AMSSubjectClassicationPrimary13D02secondary13H10KeywordsandphrasesIntersectionmultiplicitycanonicalelementcompleteintersectionalmostcompleteintersection1TheauthorwassupportedbyNSAGrantH982300 ID: 247326
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ACONNECTIONBETWEENTWOSETSOFCONJECTURESS.P.Dutta1(dutta@math.uiuc.edu)InthisarticlewearegoingtoprovideabriefsurveyofresultsonSerre'sconjecturesonintersectionmultiplicities,Hochster'scanonicalelementconjectureandaconnectionbetweenthesetwosetsofconjecturesaswell.Inparticularwewillformulateanewconnectionbetweencanonicalelementconjectureandaspecialcaseofaquestioninvolvingintersectionmultiplicityand2-propertyofSerre.Wewillprovidesketchesofproofsofsomeofourstatementsinthisshortarticle.Thedetailsandrelatednewresultsaregoingtobepublishedinseparatepapers.Ourmainaimistodrawreaders'attentiona)tothecurrentstatusofandrecentworkontheseconjecturesandb)toarelationbetweentwoapparentlycompletelyunrelatedsetsofproblems.Section1Serre'sConjectureonIntersectionMultiplicity.Let(R;m;K)bearegularlocalringi.e.Risaregularlocalring,misitsmaximalidealandK=R=mistheresidueeld.LetMandNbetwonitelygeneratedR-modulessuchthat`(M\nRN)1(\`"standsforlength).In[Se]Serredened(M;N)asP( 1)j`(TorRj(M;N))andconjecturedthefollowing:i)R(M;N)0,andii)R(M;N)=0ifandonlyifdimM+dimNdimR. AMSSubjectClassication:Primary13D02;secondary13H10Keywordsandphrases:Intersectionmultiplicity,canonicalelement,completeintersection,almostcom-pleteintersection1TheauthorwassupportedbyNSAGrantH98230-06-1-0026.TypesetbyAMS-TEX1 ThisisknownasSerre'sconjectureonintersectionmultiplicity.Notethatthiscon-jecturemakessenseonanylocalringprovidedoneofthemoduleshasniteprojectivedimensionoverthelocalring.Wedrop\R"fromthenotationwhenthereisnoscopeofambiguity.Serreprovedtheconjectureforequicharacteristicandunramiedregularlocalrings(theorem1,theorem2,ch.V,[Se]).Healsoprovedthefollowingresultsinthesamechapter:LetR;M;Nbeasaboveandlet`(M\nRN)1.Thena)dimM+dimNdimR(theorem3,ch.V,[Se]),b)ifi=dimR depthM depthN,thenTorj(M;N)=0forj.89;ऐi(theorem4,ch.V,[Se]),andc)ifdimM+dimN=dimR,andRisequicharacteristic,then(M;N)em(M)em(N)whereem(T)denotestheHilbert-SamuelmultiplicityofanitelygeneratedR-moduleT(complement,theorem1,ch.V,[Se]).Thisconjecturecanbedividedintothreeparts:(i)Vanishing:(M;N)=0whendimM+dimNdimR,(ii)Non-negativity:(M;N)0,and(iii)Positivity:(M;N).89;ऐ0whendimM+dimN=dimR.P.Roberts[R1],H.GilletandC.Soule[G-S]provedthevanishingpartindependently(inmid-eighties).TheirproofsextendtothelocalcompleteintersectionswhenbothMandNhaveniteprojectivedimension.Thehopeofgeneralizingthevalidityofthisconjecturetonon-regularringswhenonlyoneofthemoduleshasniteprojectivedimensionwasdashedbyacounterexampleduetoHochster,McLaughlinandthisauthor[D-H-M]intheearlyeighties.Thisexamplealsoledtocounter-examplestoseveralothermultiplicityrelatedconjectures.Inthemid-ninetiesGabber([B],[G])provedthenon-negativitypartoftheconjecture.ThepositivitypartofSerre'sconjecturehasbeenopenforalmostftyyears.Thefactthatpositivityornon-negativityimpliesvanishingwasprovedin[D1]intheearlyeighties.Inthemid-ninetiesGabber([G])cameupwithabrilliantideatoproveparti)ofthisconjecture.Thekeystepsinhisproofsarethefollowing:(a)usedeJong'stheoremonregularalterationtoreduce(R=P;R=q)toOX0(OY0;OZ0),whereX0=PNR,Y0;Z02 areclosedsubvarietiesofX0suchthatZ0isregularandsupportofTorOX0i(OY0;OZ0)PNKfori0,(b)extendaspectralsequenceargumentofSerretoreducetheproblemtoofintersectionofaclosedsubschemeofthenormalbundlecorrespondingtotheregularimbeddingZ0,!X0withthe0-sectionofthebundle,(c)useramicationofRandmoduleofthedierentialstoshowthattheclosedberEovers=[m]ofthisbundleisgeneratedbyglobalsections(highlyingenioustechnique!)and(d)establishnon-negativityofintersectionmultiplicityonvectorbundlesoverprojectiveschemesgeneratedbyglobalsections.ThisworkhasenergizedthestudyofOX(F;G),onaregularschemeXessentiallyofnitetypeoveraeldonanexcellentdiscretevaluationring,whereF;GarecoherentOX-modulessuchthat`(Hi(X;TorOXj(F;G)))1.InspiredbyGabber'sworkwestartedstudyingintersectionmultiplicityforbothlocalandglobalset-up.Firstweobservedthefollowingresult.Result1(Theorem2,D9)Rasaboveand,letP,qbetwoprimeidealsofRsuchthat`(R=(P+q))1.WriteX=SpecR,Y=Spec(R=P),Z=Spec(R=q).Let:eX!Xbetheblow-upofXatfmg,Etheexceptionaldivisorand:E!fmgtheinducedmap.SinceRisregularlocalofdimensionn,E=Pn 1K.LeteY,eZdenotetheblow-upsofYandZrespectivelyatfmg.Since`(R=(P+q))1,eY\eZE.Hence`(Hi(eX;TorOeXj(OeY;OeZ)))isnitefori,j0.ForanynitelygeneratedR-moduleM,em(M)denotestheHilbert-SamuelmultiplicityofMandforanypairM;Nofsuchmodules,P(G(M);G(N))denotesthealternatingsumofHilbertpolynomialsPTori(G(M);G(N))(t),fort0,onthegradedringG(R).Nowwestateourtheorem.Theorem.Withtheaboveset-upwehavethefollowing:(i)ifdimR=P+dimR=q=dimR,then(R=P;R=q)=O~X(O~Y;O~Z)+P(G(R=P);G(R=q))and(ii)ifdimR=P+dimR=qdimR,then(R=P;R=q)=OeX(OeY;OeZ):3 AsaCorollary,rstwededucethatincase(i),(R=P;R=q)=em(R=P)em(R=q)+O~X(O~Y;O~Z)andthenwededuce,viaSerre'sTheorem(result(c)),thatwhenRisequicharacteristic,O~X(O~Y;O~Z)0.ThisCorollaryhasbeenmentionedinFulton'sbook(Example20.4.3,[Fu]).Here,ourtheoremconnectslocalandglobaldirectlyviathetangentconeandtheblow-up~Xofthelocalringwithoutusinganyheavymachineryanditformulateswhathappensinthevanishingpartwhichiscompletelynew.Recallthatinordertoprovenon-negativityGabberrequiredthevectorbundleEoveraprojectiveschemetobegeneratedbyglobalsections(step(d)).Inournextresultwedescribewhathappensoveranyvectorbundleanddeducethatvanishingholdsevenwhenthebundleisnotgeneratedbyglobalsections.Result2(Proposition2.2[D9]).LetWbeaprojectiveschemeoveraeldandLbealocallyfreeOW-moduleofrankd.LetE=Spec(SymOW(L))andletVbeaclosedsubschemeE.Let:W!EdenotetheO-section;weidentifyWwith(W)whenthereisnoscopeofambiguity.Wehavethefollowing:i)OE(OV;OW)=R