Idealizer Rings and Noncommutative Projective Geometry

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

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aIDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRYDANIELROGALSKIAbstract.Westudynoetheriangradedidealizerringswhichhaveverydi er-entbehaviorontherightandleftsides.Inparticular,weconstructnoetheriangradedalgebrasToveranalgebraicallyclosed eldkwiththefollowingprop-erties:Tisleftbutnotrightstronglynoetherian;T kTisleftbutnotrightnoetherianandT kTopisnoetherian;theleftnoncommutativeprojectiveschemeT-Projisdi erentfromtherightnoncommutativeprojectiveschemeProj-T;andTsatis esleftχdforsomed≥2yetfailsrightχ1.1.IntroductionAsageneralprinciple,ringswhicharebothleftandrightnoetherianareexpectedtohaverathersymmetricpropertiesontheirleftandtherightsides.Thethemeofthispaperistoshowthatsuchintuitionfailsquiteutterlyforcertainpropertieswhichareimportantinthetheoryofnoncommutativeprojectivegeometry.Ourmainresultisthefollowing.Theorem1.1.(Theorem8.2)Foranyintegerd≥2,thereexistsaconnected nitelypresentedgradednoetheriank-algebraT,wherekisanalgebraicallyclosed eld,suchthat(1)Tisstronglyleftnoetherian,butnotstronglyrightnoetherian;(2)T kTisleftbutnotrightnoetherian,whileT kTopisnoetherian;(3)thenoncommutativeprojectiveschemesT-ProjandProj-Thaveequivalentunderlyingcategories,butnon-isomorphicdistinguishedobjects;and(4)Tsatis esχd 1butnotχdontheleft,yetTfailsχ1ontheright.Intheremainderoftheintroduction,wewillde neandbrie ydiscussalloftherelevanttermsinthestatementofthetheoremandindicatehowtheringTisconstructed.Foramoredetailedintroductiontothetheoryofnoncommutativegeometrywhichmotivatesthestudyoftheseproperties,seethesurveyarticle[18].

IfRisak-algebra,thenRiscalledstronglyleft(right)noetherianifR kBisleft(right)noetherianforeverycommutativenoetheriank-algebraB.Thestudyofthestrongnoetherianconditionforgradedringsinparticularhasrecentlybecomeimportantbecauseoftheappearanceofthispropertyinthehypothesesofseveraltheoremsinnoncommutativegeometry.Mostnotably,ArtinandZhangshowedthatifAisastronglynoetheriangradedk-algebra,thenthesetofgradedA-moduleswithagivenHilbertfunctionisparametrizedbyaprojectivescheme[3].It

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

2DANIELROGALSKI

isnotaprioriobviousthatanynoetherian nitelygeneratedk-algebrawhichisnotstronglynoetherianshouldexist;in[13],RescoandSmallgavethe rst(ungraded)suchexample.Morerecently,theauthorshowedthatthereexistnoncommutativenoetheriangradedringswhicharenotstronglynoetherian(oneitherside)[14].Theorem1.1(1)showsthatitisalsopossibleforthestrongnoetherianpropertytofailononesideonlyofanoetheriangradedring.

Itisnaturaltosuspectthataringforwhichthenoetherianpropertyfailsaftercommutativebaseringextensionmightalsohavestrangepropertieswhentensoredwithitselforitsoppositering.Theorem1.1(2)con rmssuchasuspicion.Theexistenceofapairof nitelypresentednoetheriank-algebraswhosetensorproductisnotnoetheriananswers[6,Appendix,OpenProblem16′];ourexampleshowsthatonecaneventakethealgebrasinquestiontobeN-graded. ∞WenowexplainthethirdpartofTheorem1.1.LetA=n=0AnbeanaribtraryN-gradedk-algebra,wherekisanalgebraicallyclosed eld.Inaddition,assumethatAisconnected(A0=k)and nitelygraded(dimkAn<∞foralln≥0).TheleftnoncommutativeprojectiveschemeassociatedtoAisde nedtobethepairA-Proj=(A-Qgr,A).HereA-QgristhequotientcategoryofthecategoryofZ-gradedleftA-modulesbythefullsubcategoryofmoduleswhicharedirectlimitsofmoduleswith nitek-dimension,andA,calledthedistinguishedobject,istheimageofthemoduleAAinA-Qgr.TherightnoncommutativeprojectiveschemeProj-AofAisde nedanalogously.Themotivationforthesede nitionscomesfromthecommutativecase:ifAiscommutativenoetherianandprojA=Xisitsassociatedscheme,thenA-QgrandQchX(thecategoryofquasi-coherentsheavesonX)areequivalentcategories,andAcorrespondsunderthisequivalencetothestructuresheafOX.

