilatorsoftheF(ER(k))-submodulesofER(k)asdefinedbySmithandLyubeznik.

1.Introduction

Inthispaper,westudyapositivecharacteristicanalogueofcentersoflogcanonicity,anotionfromalgebraicgeometry.Centersoflogcanonicityarenaturalobjectsthatappearinthestudyofthesingularitiesoftheminimalmodelprogram.Thesecenterssatisfymany(local)geometricproperties(seeforexample[Amb98]and[Kaw98]).Weshowthatthepos-itivecharacteristicanalogueofthisnotion,whichwecallcentersofF-purity,satisfiesthesesamegeometricpropertiesand,insomecases,vastgeneralizationsofthoselocalproperties.Wethenusereductiontocharacteristicptoliftsomeofthesegeneralizationsintocharacter-isticzero.WealsolinkcentersofF-puritytoapositivecharacteristicnotionthathasbeen

studiedbefore,annihilatorsofF-stablesubmodulesofHmd

(R)(andtheirgeneralizations);alsosee[LS01]andRemark3.14.

Considerapair(X,∆)whereXisanormalaffineschemeand∆isaneffectiveQ-Weildivisor(thatisaformalsumofprimeWeildivisorswithnonnegativerationalcoefficients)suchthatKX+∆isQ-Cartier(thatis,someintegermultipleisanintegralCartierdivisor).Thecentersoflogcanonicityofapair(X,∆)arethe(possiblynon-closed)pointsQ∈Xwhere

•forNOTeveryhaveeffectivelogcanonicalQ-CartiersingularitiesdivisorGatpassingQ.

throughQ,thepair(X,∆+G)doesInparticular,onethinksofthedivisorGashavingverysmall(butpositive)coefficients.Roughlyspeaking,thecentersoflogcanonicityarethepointswherethesingularitiesofthepair(X,∆)arethemostsevere.SeeSection2.3foranalternatedefinitionofcentersoflogcanonicity.

Inpositivecharacteristic,correspondingtothenotionoflogcanonicalpairs,thereisanotionof(sharply)F-purepairs(X,∆)whereX=SpecR,RisanF-finitenormaldomainand∆isaQ-divisoronX,[HW02].Therefore,ifoneislookingforapositivecharacteristic

analogueofcentersoflogcanonicityforapair(X,∆),itisnaturaltolookforprimeidealsQ∈SpecRsuchthatthepair(X,∆+G)isnotsharplyF-pureatQforanyeffectiveQ-CartierdivisorGthatgoesthroughQ.WeformulatethisconditionslightlydifferentlyinSection4.Infact,weformulateitfortriples(R,∆,a•)wherea•isagradedsystemofideals(ai·aj⊆ai+j);see[Har05].However,oneshouldnotethattheresultsinthispaperareinterestingeveninthecasethat∆=0andai=Rforalli≥0.

However,onceoneconsidersthisnotioninpositivecharacteristic,itbecomesclearthatthisconditionisprobablynottherightconditiontoworkwith(althoughonecaneasilyshowthatcentersoflogcanonicityreducegenerically,fromcharacteristiczero,tobecomecentersofF-purity).Therightconditiontoconsider(formulatedinthecasethat∆=0)

11

episidealsI⊂Rsuchthatforanymapφ:R⊆R

pe)⊆I),seeSection3.Wecallidealsthatsatisfythisconditionuniformly

F-compatible.ThereasonforthisnameistoremindreadersoftheconnectiontothenotionofcompatiblyF-splitidealsofMehtaandRamanathan,[MR85].Inparticular,foranF-purering,uniformlyF-compatibleidealsarecompatiblyF-split.Asimpliedabove,theprimeuniformlyF-compatibleidealsarepreciselythecentersofF-purity.ThereareseveralwaystocharacterizeuniformlyF-compatibleideals:Lemma5.1,Proposition3.12.Supposethat(S,m)isanF-finiteregularlocalringofprimecharacteristicandthatR=S/Iisaquotient.SupposethatJ′⊂SisanidealcontainingIandthatJ=J′/J⊂R.FinallyletERdenotetheinjectivehulloftheresiduefield.Thenthefollowingareequivalent.(a)IisuniformlyF-compatible.

(b)Foreverye>0andeveryf∈I,thecomposition

AnnER(I)=ER/I

iszero.

ee

(c)Foreverye>0wehave(I[p]:I)⊆(J′[p]:J′).

ER⊗RR

1

pe

pe

Characterization(b)generalizestothecontextsoftriples(R,∆,a•),andcharacterization(c)generalizestothecontextofpairs(R,a•).

Asitturnsout,thenotionofuniformlyF-compatibleidealshasbeenstudiedbySmithandLyubeznikbefore,butinadualcontext(andnotforpairsortriples).Inparticular,inthecaseofacompletelocalring(R,m)withinjectivehulloftheresiduefieldE=E(R/m),uniformlyF-compatibleidealscorrespondtotheF(E)-submodulesofE.Fordetailssee[LS01,Proposition5.2]wheretheycharacterizetheannihilatorsF(E)-submodulesofEbycondition(c)above.ThenotionofF(E)-submodulesofEisageneralizationofFrobenius

d

stablesubmodulesofHm(R);see[Smi97].

d

ThequestionofwhetherthereareonlyfinitelymanyF-stablesubmodulesofHm(R)hasbeenrecentlystudiedin[Sha07]andalsoin[EH07].Usingtheirtechniques,itimmediatelyfollowsthatthereareonlyfinitelymanyuniformlyF-compatibleidealsassociatedtoatriple(R,∆,a•)ifRisalocalring;seeSection5.

Incharacteristiczero,thereareanumberoftheoremsrelatedtocentersoflogcanon-icitythatgoundertheheadingofsubadjunction.InSection7weproveanalogues(andinsomecasesvastgeneralizations)oftheseresultsforcentersofF-purity(anduniformly

2

F-compatibleideals).Inparticular,weshowthatacenterofF-purityofmaximalheightau-tomaticallycutsoutastronglyF-regularscheme;comparewithKawamata’ssubadjunctiontheorem,[Kaw98].Infact,inareducedF-finiteF-purelocalring,thecenterofF-purityofmaximalheightisthesplittingprimeasdefinedbyAberbachandEnescu;see[AE05].Wealsoprovethefollowingtheorem:

Theorem7.1.Supposethat(R,∆,a•)issharplyF-pure.Thenany(scheme-theoretic)finiteunionofcentersofsharpF-purityfor(R,∆,a•)formanF-puresub-scheme.Ananalogous(butmuchweaker)resultincharacteristiczeroisthat,foralogcanonicalpair,any(scheme-theoretic)unionofcentersoflogcanonicityformaseminormalscheme;see[Amb98].WhenweliftTheorem7.1tocharacteristiczeroweobtainthefollowing.Corollary7.3.Suppose(X,∆)isapairoverCandKX+∆isQ-Cartier.If(X,∆)isofdensesharplyF-puretype,aclassofsingularitiesconjecturallyequivalenttobeing(semi-)logcanonical,thenanyunionofcentersoflogcanonicityalsohasdenseF-puretype.Inparticular,anysuchunionhasDuBoissingularities.

WealsoshowthatseveralcommonidealsareuniformlyF-compatible.Inparticular,weshowthattheconductoridealofanF-pureringisuniformlyF-compatible;seeCorollary7.15.AresultofEnescuandHochsterimpliesthatannihilatorsofF-stablesubmodulesofdHm(R)areuniformlyF-compatible;seeRemark5.4and[EH07,Theorem4.1].Furthermore,testideals(bothfinitisticandnon-finitistic/big;seeSection2.2)areuniformlyF-compatible.Infactwecansaymore:

Theorem6.2.Givenatriple(R,∆,a•),thenon-finitistictestidealτb(∆,a•)isthesmallestuniformly(∆,a•,F)-compatibleidealwhoseintersectionwithR◦isnon-empty.

Inthecasewhere∆=0andai=RforallR,thepreviousresultwasprovenin[LS01],althoughitwasphrasedintheduallanguage.Ontheotherhand,inthecasethatRisregular,∆=0andai=b⌈ti⌉(forsomeidealbandpositiverealnumbert),thepreviousresultisverycloselyrelatedtothecharacterizationofthetestidealgivenin[BMS06].Finally,usingthetechniqueofHaraandWatanabe,see[HW02],wealsoshowthatmultiplierideals(aswellasadjointidealsandmanyotherrelatedconstructions)areuniformlyF-compatible;seeTheorem6.6.

Acknowledgements:IwouldliketothankMelHochster,MirceaMustat¸˘aandKarenSmithforseveralvaluablediscussions.IwouldalsoliketothankShunsukeTakagiforremindingmeabouttheconnectionswithsplittingprimesandalsoforexplaininghowaresultofhis(whichwasjointwithNobuoHara)couldbeusedtorelatethenon-finitistictestideal

∆

τ󰀅(∆):=AnnR0∗ERwiththebigtestidealτb(∆)ofMelHochster(seeSubsection2.2formoredetails).

2.Definitionsandnotations

Allringswillbeassumedtobenoetherianandexcellent.IfRisareducedring,thenR◦isdefinedtobetheelementsofRnotcontainedinanyminimalprimesofR.Unlessotherwisespecified,ringswillalsobeassumedtohavecharacteristicp(thatis,theycontainafieldofcharacteristicp>0).