Vcd( V)td( V) 1( V),where VistherestrictionoftheuniversalquotientbundleonP(EL1)totheprojectiveclosure VofV;ii)ifdimVd,thenOE(OV;OW)=0;iii)ifV\W=,thenOE(OV;OW)=0;andiv)ifdimV=d,thenOE(OV;OW)=R Vcd( V)[ V]=R([V]).Partiv)oftheabovepropositionalongwithTheorem12.1(a)in[Fu]provideanewproofofGabber'spropositiononnon-negativityovervectorbundlesgeneratedbyglobalsections.Result3.WeusetheabovepropositionandGabber'stechniquetoproveournextthe-orem.Parta)ofthistheoremcanalsobededucedbyusingintersectioncycles,diagonal-izationandRiemann-RochTheorem([Fu-M]).Theorem(Theorem3,[D9]).LetXbeasmoothprojectivevarietyoveraperfecteldk.LetYandZbetwoclosedsubvarietiesofXsuchthatdimY+dimZdimX.Wehave4 thefollowing:a)IfdimY+dimZdimX,thenOX(OY;OZ)=0.b)ifTXisgeneratedbyglobalsectionsanddimY+dimZ=dimX,thenOX(OY;OZ)0.HereTXisthetangentbundleonX.Inournextresultbothvanishingandnon-negativityareestablishedonPnR;Rasabove.Result4(Theorem4[D9]).LetYandZbetwoclosedsubschemesofP=PdRsuchthatY\ZPds=PdK.Then(i)ifdimY+dimZdimP,OP(OY;OZ)=0and(ii)ifdimY+dimZ=dimP,OP(OY;OZ)0.Thenalresultofthissectionisconcernedwithintersectionmultiplicityover~X,theblow-upofX=Spec(R),Rregularlocalessentiallyofnitetypeoveraeldoranexcellentdiscretevaluationring.Let~Y;~Zbetwoclosedsubvarietiesof~Xsuchthat~Y\~ZPn 1K(=theberover[m]).Result2playsanimportantroleintheproofofparti)ofthisresult.Result5(partsi)andii)ofTheorem5,[D9]).Let~X;~Y;~Zbeasabove.Supposethatdim~Y+dim~Zdim~X.Wehavethefollowing:i)ifdim~X+dim~Zdim~Xor~Y\~Z=,thenO~X(O~Y;O~Z)=0,andii)Let~Y;~Zbeblow-upsofYandZ(notationsinResult2).Ifdim(G(R=P)\nG(R=q))1,thenO~X(O~Y;O~Z)0.Asacorollary(Result1andResult5)wederiveanewproofofSerre-vanishingandaproofofaspecialcaseofpositivityi.e.,(M;N)em(M)em(N),ifdim(G(M)\nG(N))1.InthelateseventiesTennison[T]provedthat(M;N)=em(M)em(N)ifdim(G(M)\nG(N))=0.i-ConjectureofSerre.LetRbearegularlocalringandletMandNbetwonitelygeneratedR-modulessuchthat`(M\nRN)1.SerredenedRi(M;N)fori.88;颐0,asP( 1)j`(TorRi+j(M;N)),5 j0andconjecturedthefollowing:i)Ri(M;N)0andii)Ri(M;N)=0ifandonlyifTorRj(M;N)=0forji.Thisconjecturetoomakessenseoveranylocalringprovidedoneofthemoduleshasniteprojectivedimensionoverthelocalring.Wedrop\R"fromthenotationwhenthereisnoscopeofambiguity.Serre[Se]provedtheconjectureintheequicharacteristiccase.Lichtenbaum[L]proveditoverunramiedregularlocalringsfori2;healsoshowedthatwhenbothMandNaretorsionfreetheconjectureisvalidfori=1.LaterHochster[Ho3]gaveacompleteproofforthiscase.Theconjectureisverymuchopenoverramiedregularlocalrings.Wehavebeenstudyingthisconjectureanditsrelationwithintersectionmultiplicityforquitesometime.Thefollowingresultwasprovedin[D4]:Theorem(Theorem3.2,[D4]).LetRbearegularlocalringofdimensionnandletMbeaCohen-MacaulaymoduleoverR.ThenforanymoduleNwith`(M\nN)1anddimM+dimN=dimR,(M;N)ispositiveifforeverypairofmodules(M0;N0)suchthatdimM0+dimN0dimRand`(M0\nN0)1,2(M0;N0)0.ThistheoremassurespositivitywhenoneofthemodulesisCohen-Macaulay,provided2isnon-negativeinthevanishingset-up.Theusualexpectationwasthatwhenevervanishingholds,positivityandi-conjecturesshouldalsobevalid.Recallthatin[D-H-M]thecounter-exampletovanishingoverahypersurface,whenonlyoneofthemoduleshadniteprojectivedimension,ledtocounter-examplesforbothpositivityandi-conjectureintheaboveset-up.In[D-H-M]Hochster,McLaughlinandthisauthorconstructedmodulesMofnitelengthandniteprojectivedimensionovera3-dimensionallocalhypersurfaceR(=k[x;y;u;v]m=(xy uv),mbeingthemaximalidealgeneratedbyx,y,uandv)withnegativeintersectionmultiplicity.Inthiscase(M;R=P)= 1whereP=(x;u)andthusvanishingfails.AfterwardsLevine[L]extendedittohigherdimensions(e.g.,R=k[x1;:::;xn;y1;:::;yn]m=xiyi)byusingK-theoretictechniques.RecentlyRobertsandSrinivas[R-S]provedtheexistenceofmanysuchexamplesofhypersurfaces(smoothcubic-surfacesinP3)andGorensteinrings(coordinateringsofPnPn)byusinglocalChern6 charactersandthelocalizationexactsequencefromK-theoryofThomasonandTrobaugh.Thefollowingresultswereprovedin[D8].Result1.Allthehypersurfaces,mentionedabove,possesspairsofmodulesMandN,bothhavingniteprojectivedimension,suchthat(M;N)=0and2(M;N)0.Result2.TheconclusionofResult1isvalidonany3-dimensionalGorensteinringwheregeneralizedvanishing(i.e.whenonlyonemodulehasniteprojectivedimension)fails.Tobehonest,itwasrathershockingtousaswe,likemanyotherresearchers,expectedi-conjecturetobevalidintheabovesituationsincevanishingpartofintersectionmul-tiplicityisvalidinsuchset-up.Duetothetheoremstatedearlier,onrelationbetweenpositivityand2,Wenowsuspectthatpositivitymaynotbevalidovercompleteintersec-tionsevenwhenbothmoduleshaveniteprojectivedimension.Section2CanonicalElementConjectureofM.Hochster(henceforthCEC).Let(A;m;K)beanoetherianlocalring(henceforth\local"willmean\noetherianlocal")ofdimensionnandletSidenotetheithsyzygyoftheresidueeldK.Letn:ExtnA(K;Sn)!Hnm(Sn)denotethedirectlimitmap(RecallthatforanymoduleM,Him(M)=lim~tExtiA(A=mt;M))andletA=n(classofidentitymaponSn).HochstercalledAthe\canonicalelement"ofthelocalringAandconjecturedthatA=0foreverysuchA[Ho2].Inelementarytermsthisconjectureassertsthefollowing:ForeveryfreeresolutionF:!Asi!Asi 1!!As0!K!0ofKandforeverysystemofparametersx1;:::;xnofA,ifisanymapfromtheKoszulcomplexK(x ;A)(fortheabovesystemofparameters)toF,liftingthenaturalsurjectionA=(x1;:::;xn)toK,thenn:Kn(x ;A)!Asnisnon-zero.In[Ho2]Hochsterintroducedseveralequivalentformsofthisconjectureandproveditforequicharacteristiclocalrings.Oneoftheearliestforms,thedirectsummandconjecturewasprovedadecadeearlierforthesameclassofrings.Theconjecturestatesthefollowing:7 DirectSummandConjecture(henceforthDSC)[Ho2].LetRbearegularlocalringandleti:R,!Abeamodule-niteextensionofR(i.e.AisanR-algebrasuchthatAisalsoanitelygeneratedR-module).ThenisplitsasanR-modulemap.Thegeometerswereawareofthevalidityofthisconjectureincharacteristic0formoduleniteextensionsovernormaldomains.Ananalysisofthisconjectureleadstothefollowing:MonomialConjecture(henceforthMC)[Ho1].LetAbealocalringofdimensionnandletx1;:::xnbeasystemofparametersofA.Then,foreveryintegert0,(x1:::xn)t62(xt+11;:::;xt+1n):In1980EvansandGrith[E-G],intheirproofofthesyzygyproblem,madeacrucialobservationintheequicharacteristiccasewhichcanbeviewedasanextensionoftheintersectiontheorem.Hochsterintroducedthisobservationasthefollowingconjecturein[H2]:ImprovedNewIntersectionConjecture([E-G],[Ho2])(henceforthINIC).LetAbealocalring.LetFbeacomplexofnitelygeneratedA-modulesF:0!Fs!!F1!F0!0suchthat`(Hi(F))1fori.88;颐0andH0(F)hasaminimalgeneratorannihilatedbyapowerofm.ThendimAs.In[Ho2]HochsterpointedoutthatCECimpliesINIC.In[D2]weprovedthereverseimplicationandthusCECisequivalenttoINIC.Overtheyearsseveralspecialcasesofthecanonicalelementconjecturehasbeenprovedandnewequivalentformshavebeenintroduced([D2],[D5],[D6],[D7],[Go],[Hei],[K]).Heitmann's[Hei]proofofDSCfordimension3isthemostnotableamongtheseresults.Goto[Go]provedMCforBuchsbaumringsandKoh[K]provedDSCfordegreepextensions.ThefollowingresultsplayanimportantroleinthestudyofCECfromtheperspectivepresentedinthisarticle.8 a)LetSidenotetheithsyzygyofK(=A=m)inaminimalfreeresolutionofA:dimA=n.Leti=imageof(classofidentitymaponSi)underthedirectlimitmapi:ExtiA(K;Si)!Him(Si).Theni=0for0in 1([D7]).RecallthatCECdemandsntobenon-zero.b)LetAbeacompletelocalnormaldomainofdimensionn.WriteA=S=PwhereSisacompleteintersectionsuchthatdimS=dimA.Let\n=HomS(A;S)|thecanonicalmoduleforA.ThenS=\nsatisesCEC([D7]).c)(StrookerandStuckrad)Notationsasinb).MCisvalidoverAifandonlyifforeverysystemofparametersx1;:::xnofS,\n6(x1;:::xn).d)MCisvalidforalllocalringsifandonlyifMCisvalidforalllocalalmostcompleteintersectionrings.