TheresultofTheorem1.1(3)showsthatnoncommutativeprojectiveschemesassociatedtothetwosidesofanoncommutativenoetherianringmaywellbequitedi erent.Infact,fortheringTofthetheoremwewillseethatbothT-QgrandQgr-TareequivalenttothecategoryQchXwhereX=Pdforsomed≥2.However,Proj-Tisisomorphicto(QchX,OX),whileT-Projisisomorphicto(QchX,I)whereIisanon-locally-freeidealsheaf.

Nextwediscusstheχconditions,whicharehomologicalpropertiesofgradedringswhicharoseinArtinandZhang’sworkin[2]todevelopthetheoryofnon-commutativeprojectiveschemes.Foreachi≥0,theconnected nitelygradedk-algebraAissaidtosatisfyχiontheleft(right)ifdimkExt

indicatestheExtgroupintheungradedmodulecategory.IfAsatis esχiontheleftforalli≥0thenwesaythatAsatis esχontheleft.Theχ1conditionisthemostimportantoftheseconditions:itensuresthatonecanreconstructtheringA(inlargedegree)fromitsassociatedschemeA-Proj[2,Theorem4.5].Theotherχiconditionsfori≥2areneededtoshowthe nite-dimensionalityofthecohomologygroupsassociatedtoA-Proj[2,Theorem7.4].

Althoughtheχconditionsalwaysholdforcommutativerings,Sta ordandZhangconstructednoetherianringsforwhichχ1failsonbothsides[17].Theau-thorstudiedringsin[14]whichsatisfyχ1butfailχ2onbothsides.Theorem1.1(3)demonstratesyetmorepossiblebehaviorsoftheχconditions: rst,thatχ1mayholdononesidebutnottheotherofanoetherianring;andsecond,thatforanyd≥2thereareringswhichsatisfyχd 1butnotχd(ononeside).

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY3

Finally,webrie ydescribetheconstructionoftheringsTsatisfyingTheorem1.1.RecallthatifIisaleftidealinanoetherianringS,thentheidealizerofI,writtenI(I),isthelargestsubringofSwhichcontainsIasa2-sidedideal.Explicitly,I(I)={s∈S|Is I}.NowletSbeagenericZhangtwistofapolynomialring(see§5forthede nition),whichisanoncommutativegradedringgeneratedindegree1.LetIbetheleftidealofSgeneratedbyagenericsubspaceI1 S1withdimI1=dimS1 1.TheringT=I(I) Sisthentheringofinterestwhichwillsatisfyproperties(1)-(4)ofTheorem1.1.

Ourapproachinthispaperwillbeprimarilyalgebraic.Sincethisresearchwascompleted,thearticle[10]hasdevelopedageometricframeworkforthestudyofaclassofalgebrasquitesimilartotheoneswestudyhere.Weremarkthatmanyoftheresultsbelowcanbetranslatedintothisgeometriclanguage,whichwouldallowonetoshowthatthepropertiesofTheorem1.1holdforawiderclassofideal-izerrings.Speci cally,onecouldworkwithidealizersinsidetwistedhomogeneouscoordinateringsoverarbitraryintegralprojectiveschemes,insteadofthespecialcaseofZhangtwistsofpolynomialringsweconsiderhere.Sinceourmainpurposeistoconstructsomeinterestingexamples,wewillnotattempttobeasgeneralaspossibleandwewillpreferthesimpleralgebraicconstructions.