3

IfRisaringofcharacteristicp,weuseFe:R→Rtodenotethee-iteratedFrobenius(ie.pethpower)map.WealsouseeRtodenoteRviewedasanR-moduleviathee-iterated

1

e

actionofFrobenius.IfRisreduced,thenRcanalsobeviewedasR

pe

-module,then

eF∗aM

correspondstoa

1

Definition2.6.IfSisanR-algebra,anda•isagradedsystemofidealsinR,thenwewillusea•Stodenotethegradedsystem{aiS}i≥0.

Definition2.7.[Sch07b],[Har05,Definition2.7],[Tak04a],[HW02],[HR76]SupposeRisanF-finiteringofcharacteristicp>0.Atriple(R,∆,a•)iscalledsharplyF-pureif,forasinglee>0,thereexistsanelementd∈ape−1suchthatthecomposition

ee

ξ:=F∗i◦(F∗(×d))◦Fe:R

e

F∗R

pe

e

F∗RasR

1

ifweview

R(⌈(p−1)∆⌉)⊗R

characterizationgivesamap

′

e′

)tothemapφ.Alternately,onecouldapply

,R)twice).Usingthefirst

′

TherightsideofthemapisisomorphictoR(⌈(pe−1)∆⌉)sincebothsheavesarere-flexive,andtheyarealsoisomorphicwhereverSupp(∆)isCartier(whichisoutsideofacodimension2set).Ontheotherhand,byadjointness,theleft-sideisisomorphicto

′e

F∗HomR((Fe)∗R(−⌈(pe−1)∆⌉),R(⌈(pe−1)∆⌉)).However,

′

∼=HomR(R(−pe⌈(pe−1)∆⌉),R(⌈(pe−1)∆))

′

∼=HomR(R,R(⌈(pe−1)∆⌉+pe⌈(pe−1)∆⌉))′∼=R(⌈(pe−1)∆⌉+pe⌈(pe−1)∆⌉)

e

φ:HomR(R(−⌈(pe−1)∆⌉),F∗R(⌈(pe−1)∆))→HomR(R(−⌈(pe−1)∆⌉),R).

′

HomR((Fe)∗R(−⌈(pe−1)∆⌉),R(⌈(pe−1)∆⌉))

′

since,again,allsheavesarereflexiveandisomorphicoutsideofasetofcodimensiontwo.Furthermore,sinceweobtainamap

⌈(pe−1)∆⌉+pe⌈(pe−1)∆⌉≥⌈(pe+e−1)∆⌉R(⌈(pe+e−1)∆⌉)→R(⌈(pe−1)∆⌉+pe⌈(pe−1)∆⌉).

′

′

′

′

′

e

Therefore,bycomposition,weobtainthemapψ:F∗R(⌈(pe+e−1)∆⌉)→R(⌈(pe−1)∆⌉)asdesired.Finally,notethatifφsendsanelementdto1,thensodoesψ(althoughinthatcase,bothdand1areviewedinthetotalfieldoffractions).

′

5

Lemma2.10.[Sch07b,Proposition3.3]Supposethatforsomee>0andsomed∈ape−1,

e

themapξe:R→F∗R(⌈(pe−1)∆⌉),whichsends1tod,splits.Thenforallpositiveintegersn,thereexistsdn∈apne−1suchthatthemap

ne

R→F∗R(⌈(pne−1)∆⌉)

whichsends1todn,splits.

Theproofisessentiallythesameasin[Sch07b].

e

Proof.Letususeψe:F∗R(⌈(pe−1)∆⌉)→Rtodenotethesplittingofξe.Notice,justasinRemark2.9,thatifweapplyHomR(R(−⌈(pe−1)∆⌉),

R(⌈(pe−1)∆⌉)

whichsendsdto1(atleastatthelevelofthetotalfieldoffractions).Since

pe⌈(pe−1)∆⌉+⌈(pe−1)∆⌉≥⌈(p2e−1)∆⌉

weobtainamap

e

γe:F∗R(pe⌈(pe−1)∆⌉+⌈(pe−1)∆⌉)→R(⌈(pe−1)∆⌉)

whichalsosendsdto1.

e

WethenapplythefunctorF∗

eF∗R

ψ

Definition2.13.[HW02],[Har05],[Tak04a]SupposeRisanF-finiteringofcharacteristicp>0.Atriple(R,∆,a•)iscalledstronglyF-regularif,foreveryc∈R◦thereexistse>0(equivalentlyinfinitelymanye>0)andanelementd∈ape−1suchthatthemap

R

eF∗R(⌈(pe−1)∆⌉)

whichsends1tocf,splits.

(ii)Thetriple(Rc,∆|SpecRc,a•Rc)isstronglyF-regular(hereRcisusedtodenotethe

localizationatc).Then(R,∆,a•)isalsostronglyF-regular.

Proof.Firstnotethatbyhypothesis(i),itiseasytoseethatthelocalizationmapR→Rc=R[c−1]isinjective.Nowfixc′∈R◦.Byhypothesis(ii),wecanfindane′′sothatthereexists

′′e′′

Rc(⌈(pe−1)∆|SpecRc⌉)→Rcwhichsendsc′g′tog′∈(ape′′−1Rc)givingusasplittingφ:F∗

′

′

kpe

′′

1.Notice,thatwecanwritegasg=g/cwithg∈ape′′−1sothatφsendsc′gtock(all

oftheseelementsareviewedinthetotalfieldoffractions).Thisφmustbealocalizationof

′′e′′

someψ:F∗R(⌈(pe−1)∆⌉)→Rwhichsendsc′gtocmforsomem>0.Withoutlossofgenerality,sincemakingmlargerisharmless,wemayassumethatm=p(n−1)e+···+pe+1

(n−1)e+···+pe+1

whereeisthenumberguaranteedbyhypothesis(i)sothatψsendsc′gtocp.Considerthesplittingguaranteedbyhypothesis(i).JustasintheproofofLemma2.10,

(n−1)e+···+pe+1ne

weobtainamapθ:F∗R(⌈(pne−1)∆⌉)→Rwhichsends(fc)pto1.Notethat

(n−1)ee+···+p+1

h=fp∈apne−1asintheproofofLemma2.10.SinceψisanR-homomorphism,

e′′(n−1)e+···+pe+1(n−1)e+···+pe+1

weseethatψsendshpc′gtohcp=(fc)p.NowapplythefunctorHomR(R(−⌈(pne−1)∆⌉),

andcomposewithθ.Therefore,weobtainamap

eF∗

′′+ne

R(⌈(pne+e−1)∆⌉)

′′

e′′

R

∈apne+e′′.Butthatiseasy

whichsendsc′ghpsinceasdesired.

e′′

to1.Thus,itissufficienttoshowthatghp

e′′

ghp

∈ape′′(pne−1)ape′′−1⊆ape′′(pne−1)+(pe′′−1)=apne+e′′−1.

7

󰀁

Example2.15.[LRPT06,Lemma1.1],[HW02]Suppose(X,∆)isapairandthat∆isareducedintegraldivisor.Furthersupposethat(X,∆)isF-pure.Wewishtoapplythe(naturaltransformationof)functorsHomX(OX(∆),)toasplitting

OX

e

F∗OX((pe−1)∆)

e

F∗OX((pe

OX

−1)∆)

e

F∗OX(−∆)

)wecanconcludethat(X,∆)isF-pure.

2.2.“Big”testideals.InacoursetaughtattheUniversityofMichiganinFall2007,

MelHochsterworkedoutthetheoryof“bigtestelements”and“bigtestideals”.Roughlyspeaking,thebigtestelementsarethoseelementsofR◦thatmultiplythe(non-finitistic)tightclosureofanymoduleintoitself.Fromonepointofview,thistheoryhasbeenexploredbySmithandLyubeznik[LS01]andalsobyTakagiandHara[HT04]whentheywerestudyingtheobjectτ󰀅(R):=AnnR(0∗ER)butIdonotknowofareferencewhereHochster’spointofviewisworkedout.

Definition2.16.[Tak04b],[HY03],[Har05],[HH90]SupposeRisanF-finitereducedring,X=SpecRand(X,∆,a•)isatriple.FurthersupposethatMisa(possiblynon-finitelygenerated)R-moduleandthatNisasubmoduleofM.Wesaythatanelementz∈Misin

∗∆,a•

the(∆,a•)-tightclosureofNinM,denotedNM,ifthereexistsanelementc∈R◦suchthat,foralle≫0andalla∈ape−1,theimageofzviathemap

eeF∗i◦F∗(×ca)◦Fe:M

e

M⊗RF∗R

Remark2.17.Noticethatthecompositioninthepreviousdefinitionisobtainedbysimply

e

takingthemapξ:R→F∗R(⌈(pe−1)∆⌉)whichsends1toca,andthentensoringwithM.Definition2.18.[Hoc07,LectureofSeptember17th]Anelementc∈Riscalledabigsharp

∗∆,a•

,alle≥0andalla∈ape−1onetestmultiplierforthetriple(R,∆,a•)ifforallu∈NM

ee[p]∆ee

hascazp=F∗(×c)◦F∗i◦Fe(u)∈NM.

Wesaythatsuchacisabigsharptestelementif,inaddition,c∈R◦.Definition2.19.Thesetofsharptestmultipliersforthetriple(R,∆,a•)formanideal,whichwecallthebigtestideal,anddenotebyτb(∆,a•).