[D6]e)LetAbealocalalmostcompleteintersectionringandletx1;:::xnbeasystemofparametersofA.Thenx1;:::xnsatisesMCifandonlyif`(H1(x ;A))`(A=x );herex denotestheidealgeneratedbyx1;:::;xn([D6])aswell.Section3ConnectionwithIntersectionMultiplicityandi-Property.GivenalocalringAandapairofnitelygeneratedmodulesMandNsuchthat`(M\nAN)1,weaskthefollowingquestion(Q):is`(M\nAN).88;颐`(Tor1(M;N))?itisclearthat(Q)hasobviousnegativeanswers,e.g.,whenM=N=K|theresidueeldofAorwhenM=K,N=m,themaximalidealofA.ThisfollowsfromthefactthatTorAi(K;K) siwheres=minimalnumberofgeneratorsofm([Se]).ToinvestigatethequestioninproperperspectiveweassumethatAisaregularlocalring.Inthiscase(Q)boilsdowntothefollowing:is(M;N).88;颐2(M;N)?Nowthewholemachineryofintersectionmultiplicityandi-propertymentionedinsection1comeintoplay.Wehavethefollowingpreposition:3.1Proposition.LetRbeanequicharacteristicorunramiedregularlocalringandletM;NbetwonitelygeneratedR-modulessuchthat`(M\nRN)1.Wehavethe9 following:i)ifdimM+dimNdimRanddepthM+depthN=dimR 1,then`(M\nRN)=`(TorR1(M;N)).ii)ifdimM+dimNdimRanddepthM+depthNdimR 1,then`(M\nRN)`(TorR1(M;N));iii)ifdimM+dimN=dimRanddepthM+depthN=dimR 1,then`(M\nRN).88;衰`(TorR1(M;N)).Proof.Bydenitionwehave`(M\nN) `(Tor1(M;N)=(M;N) 2(M;N).Fori),wehave(M;N)=0andTori(M;N)=0fori2(vanishing+resultb)insection1);forii),(M;N)=0and2(M;N).88;衰0(vanishing+Serre/Lichtenbaum'sresultoni);andforiii),(M;N).88;衰0andTori(M;N)=0fori2(positivity+resultb)insection1).Theabovepropositionshowsthat(Q)hasadeniteanswerinthevanishingset-upi.e.whendimM+dimNdimR.Theproblembeginswhenweconsiderthepositivityset-upi.e.whendimM+dimN=dimR.EvenwhenN=R=IandIisgeneratedbyanR-sequence,i.e.Iisacompleteintersectionideal,anydeniteanswerto(Q)isextremelydiculttoguess.3.2Thetheoreminthissectiondemonstrateswhyadeniteanswerto(Q),evenwhenoneofthemodulesisoftheformR=IwhereIisgeneratedbyanR-sequence,issodiculttocomprehend.Thisisaccomplishedbyconnectingaspecialcaseof(Q)tothemonomialconjecture.RecallthatanidealJofRiscalledanalmostcompleteintersectionidealifJisminimallygeneratedby(htJ+1)elementsandinsuchasituationwecallR=Janalmostcompleteintersectionring.Theorem.MCisvalidforalllocalringsAifandonlyifforanyregularlocalringRandforeverypairofidealsI;JofRsuchthati)Iisacompleteintersectionideal(i.e.generatedbyanR-sequence),ii)Jisanalmostcompleteintersectionideal,iii)htI+htJ=dimR10 andiv)I+Jism-primary,`(R=(I+J))`(Tor1(R=I;R=J))i.e.(Q)hasapositiveanswerforthepair(R=I;R=J).Proof.Byresultd)insection2,forvalidityofMCwecanassumethatlocalringA=S=S,whereisazero-divisorinacompleteintersectionringS;S=R=(y1;:::;yr)whereRisaregularlocalringandfy1;:::;yrgisanR-sequence.Considerasystemofparametersx1;:::;xnofA.Then,byresulte)insection2,MCisvalidoverAifandonlyif`(H1(x ;A))`(A=x ),x denotingtheidealgeneratedbyx1;:::;xn.Wecanliftx1;:::;xntoasystemofparametersx0i;:::;x0nofSandnexttoapartofasystemofparametersx00i;:::;x00nofR.Hencefx001;:::;x00ngformanR-sequence.WriteI=(x001;:::;x00n)andJ=(y1;:::;yr;).ThenTorRi(R=I;R=J)=TorSi(S=x 0;S=S)=H1(x 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