2.Idealizerringsandtheleftandrightnoetherianproperty

Asmentionedintheintroduction,themainexamplesofthispaperwillbecertainidealizerrings.Idealizershavecertainlyprovedusefulinthecreationofcounterex-amplesbefore,butitseemsthatinmanynaturalexamples(forexamplethosein

[12]or[16]),theidealizerofaleftidealisaleftbutnotrightnoetherianring.Sinceourintentionistocreatetwo-sidednoetherianexamples,inthisbriefsectionwewillgivesomegeneralcharacterizationsofboththeleftandrightnoetherianpropertiesforanidealizerring.

LetSbeanoetherianringwithleftidealI,andletT=I(I) S={s∈S|Is I}betheidealizerofI.In[16],Sta ordgivesasu cientconditionfortheleftnoetherianpropertyofT.Inthenextproposition,werestateSta ord’sresultslightlytoshowthatitcharacterizestheleftnoetherianpropertyincaseSisa nitelygeneratedleftT-module,whichoccursinmanyexamplesofinterest.

Proposition2.1.LetTbetheidealizeroftheleftidealIofanoetherianringS,andassumeinadditionthatTSis nitelygenerated.Thefollowingareequivalent:

(1)Tisleftnoetherian.

(2)HomS(S/I,S/J)isanoetherianleftT-module(orT/I-module)forallleft

idealsJofS.

Proof.By[16,Lemma1.2],ifHomS(S/I,S/J)isanoetherianleftT-moduleforallleftidealsJofScontainingI,thenTisleftnoetherian.Soifcondition(2)holds,thenTiscertainlyleftnoetherian.

Ontheotherhand,ifTisleftnoetherian,thensinceTSis nitelygenerated,TSisalsonoetherian.GivenanyleftidealJofS,wecanidentifytheleftT-moduleHomS(S/I,S/J)withthesubfactor{x∈S|Ix J}/JofTS,soHomS(S/I,S/J)isanoetherianT-module.

Next,wegiveacharacterizationoftherightnoetherianpropertyforidealizersofleftideals.ItisformallyquitesimilartothecharacterizationofProposition2.1,

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

4DANIELROGALSKI

andmaybeofindependentinterest.Infact,theresultappliesmoregenerallytoallsubringsofSinsideofwhichIisanideal.

Proposition2.2.LetSbeanoetherianringwithleftidealI,andletTbeasubringofSsuchthatI T I(I).Thefollowingareequivalent:

(1)Tisrightnoetherian.

(2)T/Iisarightnoetherianring,andTorS1(S/K,S/I)=(K∩I)/KIisa

noetherianrightT-module(orT/I-module)forallrightidealsKofS.

Proof.Theidenti cationofTorS1(S/K,S/I)withthesubfactor(K∩I)/KIofTTfollowsfrom[15,Corollary11.27(iii)],anditisimmediatethat(1)implies(2).

Nowsupposethatcondition(2)holds.SinceSisrightnoetherian,Tisrightnoetherianifandonlyif(JS∩T)/JisanoetherianrightT-moduleforall nitelygeneratedrightT-idealsJ—see[14,Lemma6.10]foraproofofthisinthegradedcase;theproofintheungradedcaseisthesame.LetJbeanarbitrary nitelygeneratedrightidealofT.SinceT/Iisrightnoetherian,(JS∩T)/(JS∩I)andJ/JIarenoetherianrightT/I-modules(the rstinjectsintoT/I,andJsurjectsontothesecond.)Then(JS∩T)/JisrightnoetherianoverTifandonlyif(JS∩I)/JIis.By[15,Corollary11.27(iii)]andthefactthatJSI=JI,wemayidentify(JS∩I)/JIwithTorS1(S/JS,S/I),whichisanoetherianrightmoduleoverTbyhypothesis.ItfollowsthatTisarightnoetherianring.