Asmentionedbefore,thereisanothernon-finitistic“test-ideal”whichhasbeenheavilystudiedinthelastdecade;seeforexample[LS01]or[HT04].Thistestideal,usuallydenotedbyτ󰀅(∆,a•),isobtainedbytakingtheannihilatorofthe(∆,a•)-tightclosureofzerointheinjectivehulloftheresiduefield(assumingRisalocalring).In[Hoc07,LectureofNovember30th],Hochstershowsthatτ󰀅(R)∼=τb(R)(alsocomparewith[HH90,Proposition8.23(c)]).However,Hochster’sapproachdoesn’tworkinthecaseoftriples(X,∆,a•),sotechniquesofTakagiwillbeemployedinstead.OneshouldnotethatHochster’sargumentdoesgothroughessentiallyunchangedforpairs(R,a•)

∗∆,a•

).ThedifficultyisthatWewouldliketoprovethatτb(∆,a•)=τ󰀅(∆,a•):=AnnR(0ER

whenworkingwithtightclosureofpairs(X,∆),onedoesnothaveananalogueoftheeasy

′′

[pe][pe][pe+e]∼fact(NM)M⊗RF∗(thereisamap,butitgoesthe“wrongway”forourpurposes).eR=NM

ShunsukeTakagihasexplainedtomeanalternatewayaroundthisproblem,andwhatisrecordedfortherestofthesubsectionishisargument.FirstwerecallthefollowingresultofHaraandTakagi.Lemma2.20.[HT04,Lemma2.1]Supposethat(R,m)isanF-finitelocalringofchar-acteristicp>0andthatX=SpecR.Furthersupposethat(X,∆,a•)isatriple.Fix

(e)(e)

asystemofgeneratorsx1,...,xreofape−1foreache≥0.Thenanelementc∈R

∗∆,a•

)ifandonlyif,foranynon-zeroelementd∈Riscontainedinτ󰀅(∆,a•):=AnnR(0ER

(e)

andanypositiveintegere0,thereexistsanintegere1≥e0andR-homomorphismsφi∈

e

HomR(F∗R⌈(pe−1)∆⌉,R)fore0≤e≤e1and1≤i≤resuchthat

c=

ree1󰀁󰀁

φi(dxi)

󰀁

(e)(e)

e=e0i=1

Proof.Theproofisthesameas[HT04,Lemma2.1],alsocomaprewith[Tak06a].

Remark2.21.SinceHomlocalizeswell(andthussodoesτ󰀅(∆,a•),see[HT04]),itisnaturaltousethecharacterizationofthepreviouslemmatodefineτ󰀅(∆,a•)inthecasethatRisnotalocalring.

WearenowinapositionexplainTakagi’sargumentrelatingτb(∆)withτ󰀅(∆).BecauseIdonotknowofareferencewherethisispublished,Iincludetheargumentbelow.Alsocomparewith[Hoc07,LectureofNovember30th]and[HH90,Proposition8.23(c)].Theorem2.22.SupposethatRisaringofpositivecharacteristic,andthat(R,∆,a•)isatriple,inparticularRisF-finite.Thenτb(∆,a•)=τ󰀅(∆,a•).

9

ThefollowingargumentisduetoShunsukeTakagi.

Proof.Sincethecontainment⊆isobvious,letuschoosec∈τ󰀅(∆,a•)andalsofixe≥0.Fix

∗∆,a•

N⊆MtobeacontainmentofR-modulesandsupposethatz∈NM.Thusthereexistsd∈Randanintegere0≥0suchthatdaz∈foralle′≥e0andalla∈ape′−1.Fix

(k)(k)

asystemofgeneratorsx1,...,xrkofapk−1foreachk≥0.ByLemma2.20,thereexists

(k)k

e1≥e0andmapsφi∈HomR(F∗R(⌈(pk−1)∆⌉),R)fore0≤k≤e1and1≤i≤rksuchthat

rke1󰀁󰀁(k)(k)

φi(dxi).c=

k=e0i=1

◦

pe

′

[pe]∆NM

′

UsingthesametechniqueasinRemark2.9,eachφi

(k)

inducesamap

e

WecantensorthismapwithbothMandNtoobtainthefollowingF∗R-lineardiagramofmaps

[pe+k]∆NM[pe]∆NMe+keF∗R(⌈(pe+k−1)∆⌉)→F∗R(⌈(pe−1)∆⌉).

M

e

⊗RF∗R(⌈(pe(k)

−1)∆⌉)

Noticethatwelabeledthebottomhorizontalarrowψi.

e(k)(k)ke+k(k)(k)

Nowfixanarbitrarya∈ape−1andnotethatψisendsdxiapzptoφi(dxi)azp.Butalsoobservethat

xiap∈apk−1appe−1=apk−1ape+k−pk⊆ape+k−1.

Thus,

thisimpliesthat

(k)kedxiapzp+k

(k)

k

k

∈

[pe+k]∆NM

sincee+k≥e0.Bythecommutativityoftheabovediagram,

e(k)(k)

φi(dxi)azp

Therefore,weobtainthat

caz

asdesired.

pe

∈

[pe]∆NM.

=

rke1󰀁󰀁k=e0i=1

φi(dxi)azp∈NM

(k)(k)

e

[pe]∆

󰀁

Wealsoobtainthefollowingcorollary(whichisstraightforwardtoprovedirectlyinthe

caseofapair(R,a•)orintheclassical(non-pair)setting).

Corollary2.23.SupposethatRisaring,X=SpecRandthat(X,∆,a•)isapair.Then

󰀃∗∆,a•

τb(∆,a•)=N⊆M(N:NM)wheretheintersectionrunsoverallR-modulesN⊆M.

󰀃󰀃∗∆,a•

Proof.Itisclearthatτb(∆,a•)⊆N⊆M(N:NM).Ontheotherhand,N⊆M(N:∗∆,a•NM)⊆τ󰀅(∆,a•).󰀁

10

Remark2.24.Usingthesameargumentsasin[HT04,Proposition3.1,Proposition3.2],oneseesthattheformationofτb(∆,a•)commuteswithlocalizationandcompletion.Fur-thermore,sincetheformationofthebigtestidealcommuteswithlocalization,weseethatτb(∆,a•)∩R◦=∅.

2.3.Characteristiczerosingularities.Inthissubsection,wegiveabriefdescriptionoflogcanonicalsingularitiesandcentersoflogcanonicity.Foramorecompleteintroduction,pleasesee[Kol97]or[KM98].Wewillworkwithpairs(X=SpecR,∆)inthecharacteristiczerosetting.

Supposethat(X,∆)isapairsuchthatXisanormalaffineschemeoffinitetypeoverafieldofcharacteristiczeroandKX+∆isQ-Cartier.Byalogresolutionof(aproperbirationalmapπ:Xπisproperandbirational,󰀅X,∆),wemean

→Xthatsatisfiesthefollowingconditions.(i)

(ii)X

(iii)

The󰀅issmooth,supportofthedivisorπ−1discussionofπ∗∆isasmoothsubschemeofX

−(iv)ThedivisorSupp(∗1

∆,thestricttransformof∆),

󰀅(see[KM98]formoreπ∗(∆+KX))∪exc(π)hassimplenormalcrossings.

LogresolutionsalwaysexistifXisoffinitetypeoverafieldofcharacteristiczeroby[Hir].

Afterpickingsuchalogresolutionπ:X󰀅→X,wecanfixacanonicaldivisorKeonX󰀅X

andwemayassume(bychangingKXwithinitslinearequivalenceclassifneeded)thatπ∗KXe=KX.SinceKX+∆isQ-Cartier,thereexistsanintegerm>0suchthatm(K∗X+∆)isCartier,andsoweusethenotationπ(KX+∆)todenotetheQ-divisor1

Remark2.29.Alargeamountoftheinterestinthesesingularitieshasbeenunderthehy-pothesisthatXisprojective(andnotaffine).Inthispaperhowever,werestrictourselvestotheaffinecase.

Thereisafurthergeneralizationoflogcanonicalsingularities(withoutthenormalityhypothesis)thatwewillneed.

Definition2.30.[KSB88],[K+92],[KSS08]LetXbeanreducedaffineseminormalS2schemeandsetDtobetheeffectivedivisoroftheconductoridealonXN(thenormalizationofX).NotethatDisareduceddivisorby[GT80,Lemma7.4]and[Tra70,Lemma1.3].WesaythatXhasweaklysemi-logcanonicalsingularitiesifthepair(XN,D)islogcanonical.

3.UniformlyF-compatibleideals

Inthissection,weconsideridealsofRthataresentbackintothemselvesviaeverymap(or,

e

inthecaseofpairs,acertaincollectionofmaps)F∗R→R.WecalltheseidealsuniformlyF-compatibleinordertoremindreadersoftheconnectionwithcompatiblyF-splitsubschemes(ideals)asdefinedbyMehtaandRamanathan;see[MR85]andalsoDefinition2.12.WewillalsoseethattheseobjectsareessentiallydualtotheF(ER(k))-submodulesofER(k)asdefinedin[LS01].WewillnotrelyonLyubeznikandSmith’sofviewfortheremainderofthepaper,butwewillpointoutrelationsastheyappear.Inparticular,somestatementsinthefollowingsectionwillpartiallyoverlapwithresultsof[LS01].