3.NoncommutativeProjofGradedidealizerrings

Startingwiththissection,wefocusourattentiononidealizerringsinsidecon-nected nitelygradedk-algebrasinparticular.Our rsttaskistostudytheprop-ertiesoftheleftandrightnoncommutativeschemesassociatedtosuchidealizerrings,andsowebeginwithareviewofsomeoftherelevantde nitions.

Below,Awillalwaysbeaconnected nitelygradedk-algebra,andwewriteA-GrforthecategoryofallZ-gradedleftA-modules.AmoduleM∈A-Griscalledtorsionifforeverym∈Mthereissomen≥0suchthat(A≥n)m=0.LetA-TorsbethefullsubcategoryofA-Grconsistingofthetorsionmodules,andde neA-QgrtobethequotientcategoryA-Gr/A-Tors,withquotientfunctorπ:A-Gr→A-Qgr.ForaZ-gradedA-moduleMwede neM[n]foranyn∈ZtobeMasanungradedmodule,butwithanewgradinggivenbyM[n]m=Mn+m.TheshiftfunctorM→M[1]isanautoequivalenceofA-GrwhichnaturallydescendstoanautoequivalenceofA-Qgrwecalls,thoughweusuallywriteM[n]insteadofsn(M)foranyM∈A-Qgrandn∈Z.

Ingeneral,anycollectionofdata(C,F,t)whereCisanabeliancategory,FisanobjectofC,andtisanautoequivalenceofCiscalledanArtin-Zhangtriple.ForeveryconnectedgradedringAthedata(A-Qgr,πA,s)givessuchatriple.Anisomorphismoftwosuchtriplesisanequivalenceofcategorieswhichcommuteswiththeautoequivalencesandunderwhichthegivenobjectscorrespond;see[2,p.237].Forexample,ifAisaconnectedgradedcommutativeringandX=projAistheassociatedscheme,thenbyatheoremofSerreonehasthat(A-Qgr,πA,s)isisomorphicto(QchX,OX, O(1)).Motivatedbythis,foranyconnectedgradedringAonecallsthepairA-Proj=(A-Qgr,πA)theleftnoncommutativeprojectiveschemeassociatedtoA,theobjectπAthedistinguishedobject,andtheautoequivalencesofA-Qgrthepolarization.Wede neanalogouslytheright-sidedversionsQgr-A,Proj-A,etceteraofallofthenotionsabove.

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY5

Ouranalysisofthenoncommutativeschemesforidealizerringswillberestrictedtoringswhichsatisfythefollowinghypotheses,whichwillholdforalargeclassofexampleswestudylater.

Hypothesis3.1.Letkbea eld.LetSbeanoetherianconnected nitelyN-gradedk-algebra,letIbesomehomogeneousleftidealofSsuchthatdimkS/I=∞,andputT=I(I).AssumeinadditionthatTSisa nitelygeneratedmodule,andthatdimkT/I<∞.

Undertheassumptionsof3.1,

boththeleftandrightnoncommutativeschemesfortheidealizerringTarecloselyrelatedtothosefortheringS,asweseenow.Lemma3.2.AssumeHypothesis3.1.

(1)Thereisanisomorphismoftriples(S-Qgr,πI,s)～=(T-Qgr,πT,s).

(2)Thereisanisomorphismoftriples(Qgr-S,πS,s)～=(Qgr-T,πT,s).

Proof.(1)SupposethatM∈S-Gr.ThenweclaimthatifTM∈T-Tors,thenSM∈S-Tors.Toprovethisfact,note rstthatifTMis nitelygenerated,thenMif nite-dimensionaloverk,soobviouslySM∈S-Tors.Ingeneral,TMisadirectlimitof nite-dimensionalT-modules,soM′=S TMisadirectlimitof nite-dimensionalS-modulesandthusM′∈S-Tors.SincethereisanS-modulesurjectionM′→M,thiscompletestheproofoftheclaim.

Nowwede netwofunctorsbytherules

F:T-Gr →S-Gr

→S(I TM)TM

G:S-Gr →T-Gr

→TNSN

togetherwiththeobviousactionsonmorphisms.IfTM∈T-Gr,thensincedimkT/I<∞itfollowsbycalculatingusingafreeresolutionofMthatTor

F:T-Qgr→S-Qgr.Similarly,itisclear

thatifN∈S-TorsthenG(N)=N∈T-Tors.ThenG′=π G:S-Gr→T-QgrisanexactfunctorwithG′(N)=0forallN∈S-Tors,soG′descendstoafunctor

G

F

Fand～F(πT)=πI,andallofthemapsarecompatiblewiththeshiftfunctors

s,sinceFandGarecompatiblewiththeshiftfunctorsinthecategoriesS-GrandT-Gr.

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

6DANIELROGALSKI

(2)BecauseSI=I T,wehave(S/T)I=0andsosinceT/Iis nitedimen-sionalweseethat(S/T)Tistorsion.ByassumptionwealsoknowthatT(S/T)is nitelygenerated.Nowtheproofofthistripleisomorphismisentirelyanal-ogoustotheproofof[17,Proposition2.7],withtheexceptionthattheauthorsassumetherethatTisnoetherianandthenprovetherequiredequivalenceforthesubcategoriesofnoetherianobjects.Weleaveittothereadertomaketheob-viousadjustmentstotheprooftoshowwithoutthenoetherianassumptionthat(Qgr-S,πS,s)～ =(Qgr-T,πT,s).

Remark3.3.ThegradedidealizerringsstudiedbySta ordandZhangin[17]havethespecialpropertythattheidealIisaprincipalidealgeneratedbyanelementofdegree1inagradedGoldiedomainS.Inthatcase,T=I(I)isisomorphictoitsoppositering,andthusthedi erencesbetweenparts(1)and(2)ofProposition3.2mustdisappear(indeed,inthiscaseπI～=πS[ 1]).Inthegeneralcase,however,itisclearfromProposition3.2thatweshouldexpectthenoncommutativeschemesT-ProjandProj-Ttobenon-isomorphic.

TheinformationprovidedbyLemma3.2willallowustoprovewitheaseseveralfurtherresultsaboutthenoncommutativeprojectiveschemesofidealizerrings.First,wemayshowinwidegeneralitythatpassingtoaVeroneseringofTdoesnota ecttheassociatednoncommutativeprojectiveschemes.Recall thatforanN-gradedringAthenthVeroneseringofAisthegradedringA(n)=∞

i=0Ain.

Proposition3.4.AssumeHypothesis3.1,andinadditionletSbegenerated ∞in′(n)′(n)′(n)degree1.Choosen≥1andwriteT=T,S=S,andI=I=i=0Iin.′′′′LetR SbetheidealizeroftheleftidealIofS.

(1)T′andR′areisomorphicinlargedegree.

(2)ThereareisomorphismsofnoncommutativeprojectiveschemesT-Proj～=′′T-ProjandProj-T～=Proj-T.

Proof.(1)Asungradedrings,wemayidentifyR′,T′andS′withsubringsofS.′Supposethatx∈(Rm),sothatI′x I′.ThensincetheleftidealIofSisgeneratedinsome nitedegree,weseethatintheringSwehave(I≥p)x Iforsomep≥0,wherex∈Snm.SincethetorsionsubmoduleofS(S/I)is nitedimensional,ifm 0thenIx Iandhencex∈T.ThenasanelementofS′,x∈T′.SincetheinclusionT′ R′isobvious,T′andR′mustagreeinlargedegree.