Definition3.1.SupposeRisF-finiteandthat,(R,∆,a•)isatriple.WesaythatIisuniformly(∆,a•,F)-compatibleif,foreverye>0,everya∈ape−1andeveryR-linearmap

eeee

R(⌈(pe−1)∆⌉).Ifai=RforeveryR⊆F∗(aI))⊆I.NoticethatF∗(aI)⊆F∗wehaveφ(F∗

i≥0and∆=0,thenwesaysimplythatIisuniformlyF-compatible.

e

φ:F∗R(⌈(pe−1)∆⌉)→R,

Remark3.2.IfthereaderismorecomfortablewiththenotationR

pe

1

→R,wehaveφ((aI)

1

I).

e

Proof.Fixsomee>0,anelementa∈ape−1andfixamapφ:F∗R(⌈(pe−1)∆⌉)→Rsuchthatφ(a)=1.Wehavethefollowingdiagramwithsplitrows:

I

eF∗I

I

e

F∗R(⌈(pe

−1)∆⌉)

φ

12

R/IR/I

ThisimpliesthatR/IisF-pure,andinparticularthatIisaradicalideal.󰀁

Remark3.5.Itmightbetemptingtotrytoconcludethat(R/I,(a•R/I))isasharplyF-purepair,butthatdoesnotfollowsincethemapφintheproofabovedoesnotnecessarilysendeF∗IintoI.However,ifIisalsouniformlyF-compatiblesimplyfortheringR(withoutanypairortriple),and(R,a•)issharplyF-pure,thenitfollowsthat(R/I,(a•R/I))issharplyF-pure.Similarstatementscanbemadefortriples(R,∆,a•).Wealsohavethefollowinglemmas,whichwillbeusefullater.

Lemma3.6.SupposeRisF-finite,(R,∆,a•)isatripleandthat{Ii}isacollectionofuni-󰀃

formly(∆,a•,F)-compatibleideals.TheniIiandΣiIiareuniformly(∆,a•,F)-compatible.

e

φ:F∗R(⌈(pe−1)∆⌉)→R.

󰀃

Firstchoosex∈iIi.Thenbyassumption,foreverya∈ape−1,we󰀃haveφ(ax)∈Ii.Thisimplesthatforanarbitrary(butfixed)a∈ape−1,wehaveφ(ax)∈iIi.Likewise,considerxi1+...+xin∈ΣiIi(wherexij∈Iij).Thenφ(a(xi1+...+xin))=φ(axi1)+...+φ(ayin)∈ΣiIiasdesired.󰀁

Proof.Fixamap

ItisalsonotdifficulttoseethatuniformlyF-compatibleidealslocalizewell;comparewith[LS01,Proposition5.3].

Lemma3.7.SupposethatRisF-finite,I⊆Risanidealand(R,∆,a•)isatriple.FurthersupposethatT⊂Risamultiplicativesystem.IfIisuniformly(∆,a•,F)-compatible,thenT−1I⊆T−1Risuniformly(∆|SpecT−1R,(T−1a)•,F)-compatible.

e

Proof.SetS=T−1Randset∆′=∆|SpecS.Consideramapψ:F∗S(⌈(pe−1)∆′⌉)→S.

e

SinceRisF-finite,weknowthatthereexistssomemapφ:F∗R(⌈(pe−1)∆⌉)→Rsuchthatψcanberepresentedby1

1

umtn

ψ(wz).Butweknowthatψ(wz)isidentifiedwith

Thesameproofalsogivesusthefollowingstatementsubstitutingcompletionforlocaliza-tion.

Lemma3.9.SupposeIˆthatRisanF-finitelocalringandsuppose(R,∆,a•)isatriple.Let

andletusdenotebyfthe(∆ˆbeanidealofR

naturalmapf:R→Rˆ.IfI⊆Rˆisuniformly,ˆa•,F)-compatiblethenf−1(I)isuniformly(∆,a•,F)-compatible.Lemma3.10.Supposethattriple.LetIbeanidealofR

ˆRisanF-finitelocalringandsupposethat(R,∆,a•)isa

andletusdenotebyfthenaturalmapf:R→Rˆ.IfIisuniformly(∆,a•,F)-compatible,thenIRˆ⊆Risuniformly(∆ˆ,ˆa•,F)-compatible.

Proof.SinceRisF-finite,noticethatHomR(F∗eR(⌈(peˆisisomorphictoHom,R).Let{x1,...,xn−1)∆⌉),R)⊗RR

Rˆ(F∗eRˆ(⌈(peThisimpliesthattheir−1)∆ˆ⌉)ˆeimagesaregeneratorsfor}beFgeneratorsforF∗IasanR-module.

thatforeverye>0,everya∈ˆa∗eIˆasanR

ˆ-module.Butthen,itisenoughtocheckpe−1,andeveryhomomorphismφ∈

HomRˆ(F∗eRˆ(⌈(peap−1)asanR

−1)∆ˆ⌉,Rˆ),wehaveφ◦F∗e(×a)(xi)⊆Iˆ.However,again,wemaygenerate

Fthe∗e(ˆeˆ-module,bytheimagesofelementsofape−1,andsowemayassumethataabovemaybetakentobeoneoftheseelements.Butthenitbecomesclearthatφ◦F∗e(×a)(f(xi))⊆IˆsincetheresultholdsforφcomingfromHomR(F∗

eR(⌈(pe−1)∆⌉),R).󰀁WecanalsolinkuniformlyF-compatibleidealswithFedder-typecriteria(infactweuseFedder’soriginalmachinery).WefirstrecallalemmaofFedder(whichwasstatedinslightlymoregeneralcontextoriginally).Alsocomparewith[LS01,Proposition5.2].

Lemma3.11.[Fed83,Lemma1.6]SupposethatSisanF-finiteregularlocalringandR=S/I.Then,

(a)HomS(F∗eS,S)ee

generator∼=FSasanF(b)LetTbea∗

forHom∗S-moduleS(FbeanelementofS.Thentheimage∗eS,S)asanFofF∗e

S-module,andJanideal.Let

xee

T◦F(×x):F∗J⊂F∗Sunderthehomomorphism

e

(c)There∗

∗eS→SiscontainedinJifandonlyifx∈(J[pe]existsanisomorphismφ:F∗e((I[pe]:I)/I[pe:J).])→HomR(F∗

e

R,R)definedbyψ(T◦Fe(×s)whereTisanychoiceofaFeS-modulegeneratorforthemoduleHomS(F∗e

S,S∗).∗Proposition3.12.SupposethatSisanF-finiteregularringandthatR=S/I.Furthersupposethata•isagradedsystemofidealssuchthat(S,a•)and(R,

J⊆Risuniformly(

J⊆Risuniformly(

F∗eR

α

e

s∈(I[p]:I).ButthenφcanberepresentedbyT◦F∗(×as).Butas∈(J[p]:J),which

e

impliesthatφmustsendF∗(J,andsoJisuniformly(a•,F)-compatible.Conversely,supposethat

ee

eF∗R

α

J)⊆

J)⊆

F∗(×fa)eF∗RP

e

doesnotsplit.If∆=0andai=Rforalli≥0,thenwesimplysaythatsuchaP∈Xisa

centerofF-purity.

Remark4.2.Aswewillsee,thesenotionshavethemostinterestingimplicationsunderthehypothesisthatthetriple(R,∆,a•)issharplyF-pure.

Remark4.3.Supposethat(R,m)isanF-finiteF-purereducedlocalring.AberbachandEnescupreviouslydefinedthesplittingprimeofR,denotedP=P(R),tobethesetof

e

elementsc∈RsuchthatthemapR→F∗Rwhichsends1toc,doesnotsplit;see[AE05].Furthermore,theyprovedthatPisaprimeidealandiscompatiblewithlocalizationatprimescontainingP,see[AE05,Theorem3.3,Proposition3.6].Therefore,bylocalizingatPitself,weimmediatelyseethatPisacenterofF-purity.Ontheotherhand,suppose

e

therewasacenterofF-purityQwithP󰀅Q.Thenforc∈Q\\P,themapR→F∗Rwhichsends1toccannotsplit(sinceitdoesn’tsplitafterlocalizingatQ).Butthatisimpossible

15

eF∗RP(⌈(pe−1)∆|SpecRP⌉)

sinceitwouldimplythatc∈P.ThereforeweseethatPistheuniquelargestcenterofF-purityin(R,m).

Remark4.4.Inthepreviousdefinition,itisequivalenttoconsideralla∈(ape−1RP)orallf∈P.Wecandothisforthefollowingreason.Supposethatforsomea∈(ape−1RP),themap

RP

eF∗RP

e(×fa′)F∗

eF∗RP

e

F∗RP

e(×fa′)F∗

eF∗RP

splits.Thesameargumentshowsthatitisequivalenttopickf∈Pinsteadoff∈PRPCentersofsharpF-purityareanaloguesofcentersoflogcanonicityincharacteristiczero.

Thefollowingsimpleobservationgivesacertainamountofevidenceforthis.

Proposition4.5.SupposethatRisF-finiteandreducedandthat(R,∆,a•)isatriple.Thentheminimalprimesofthenon-stronglyF-regularlocusofthetriplearecentersofsharpF-purity.

Proof.SupposePissuchaminimalprime.SincebeingstronglyF-regularisalocalcon-dition,weseethat(RP,∆|SpecRP,(aRP)•)isstronglyF-regularonthepuncturedspectrum(thatis,exceptattheuniqueclosedpointP)butisnotstronglyF-regularatP.IfPisnotacenterofsharpF-purity,thenthereexistsanf∈POX,Pandana∈ape−1suchthatthecomposition

RP

eF∗RP

e

F∗RP(⌈(pe−1)∆|SpecRP⌉)

Proof.FirstsupposethatPisacenterofsharpF-purityandsomeψsendssomeelementofaPOX,PtoanelementnotinP,orinotherwords,toaunit(whichcanthenbesentto1).ButthisisimpossiblesincePisacenterofsharpF-purityfor(R,∆,a•).Conversely,supposethatPisnotacenterofsharpF-purity.Thenforsomef∈PRPsomecomposition

RP

eF∗RP

Iisalso(∆,a•,F)-compatible.

Proof.ItwillbeenoughtoshowthattheminimalassociatedprimesofIareuniformly

(∆,a•,F)-compatible,sincewecanthenuseLemma3.6.Therefore,letQbeaprimecorre-spondingtoaminimalprimarycomponentoftheprimarydecompositionofI.Noticethat󰀄byassumption

UsingthetechniqueofFedder’scriterion,wecanalsoeasilycharacterizecentersofsharpF-purityforpairs(R,a•).

Proposition4.13.SupposethatSandRareinthelemmaandthata•isagradedsystemofidealsonSsuchthatboth(S,a•)and(R,

R

e

e

1

pe

pe

P

splitsifandonlyifs(I[p]SP:ISP)󰀇P[p]SP.See[Fed83]andalso[Tak04a,Lemma3.4,Lemma3.9].

d

5.RelationstoF-stablesubmodulesofHm(R)andfinitenessofcentersof

sharpF-purity

InthissectionwewillshowthatthereareonlyfinitelymanycentersofF-purityinthecaseofanF-purelocalring,andgiveanalternateexplanationofwhythenotionswehave

d

beenconsideringareverycloselyrelatedtothetheoryofF-stablesubmodulesofHm(R)(asstudied,forexample,in[Smi97]).Infact,itisbyapplyingthemachinerythatwasusedto

d

studytheF-stablesubmodulesofHm(R)byEnescu-HochsterandSharpthatweareabletoprovethatthereareonlyfinitelymanycentersofF-purityforanF-purepair.

Lemma5.1.SupposethatRisanF-finiteringandthat(R,∆a•)isatriple.FixanidealI⊆R.Thefollowingareequivalent:

(a)Iisuniformly(∆,a•,F)-compatible.

(b)Foreverye>0andeverya∈ape−1andf∈I,thecomposition

eR(⌈(pe−1)∆⌉),R)HomR(F∗

eR,R)HomR(F∗

R/I

iszero.HerethefirstthreemapsinthecompositionaresimplyHomR(

e(×af)F∗

eF∗R

e

F∗R(⌈(pe−1)∆⌉).

(*)If,inaddition,weassumethatRisalocalringwithmaximalidealmandweuse

ERtodenotetheinjectivehulloftheresiduefieldR/m,thencondition(c)belowisalsoequivalentto(a)and(b).

(c)Foreverye>0andeverya∈ape−1,thecompositionER/I

e

ER⊗RF∗R

e(×af)F∗

iszero.

e

ER⊗RF∗R(⌈(pe−1)∆⌉)

18

Proof.BytheusualapplicationofMatlisduality,itisclearthatconditions(b)and(c)areequivalent;seeforexample[Tak04a,Lemma3.4].Ontheotherhand,Iisuniformly

e

(∆,a•,F)-compatibleifandonlyifforeverymapφ:F∗R(⌈(pe−1)∆⌉)→Randevery

e

a∈ape−1wehaveφ(F∗aI)⊆I.Butthisimpliesthatforanyf∈I,theimageofanycomposition

R

eF∗R

R

iscontainedinI,whichcertainlyimpliesthatthemapin(b)iszero.Conversely,if(a)is

e

false,thenthereexistssomea∈ape−1,f∈Iandφ:F∗R(⌈(pe−1)∆⌉)→Rsuchthatφ(af)∈/I.Butthentheassociatedcompositionfrom(b)isnon-zero.󰀁Corollary5.2.LetRbeareducedlocalring.Supposefurtherthat(R,∆,a•)issharplyF-pure.Thenthereareonlyfinitelymanyuniformly(∆,a•,F)-compatibleidealsI.Inparticular,thereareonlyfinitelymanycentersofF-purity.

Proof.Wehaveobservedpreviouslythatuniformly(∆,a,F)-compatibleidealsarereduced,seeCorollary3.4,andalsothattheyareclosedundersumandintersection,seeLemma3.6.Proposition4.10impliesthatthesetofuniformly(∆,a•,F)-compatibleidealsareclosedundertakingprimarydecomposition.Butthenitfollowsthattherecanbeatmostfinitelymanysuchidealsby[EH07,Theorem3.1]orby[Sha07].󰀁Remark5.3.Supposethat(R,m)isaquasi-Gorensteinlocal,∆=0,ai=Rforalli≥0and

d

(R).ThenweseethatIisuniformlyI⊂Risanideal.AlsosetN=AnnERI⊂ER∼=Hm

d

F-compatibleifandonlyifNisaF-stablesubmoduleofHm(R).

dRemark5.4.Supposethat(R,m)isanF-finitelocalringandsupposethatN⊆Hm(R)isan

d

F-stablesubmoduleofHm(R).Theproofof[EH07,Theorem4.1]showsthatifI=AnnRN,thenIisuniformlyF-compatible(theyonlystatethatIis“compatible”withsplittings,buttheproofofthegeneralcaseisthesame).Infact,thesameproofcanbeusedtoshowthefollowingresult(alsocomparewiththeproofofTheorem2.22).

Proposition5.5.SupposethatMisanR-moduleand(R,∆,a•)isatriple.Foralle≥0

e

andalla∈ape−1,useφe,a:R→F∗R(⌈(pe−1)∆⌉)todenotethemapwhichsends1toa.SupposethatN⊆MandthatJ=AnnRN.FinallysupposethattheimageofNunderthemap

e

γe,a:=M⊗φe,a:M→M⊗RF∗R(⌈(pe−1)∆⌉)e

isannihilatedbyF∗Jforalleanda.ThenJisuniformly(∆,a•,F)-compatible.Theargumentoftheproofisessentiallythesameasin[EH07,Theorem4.1],wesimplygeneralizeittothecontextoftriples(R,∆,a•).Weprovideitfortheconvenienceofthereader.

e

Proof.Consideramapψ:F∗R(⌈(pe−1)∆⌉)→Randnotethatψinducesamap

δ:M⊗RFeR(⌈(pe−1)∆⌉)→M⊗RR∼=M

∗

e

(ape−1J))⊆J=AnnR(N).Thereforechoosen∈N,a∈ape−1Weneedtoshowthatψ(F∗

eee

andx∈F∗J⊂F∗R⊆F∗R(⌈(pe−1)∆⌉).Wenotethatδ(n⊗ax)=n⊗ψ(ax)=ψ(ax)n∈

e

M∼R(⌈(pe−1)∆⌉)byhypothesis.󰀁=M⊗R.Butn⊗ax=(n⊗a).x=0∈M⊗RF∗

19

6.Relationstobigtestidealsandmultiplierideals

Inthissection,weshowthatbig(non-finitistic)testidealsareuniformlyF-compatibleandthatthebigtestidealisthesmallestuniformlyF-compatibleidealwhoseintersectionwithR◦isnon-trivial.Notethattheusual(finitistic)testidealisuniformlyF-compatibleby[Vas98]or,inthecaseofpairs(R,at),by[Sch07b](thesameargumentworksforpairs(R,a•)aswell).SeeSection2.2forbasicdefinitionsofbig(non-finitistic)testideals,orsee[HY03]or[LS01]foranalternatepointofview.

Proposition6.1.[LS01,Theorem6.2],[Smi95]Givenatriple(R,∆,a•),theidealτb(∆,a•)isuniformly(∆,a•,F)-compatible.

Proof.ByLemmas3.7and3.8andthefactthatthebigtestidealbehaveswellwithrespecttolocalization,weseethatitisharmlesstoassumethatRisalocalring.Likewise,byLemmas3.9and3.10andsincethebigtestidealbehaveswellwithrespecttocompletion,see[HT04],weseethatitisharmlesstoassumethatRiscomplete.Weneedtoshowthatthecomposition

ER/τb(∆,a•)

e(×af)F∗

eER⊗RF∗R

e

ER⊗RF∗R(⌈(pe−1)∆⌉)

iszeroforeverye≥0andeveryf∈τb(∆,a•).Butthisfollowsimmediatelyfromthe

∗∆,a•

=factthatτb(∆,a•)ismadeupofbigsharptestelementsfor(R,∆,a•)andthat0ER

ER/τb(∆,a•).󰀁Theorem6.2.Givenatriple(R,∆,a•),theidealτb(∆,a•)isthesmallestidealwhichisuniformly(∆,a•,F)-compatibleandwhoseintersectionwithR◦isnon-empty.

Proof.Asabove,wemayassumethatRislocal.IftheconclusionofthetheoremisfalsethenbyLemma3.6wemayassumethatthereexistssomeI󰀅τb(∆,a•)whichisuniformly(∆,a•,F)-compatibleandwhichsatisfiesI∩R◦=∅.Thisimpliesthatforeverye≥0andeveryf∈I∩R◦,thecompositionER/I

e

ER⊗RF∗R

e(×af)F∗

e

ER⊗RF∗R(⌈(pe−1)∆⌉)

iszero.ButthisimpliesthatER/Iiscontainedinthe(∆,a•)-tightclosureofzeroinER.