(2)SinceTSis nitelygeneratedanddimkT/I<∞,weseethatT′S′is nitelygeneratedanddimkT′/I′<∞.ThenbecauseT′andR′agreeinlargedegreebypart(1),itfollowsthatR′S′is nitelygeneratedandthatdimkR′/I′<∞.Also,sinceSisnoetherian,S′mustbenoetherian[2,Proposition5.10(1)].

Nowweclaimthatwehaveisomorphismsofnoncommutativeprojectiveschemes(Qgr-T,πT)～=(Qgr-S,πI)～=(Qgr-S′,πI′)～=Qgr-(R′,πR′)～=Qgr-(T′,πT′).Toseethis,notethatsinceSisgeneratedindegree1,thereisanisomorphismProj-S～=Proj-S′[2,Proposition5.10(3)];theassociatedequivalenceofcategoriesQgr-S Qgr-S′sendsπItoπI′.Thesecondisomorphismfollows,andthe rstandthirdfollowfromProposition3.2(1),appliedtoT SandtoR′ S′,st,the nalisomorphismfollowsfrompart(1).AltogetherthischainofisomorphismssaysthatProj-T～=Proj-T′.

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY7

Theargumentontheleftsideisverysimilar,exceptusingtheothertripleisomorphismofProposition3.2,andislefttothereader.

Nextwewillshowthatundermildhypothesesthenoncommutativeprojec-tiveschemesassociatedtoSandT(oneitherside)havethesamecohomo-logicaldimension;wereviewthede nitionofthispropertynow.CohomologygroupsiforthenoncommutativeprojectiveschemeA-Projarede nedbysettingH(M)=Exti

ofA-ProjA-Qgr(πA,M)forallM∈A-Qgr.Thenthecohomologicaldimen-sionis

cd(A-Proj)=max{i|Hi(M)=0forsomeM∈A-Qgr}

andtheglobaldimensionofthecategoryA-Qgris

gd(A-Qgr)=max{i|ExtiA-Qgr(M,N)=0forsomeM,N∈A-Qgr}.

Theright-sidedversionsofthesenotionsarede nedsimilarly.

Proposition3.5.AssumeHypothesis3.1.

(1)cd(Proj-T)=cd(Proj-S).

(2)AssumeinadditionthatSisadomainwithgd(S-Qgr)=cd(S-Proj)<∞.

Thencd(T-Proj)=cd(S-Proj).

Proof.(1)ThispartisimmediatefromthetripleisomorphismofProposition3.2(2).

(2)Byproposition3.2(1),wehavetheisomorphismoftriples(T-Qgr,πT,s)～=(S-Qgr,πI,s).Fromthisitquicklyfollowsthat

cd(T-Proj)≤gd(T-Qgr)=gd(S-Qgr)=cd(S-Proj).

Letd=cd(S-Proj).To nishtheproofthatcd(T-Proj)=cd(S-Proj)wehaveonlytoshowthatthereissomeF∈S-QgrsuchthatExtd

choosesomeinjectionS[ m]→SI-Qgr(πI,F)=0.Since

Sisadomain,wemayforsomem≥0,andpassingtoS-Qgrwehaveashortexactsequence0→πS[ m]→πI→N→0forsomeN.SinceS-ProjhascohomologicaldimensiondwemaychoosesomeF∈S-QgrwithExtdS-Qgr(πS[ m],F)=0.ButExtdS+1-Qgr(N,F)=0sincetheglobaldimensiondofS-Qgrisd,soweconcludefromthelongexactsequenceinExtthatExtS-Qgr(πI,F)=0.

4.Theχconditionsforgradedidealizers

Thegoalofthissectionistobeginananalysisoftheχconditions,whichwede nedintheintroduction,forthecaseofgradedidealizerringsTsatisfyingHy-pothesis3.1.ThemainresultbelowwillshowthatifSitselfsatis esleftχ,thentheleftχconditionsfortheidealizerringTmaybecharacterizedintermsofhomo-logicalalgebraoverSonly.WealsostudytherightχconditionsforT;theanalysisoftheseturnsouttobeamuchsimplermatter.