∗∆,a•

=τb(∆,a•),whichisimpossible.󰀁However,I=AnnRER/I⊇AnnR0ER

Corollary6.3.SupposethatRisaquotientofaregularlocalringSbyanidealIanddenotetheprojectionS→S/I=Rbyπ.Furthersupposethata•isagradedsystemofidealsofSsuchthatboth(S,a•)and(R,

Rispossiblysingular),thecharacterizationofthebigtestidealinCorollary6.3coincidespreciselywithacriteriongivenin[LS01].

InthecasethatR=S,thisgivesanalternateproofofthefollowingresultofTakagi;see[Tak06b,ProofofTheorem2.4].

Corollary6.5.SupposethatRisaregularlocalringand(R,a•)isapair.Thenτb(a•)∗a•=R.

Sincetestidealsarecloselyrelatedtomultiplierideals(see[Smi00],[Har01],[HY03],[Tak04b])itisnaturaltoaskwhethermultiplierideals(orevenmultiplierideal-likecon-structionssuchasadjointideals,see[Laz04]and[Tak06a])alwaysyielduniformly(∆,F)-compatibleideals.Aswewillsee,theanswertothisisaffirmative.

Theorem6.6.SupposeRisanormaldomainandaquotientofaregularringessentiallyoffinitetypeoveraperfectfield.SetX=SpecRandsupposethat(X,∆)isapairsuchthat

󰀅→XisaproperbirationalmapandthatKX+∆isQ-Cartier.Furthersupposethatπ:X

󰀅isnormal.IfGisanyeffectiveintegraldivisoronX󰀅suchthatπ∗Oe(⌈Ke−π∗(KX+XXX

∆)⌉+G)isnaturallyasubmoduleofOX(inparticular,thishappensifGisexceptional),

∗

thenπ∗OXe(⌈KXe−π(KX+∆)⌉+G)isuniformly(∆,F)-compatible.Theproofusesthesametechniqueastheproofofthemaintheoremof[HW02].Proof.Firstnote,thatwemayassumethatπ∗KXe=KX(asdivisors)andthatKXe(andthusKX)isanintegralWeildivisor.

e

Chooseamapφ∈HomR(F∗R(⌈(pe−1)∆⌉),R).Byusing[HW02,Lemma3.4],wemayidentifyφwithanelementf∈R(⌊(1−pe)(KR+∆)⌋).Chooser>0tobeanintegersuchthatr(KR+∆)isCartier.Thus

Thereforewecanviewfrasaglobalsectionof

fr∈R(r⌊(1−pe)(KX+∆)⌋)⊆R((1−pe)(r(KX+∆))).

1

ee∗

OXe(r⌈(1−p)e((1−p)π(r(KX+∆)))⊆OX

r

π∗(r(KX+∆))=KXe−ΣaiEi(herewemeanequalityasQ-Weildivisors).Note

thatnotalloftheEiareexceptionaldivisors;somecorrespondtocomponentsofthestrict

∗

transformof∆.AlsonotethatΣaiEi=KXe−π(KX+∆).Thus,wemayviewfasaglobalsectionof

ee

whichweviewasOXe((1−p)KX−(1−p)(⌈ΣaiEi⌉+G)).Byusing[HW02,Lemma3.4]intheoppositedirectionasbefore,wecanviewfassomeψintheglobalsectionsof

ee

HomOX(F∗OXe((1−p)(⌈ΣaiEi⌉+G)),OX)e

eee

OXe(⌈(1−p)(KXe−ΣaiEi)⌉)=OXe((1−p)⌊KXe−ΣaiEi⌋)⊆OXe((1−p)(KXe−⌈ΣaiEi⌉−G)),

AtthelevelofthefractionfieldofR(orsimplyoutsideoftheexceptionalsetofπandoutsidethesupportof∆),φandψinducethesamemap.Therefore,consideringtheaction

∗

ofψonglobalsections,weseethatπ∗OXe(⌈KXe−π(KX+∆)⌉+G)=π∗OXe(⌈ΣaiEi⌉+G)isuniformly(∆,F)-compatible.󰀁

21

e∼(F∗OX=HomOXe(⌈ΣaiEi⌉+G),OXe(⌈ΣaiEi⌉+G)).e

Insomesensewecanviewtheidealsobtainedinthiswayasageneralizationofcentersoflogcanonicity(inparticular,anycenteroflogcanonicitymaybeobtainedforanappropriatechoiceofG).Alsonotethatalloftheidealsobtainedthiswayarenecessarilyintegrallyclosed,buttherearetestidealswhicharenotintegrallyclosed.

Inparticular,ifwereducethemultiplieridealofapair(X,∆)fromcharacteristiczero,thecorrespondingidealincharacteristicp≫0isuniformly(∆p,F)-compatible(where∆pisthecorrespondingdivisorincharacteristicp).Thesameargumentworksforadjointidealsaswell,sincewehaveagreatdealoffreedominhowwechoosethedivisorG.Wealsonote,asmentionedabove,thatforanyexceptionaldivisorEiofπwithdiscrepancy≤−1,wecan

∗

chooseGsothat⌈KXe−π(KX+∆)⌉+G≥−Ei.Thusweobtainthefollowingresult.

Theorem6.7.Supposethat(X,∆)isapairwhereXisvarietyoffinitetypeoverafieldofcharacteristiczero.FurthersupposethatKX+∆isQ-Cartier.SupposethatQ∈XisacenteroflogcanonicitycorrespondingtoanexceptionaldivisorEinsomeproperbirational

󰀅→X(oranon-exceptionalcenteraswell).Thenaftergenericreductiontomapπ:X

characteristicp>0,thecorrespondingidealsQpareuniformly(Xp,∆p)-compatibleandinparticular,theyaredisjointunionsofcentersofsharpF-purityfor(Rp,∆p).

7.ResultsrelatedtoF-adjunctionandconductorideals

Inthissection,weproveanumberoflocalgeometricpropertiesofcentersofF-purity.Therehavebeenanumberofrelatedresultsinthepast,seeforexample[Vas98],[Sch07b]and[EH07,Theorem4.1].Manyoftheseresultscanalsobeprovendirectly,usingtheFedder-typecriteria,forsharplyF-purepairs(R,a•)underthehypothesisthatRisaquotientofaregularring.

Theorem7.1.Supposethat(R,∆,a•)issharplyF-pure.Thenanyfinite(scheme-theoretic)unionofcentersofsharpF-purityfor(R,∆,a•)formanF-puresub-scheme.

Proof.AunionofcentersofsharpF-purityis,bydefinition,areducedschemesuchthateachirreduciblecomponentisacenter󰀃nofsharpF-purity.LetIdenotethereducedidealdefiningthisscheme.NotethatI=i=1PiwherethePiareprimecentersofsharpF-purity.ButthenIisuniformly(∆,a•,F)-compatiblebyLemma3.6,whichimpliesthatR/IisF-purebyCorollary3.4.󰀁Remark7.2.Actually,thesameproofshowsthattheclosureofany(possibly-infinite,scheme-theoretic)unionofcentersofsharpF-purityfor(R,∆,a•)formanF-puresub-scheme.However,wedonotknowifthesetofcentersofsharpF-purityfor(R,∆,a•)canbeinfinite(inthelocalcase,itisalwaysfiniteaswehaveshown).

Corollary7.3.Suppose(X,∆)isapairoverCandKX+∆isQ-Cartier.If(X,∆)isofdensesharplyF-puretype,thenanyunionofcentersoflogcanonicityalsohasdenseF-puretype.Inparticular,anysuchunionhasDuBoissingularities.

Proof.ThemainstatementisadirectresultofTheorem7.1.ThestatementaboutDuBoissingularitiesfollowsimmediatelyfromthefactthatF-puresingularitiesareF-injectiveandfrom[Sch07a,Theorem6.1].󰀁Theorem7.4.Supposethat(R,∆,a•)issharplyF-pure.Thenanyfinite(scheme-theoretic)intersectionofcentersofsharpF-purityisaunionofcentersofsharpF-purity,andsotheintersectioncutsoutanF-puresubscheme.

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Proof.ThisimmediatelyfollowsfromLemma3.6.Alternately,ifRisaquotientofaregularlocalringand∆=0,thenonecanalsoprovetheresultusingProposition4.13.󰀁ThefollowingpropositionshowsthatcentersofsharpF-puritythemselvesforcetheexis-tenceofothercentersofsharpF-purity.

Proposition7.5.Supposethat(R,∆,a•)isatripleandthatIisauniformly(∆,a•,F)-compatibleideal.FurthersupposethatP⊃IcorrespondstoacenterofF-purityforR/I.ThenPisacenterofsharpF-purityfor(R,∆,a•)aswell.

e

Proof.Chooseanarbitrarye>0,anelementa∈ape−1,andamapγ:F∗R(⌈(pe−1)∆⌉)→R.SinceIisuniformly(∆,a•,F)-compatible,wehaveacommutativediagram

eF∗R

e

F∗R

R

R/I

wheretheverticalarrowsarethenaturalmaps.Notethatthereexistsacommutative

diagram

eF∗R/I

φ

R/I

R/P.

Butthisgivesusacommutativediagram

e

F∗R

e

F∗R

R

β

α

R/P.

ee

Notethatker(α)=F∗Pandkerβ=P,soweseethatγ(F∗(aP))⊆P,whichprovesthatPisacenterofsharpF-purityfor(R,∆,a•)sinceγwasarbitrary.󰀁

Remark7.6.InthestatementofProposition7.5,Pneednotbeprime,itsimplymustbeanidealcontainingIwhichcorrespondstoauniformlyF-compatibleidealofR/I.Theproofisunchanged.

Remark7.7.Theconversetothepreviouspropositionisnottrue.Inparticular,thereexistsaringR,anidealI⊂RandaprimeP⊃IsuchthatPisacenterofF-purityforRbutthatPdoesnotcorrespondtoacenterofF-purityforR/I.Forexample,considerthering

R=k[a,b,c]/(a3+abc−b2)=k[xy,x2y,x−y]⊂k[x,y]

wherekisaperfectfieldofpositivecharacteristic.(Geometrically,thisexampleisobtainedbytakingthetwoaxesinA2andgluingthemtogether).

Itiseasytosee,usingFedder’scriterion,thatthisringisF-pure;see[Fed83].Thesingularlocusisdefinedbytheheightoneprimeideal(a,b).ThussincedimR=2,theideal(a,b)isacenterofF-purity(sincestronglyF-regularringsarenormal).AlsonotethatR/(a,b)∼=k[c]isregular(andsoithasnocentersofF-purity).Ontheotherhand,Iclaimthattheideal

23

(a,b,c)isalsoacenterofF-purity.Toseethis,simplynotethatforanyf∈(a,b,c),wehavethat

f(a3(p

e−1)

+(abc)p

e−1

fore>0simplybecauseofthemiddleterm(abc).

Ontheotherhand,letICbetheconductoridealofRinitsnormalizationRN=k[x,y].InsideRN,theconductoridealsimplycutsoutthetwoaxes,thatisIC=(xy).Itistheninterestingtonotethattheorigin(x,y)isacenterofsharpF-purityforthepair(RN,IC)=(k[x,y],div(xy)).

ThefollowingcorollaryiscloselyrelatedtoKawamata’ssubadjunctiontheorem,[Kaw98].NotethatwedonotneedtoassumethattheambientringRisstronglyF-regular(thecharacteristicpanalogueofKawamatalogterminal).Thefollowingresultgeneralizes[AE05,Theorem4.7]totriples(R,∆,a•),tothenon-localcase,andtothesituationwhereRisnot(necessarily)thequotientofaregularring.

Corollary7.8.[AE05,Theorem4.7]Supposethat(R,∆,a•)isatripleandPisacenterofsharpF-purityfor(R,∆,a•)thatismaximalamongthecentersofF-purityfor(R,∆,a•)withrespecttoidealcontainment.ThenR/PisstronglyF-regular.

Proof.SupposethatR/PisnotstronglyF-regular,thenthereexistsaprimeQ󰀆PsuchthatQ/PisacenterofF-purityforR/P(forexample,Qcouldcorrespondtoaminimalprimeofthenon-stronglyF-regularlocusofR/P).ButthenQisacenterofsharpF-purityfor(R,∆,a•)aswell,byProposition7.5,contradictingthemaximalityofthechoiceofP.󰀁Remark7.9.Notethatthepreviousresultisnotveryinterestingunlessthetriple(R,∆,a•)issharplyF-pure.Thisisbecauseinalocalring(R,m)suchthat(R,∆,a•)isnotsharplyF-pure,themaximalcenterofF-purityfor(R,∆,a•)isthemaximalideal.

SupposethatRisaringandRNisitsnormalization.SincetheheightoneassociatedprimesoftheconductoridealofR⊂RNareamongtheminimalprimesofthenon-stronglyF-regularlocus,weseethattheintersectionoftheheightoneassociatedprimesoftheconductoridealareuniformlyF-compatible.Thissuggeststhatwemightaskthefollowingquestion:“IstheconductoruniformlyF-compatibleasanidealinR?”

Theauthordoesnotknowtheanswertothisquestion.However,wecanprovethefollowingrelatedresults.Firsthowever,wenotethefollowing:IfRisS2andseminormalthentheconductoridealisradicalandheightoneinbothRandRN,see[Tra70,Lemma1.2]and[GT80,Lemma7.4].ThusitfollowsthattheconductoridealcorrespondstoareducedintegraleffectivedivisorConXN=SpecRNandacodimension1subschemeBonX=SpecR.

Proposition7.10.SupposethatX=SpecRisseminormalandS2.LetXN=SpecRNdenotethenormalizationofXandletCdenotethe(reduced)divisorcorrespondingtothe

eeN

conductoronXN.Then,everymapF∗R→RinducesamapF∗R((pe−1)C)→RN.Furthermore,ifRisF-purethenthepair(RN,C)isF-pure.

Proof.WeseethatthecomponentsofBarecentersofF-purityforR;thereforetheconductoridealitselfIB⊆R,isuniformlyF-compatibleinR.Thisimpliesthatforeverymap

24

−b2(p

e−1)

)⊆(ap,bp,cp)

eee

e

γ:F∗R→Rwehavethefollowingdiagram.

IB

α

e

F∗IB

β

IB

eF∗R

γ

eF∗IB

).Thisgivesusacomposition

β

RN

RN.

Iftheoriginalcompositionsent1to1,thensodoesthis,whichprovesthatthepair(XN,C)isF-pure.󰀁Remark7.11.Notethatthesplittingobtainedinthepreviousproof,iflocalizedattheminimalprimesofR,againinduces(byrestriction)thesamesplittingwestartedwith,

RR.Corollary7.12.SupposethatXisanS2affineschemeoffinitetypeoverafieldofchar-acteristiczero,andsupposethatXhasdenseF-puretype.IfKXN+CisQ-CartierthenX

isweaklysemi-logcanonical.

Proof.Theproofofthisresultfollowsfromthemainresultof[HW02].OnesimplyreducesbothXanditsnormalizationXNtocharacteristicp>0.󰀁Corollary7.13.Supposethat(X,∆)isapairoffinitetypeoverafieldofcharacteristiczero,andfurthersupposethat(X,∆)hasdensesharplyF-puretype.SupposethatW⊂SpecRisascheme-theoreticunionofcentersoflogcanonicityandthatWisS2.SetWNtobethenormalizationofWandsetCWisthedivisorcorrespondingtotheconductoronWN.IfKWN+CWisQ-CartierthenWisweaklysemi-logcanonical.

Remark7.14.Inthenextcorollary,wewillusetheprocessofS2-ification;see[Gro67,Part2,Section5.10],andalso[HH94b,Section2],[Aoy83],[Har07,Section1]formoredetailson

e

thisoperation.AkeypointisthatgivenamapF∗R→RwhereRisF-finite,ifweapply

eS2

theS2-ificationfunctorthenweobtainamapF∗R→RS2whichagreeswiththeoriginalmapattheleveloftotalfieldoffractions.ThisisrelativelytoseefromthecharacterizationofS2-ificationfoundin[Gro67],butalsofollowsfromtheobservationthat

HomR(HomR(FeR,ωR),ωR)∼=FeRS2.=FeHomR(HomR(R,ωR),ωR)∼

∗

∗

∗

ThusifRisF-pure,thenitsS2-ifictionisalsoF-pure.

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Corollary7.15.IfRisareducedF-finiteringwhoseS2-ificationisseminormal(inpar-ticular,thishappensifRisF-pure),thentheconductoridealIB⊆RofthenormalizationRNisuniformlyF-compatible.

e

Proof.Supposewearegivenamapγ:F∗R→R.Byusingthepreviousremarkand

eN

Proposition7.10,weseethatthismapinducesamapF∗R→RNonthenormalization.Inparticularwehavethefollowingdiagram:

eF∗R

γ

R

RN

e

LetususeICtodenotetheidealoftheconductorviewedinRN.Noticethatγ(F∗IB)⊂R.

e′eNe

Butγ(F∗IB)=γ(F∗IC)isalsoanidealofR.Thereforeγ(F∗IB)isanidealofbothRandRNandsoitiscontainedintheconductorIB.󰀁eeNRemark7.16.IfoneknewthateverymapF∗R→RinducedamapF∗R→RNthenthepreviousargumentwouldalsoimplythattheconductorisuniformlyF-compatiblewithoutthehypothesisontheS2-ification.

8.FurtherQuestions

Thereareanumberofopenquestionsthatonecanaskaboutthistheory.

e

Question8.1.Givenafixedmapφ:F∗R→R,isthereageometricwaytointerpretwhat

e

idealsarecompatiblewiththismap(thatis,whichidealsIsatisfyφ(F∗I)⊆I)?

Question8.2.GivenanF-finiteF-purering(butnon-local),arethereonlyfinitelymanyuniformlyF-compatibleideals?

Question8.3.IstheconductoridealalwaysuniformlyF-compatible?

Question8.4.CanonegiveadirectproofthatuniformlyF-compatibleidealscorrespondtoF(E)submodulesofEwithoutpassingthroughaFeddertypecriterion?(Thismayberelativelyeasyifsetupproperly).

Question8.5.IstheintegralclosureofauniformlyF-compatibleidealstilluniformlyF-compatible?

Question8.6.Isthereageneralizationoflogcanonicalsingularities(withoutthenormalorQ-Cartierhypotheses)andapurelycharacteristiczeroproofofthefollowing?

Conjecture:Supposethat(X,∆)isalogcanonicalpair.Thenanyscheme-theoreticunionofcentersoflogcanonicityis“generalizedlogcanonical”

RecentworkofHaconanddeFernex,see[DH08],givesadefinitionofnormallogcanonicalsingularitieswithoutthehypothesisthatKX+∆isQ-Cartier,sothismaybeagoodplacetostart.

26

References

[AE05][Amb98]I.M.AberbachandF.Enescu:ThestructureofF-purerings,Math.Z.250(2005),no.4,791–806.MR2180375

F.Ambro:Thelocusoflogcanonicalsingularities,PreprintavailableatarXiv:math.AG/9806067[Aoy83][BMS06][DH08][EH07][Fed83][GT80][Gro67]

[Har01][Har05][HT04][HW02][HY03][Har94][Har07][Hir][Hoc07][HH90][HH94a][HH94b]

[HR76][Kaw98][KSB88]

(1998).

Y.Aoyama:Somebasicresultsoncanonicalmodules,J.Math.KyotoUniv.23(1983),no.1,85–94.MR692731(84i:13015)

M.Blickle,M.Mustat¸˘a

,andK.Smith:DiscretenessandrationalityofF-thresholds,arXiv:math/0607660.

T.DeFernexandC.Hacon:Singularitiesonnormalvarieties,arXiv:0805.1767.

F.EnescuandM.Hochster:Thefrobeniusstructureoflocalcohomology,arXiv:0708.0553.R.Fedder:F-purityandrationalsingularity,Trans.Amer.Math.Soc.278(1983),no.2,461–480.MR701505(84h:13031)

S.GrecoandC.Traverso:Onseminormalschemes,CompositioMath.40(1980),no.3,325–365.MR571055(81j:14030)

A.Grothendieck:El´´ementsdeg´eom´etriealg´ebrique.IV.Etude´localedessch´emasetdes

morphismesdesch´emasIV,Inst.HautesEtudes´Sci.Publ.Math.(1967),no.32,361.MR0238860

(39#220)

N.Hara:Geometricinterpretationoftightclosureandtestideals,Trans.Amer.Math.Soc.353

(2001),no.5,1885–1906(electronic).MR1813597(2001m:13009)

N.Hara:Acharacteristicpanalogofmultiplieridealsandapplications,Comm.Algebra33(2005),no.10,3375–3388.MR2175438(2006f:13006)

N.HaraandS.Takagi:Onageneralizationoftestideals,NagoyaMath.J.175(2004),59–74.

MR2085311(2005g:13009)

N.HaraandK.-I.Watanabe:F-regularandF-pureringsvs.logterminalandlogcanonicalsingularities,J.AlgebraicGeom.11(2002),no.2,363–392.MR1874118(2002k:13009)

N.HaraandK.-I.Yoshida:Ageneralizationoftightclosureandmultiplierideals,Trans.Amer.Math.Soc.355(2003),no.8,3143–3174(electronic).MR1974679(2004i:13003)

R.Hartshorne:GeneralizeddivisorsonGorensteinschemes,ProceedingsofConferenceonAlgebraicGeometryandRingTheoryinhonorofMichaelArtin,PartIII(Antwerp,1992),vol.8,1994,pp.287–339.MR1291023(95k:14008)

R.Hartshorne:Generalizeddivisorsandbiliaison,IllinoisJ.Math.51(2007),no.1,83–98(electronic).MR2346188

H.Hironaka:Resolutionofsingularitiesofanalgebraicvarietyoverafieldofcharacteristiczero.I,II,Ann.ofMath.(2)79(19),109–203;ibid.(2)79(19),205–326.MR0199184(33

#7333)

M.Hochster:Foundationsoftightclosuretheory,;ecturenotesfromacoursetaughtontheUniversityofMichiganFall2007(2007).

M.HochsterandC.Huneke:Tightclosure,invarianttheory,andtheBrian¸con-Skodatheo-rem,J.Amer.Math.Soc.3(1990),no.1,31–116.MR1017784(91g:13010)

M.HochsterandC.Huneke:F-regularity,testelements,andsmoothbasechange,Trans.Amer.Math.Soc.346(1994),no.1,1–62.MR1273534(95d:13007)

M.HochsterandC.Huneke:Indecomposablecanonicalmodulesandconnectedness,Com-mutativealgebra:syzygies,multiplicities,andbirationalalgebra(SouthHadley,MA,1992),Contemp.Math.,vol.159,Amer.Math.Soc.,Providence,RI,1994,pp.197–208.MR1266184

(95e:13014)

M.HochsterandJ.L.Roberts:ThepurityoftheFrobeniusandlocalcohomology,AdvancesinMath.21(1976),no.2,117–172.MR0417172(54#5230)

Y.Kawamata:Subadjunctionoflogcanonicaldivisors.II,Amer.J.Math.120(1998),no.5,3–9.MR16046(2000d:14020)

J.Koll´a

randN.I.Shepherd-Barron:Threefoldsanddeformationsofsurfacesingularities,Invent.Math.91(1988),no.2,299–338.MR922803(88m:14022)

27

´r:Singularitiesofpairs,Algebraicgeometry—SantaCruz1995,Proc.Sympos.PureJ.Kolla

Math.,vol.62,Amer.Math.Soc.,Providence,RI,1997,pp.221–287.MR1492525(99m:14033)

´rand14coauthors:Flipsandabundanceforalgebraicthreefolds,Soci´[K+92]J.Kollaet´e

Math´ematiquedeFrance,Paris,1992,PapersfromtheSecondSummerSeminaronAlgebraicGeometryheldattheUniversityofUtah,SaltLakeCity,Utah,August1991,Ast´erisqueNo.211(1992).MR1225842(94f:14013)

´randS.Mori:Birationalgeometryofalgebraicvarieties,CambridgeTractsinMath-[KM98]J.Kolla

ematics,vol.134,CambridgeUniversityPress,Cambridge,1998,WiththecollaborationofC.H.ClemensandA.Corti,Translatedfromthe1998Japaneseoriginal.MR16559(2000b:14018)

´cs,K.Schwede,andK.Smith:Cohen-Macaulaysemi-logcanonicalsingularitiesare[KSS08]S.Kova

DuBois,arXiv:0801.1541.

[LRPT06]N.Lauritzen,U.Raben-Pedersen,andJ.F.Thomsen:GlobalF-regularityofSchubertva-rietieswithapplicationstoD-modules,J.Amer.Math.Soc.19(2006),no.2,345–355(electronic).[Kol97]

MR2188129(2006h:14005)

[Laz04]

[LS01][MR85][Sch07a][Sch07b][Sha07][Smi95][Smi97][Smi00][Tak04a][Tak04b][Tak06a][Tak06b][Tra70][Vas98]

R.Lazarsfeld:Positivityinalgebraicgeometry.II,ErgebnissederMathematikundihrerGren-zgebiete.3.Folge.ASeriesofModernSurveysinMathematics[ResultsinMathematicsandRe-latedAreas.3rdSeries.ASeriesofModernSurveysinMathematics],vol.49,Springer-Verlag,Berlin,2004,Positivityforvectorbundles,andmultiplierideals.MR2095472(2005k:14001b)

G.LyubeznikandK.E.Smith:Onthecommutationofthetestidealwithlocalizationandcompletion,Trans.Amer.Math.Soc.353(2001),no.8,3149–3180(electronic).MR1828602

(2002f:13010)

V.B.MehtaandA.Ramanathan:FrobeniussplittingandcohomologyvanishingforSchubertvarieties,Ann.ofMath.(2)122(1985),no.1,27–40.MR799251(86k:14038)K.Schwede:F-injectivesingularitiesareDuBois,arXiv:0806.3298,2007.

K.Schwede:Generalizedtestideals,sharpf-purity,andsharptestelements,arXiv:0711.3380,toappearinMathResearchLetters.

R.Y.Sharp:GradedannihilatorsofmodulesovertheFrobeniusskewpolynomialring,andtightclosure,Trans.Amer.Math.Soc.359(2007),no.9,4237–4258(electronic).MR2309183

(2008b:13006)

K.E.Smith:TheD-modulestructureofF-splitrings,Math.Res.Lett.2(1995),no.4,377–386.

MR1355702(96j:13024)

K.E.Smith:F-rationalringshaverationalsingularities,Amer.J.Math.119(1997),no.1,159–180.MR1428062(97k:13004)

K.E.Smith:Themultiplieridealisauniversaltestideal,Comm.Algebra28(2000),no.12,5915–5929,SpecialissueinhonorofRobinHartshorne.MR1808611(2002d:13008)

S.Takagi:F-singularitiesofpairsandinversionofadjunctionofarbitrarycodimension,Invent.Math.157(2004),no.1,123–146.MR2135186

S.Takagi:Aninterpretationofmultiplieridealsviatightclosure,J.AlgebraicGeom.13(2004),no.2,393–415.MR2047704(2005c:13002)S.Takagi:Acharacteristicpanalogueofpltsingularitiesandadjointideals,arXiv:math.AG/0603460.

S.Takagi:Formulasformultiplieridealsonsingularvarieties,Amer.J.Math.128(2006),no.6,1345–1362.MR2275023(2007i:14006)

C.Traverso:SeminormalityandPicardgroup,Ann.ScuolaNorm.Sup.Pisa(3)24(1970),585–595.MR0277542(43#3275)

J.C.Vassilev:TestidealsinquotientsofF-finiteregularlocalrings,Trans.Amer.Math.Soc.350(1998),no.10,4041–4051.MR1458336(98m:13009)

DepartmentofMathematics,UniversityofMichigan,EastHall530ChurchStreet,AnnArbor,Michigan,48109

E-mailaddress:kschwede@umich.edu

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