Wereviewseveralde nitionswhichwewillneedbeforeprovingthemainre-sultofthissection.AmoduleM∈A-GrisrightboundedifMn=0forn 0,leftboundedifMn=0forn 0,andboundedifitisbothleftandrightbounded.Mis nitelygradedifdimkMn<∞foralln∈Z.ForM,N∈A-Gr,HomA(M,N)meansthegroupofdegree-preservingmodulehomo-morphisms,andExtiA(M, )istheithrightderivedfunctorofHomA(M, ).WealsosetHom

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

8DANIELROGALSKI

homomorphismsintheungradedcategoryifMis nitelygenerated.Moregener-ally,wewriteExt

i

A-grbethesubcategoryofAall-Qgr(M,N)=

noetherianmodules n∈ZExti

inAA-Qgr(M,N[n]).Finally,let

-Gr.

Notethatwehavede nedtheχconditionsfornotnecessarilynoetherianalge-bras;itiseasytoprove,however,thattheleftχ0conditionforaconnectedgradedringAisequivalenttotheleftnoetherianpropertyforA.RecallalsothatifAisconnectedgradedleftnoetherianwithmodulesM∈A-grandN∈A-Gr,thenforanyj≥0wehaveExtj

A(M≥n,N)[2,Propo-

sition7.2(1)].Inparticular,inthiscasethereisanaturalmapofvectorspacesExtj

A-Qgr(πM,πN).Intheproofofthefollowingpropositionwe

willuseseveralresultsofArtinandZhangfrom[2]whichinterprettheχconditionsintermsofthepropertiesofsuchmaps.

Proposition4.1.AssumeHypothesis3.1,andassumealsothatSsatis esχontheleft.ThenTsatis esχiontheleftforsomei≥0ifandonlyifdimkExt

S(S/I,M)isanoe-

therianleftT/I-module(equivalently,of nitek-dimension)forallM∈S-gr.SincetheleftnoetherianpropertyforTisequivalenttoleftχ0forT(asweremarkedbeforetheproposition),thecharacterizationofthepropositionholdswheni=0.

NowassumethatTisleftnoetherian.Thereisanisomorphismoftriples(S-Qgr,πI,s)～=(T-Qgr,πT,s)byProposition3.2(1).ForanyM∈S-grwehaveadiagram

MγHom

HomHom

.Itis

straightforwardtocheckthatthisdiagramcommutes.NowsinceShasχ,themapβisanisomorphisminlargedegree[2,Proposition3.5(3)].Furthermore,χ1holdsontheleftforTifandonlyifthemapαhasrightboundedcokernelforallM∈T-gr[2,Proposition3.14(2a)].NotethatitisequivalenttorequirethatαhaveboundedcokernelforallM∈S-gr,asfollows:ifM∈T-gr,thenIM∈S-grwithdimkM/IM<∞andthusπM=πIM;conversely,ifM∈S-grthenM∈T-grsinceTSis nitelygeneratedandTisleftnoetherian.Thusfromthediagramitfollowsthatχ1holdsforTontheleftifandonlyifγhasrightboundedcokernelforallM∈S-gr.ButthecokernelofγisExt

j(πM)=Ext

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY9

torequirethisconditionforallM∈S-gr.NowforeveryM∈S-grandj≥1wehaveasequenceofmaps

ExtExtExtExt

,thenat-

uralmapαisanisomorphisminlargedegreesinceSsatis esχ[2,Proposition

3.5(3)],andthe nalisomorphismcomesfromtheisomorphismoftriplesinPropo-sition3.2(1).Inaddition,Ext

j

S(S/I,M)isrightbounded

(equivalently, nitedimensionaloverksinceitisalwaysleftboundedand nitelygraded)forall2≤j≤iandallM∈S-gr.Thisprovesthecharacterizationofχifori≥2,andconcludestheproofoftheproposition.

IncontrasttoProposition4.1,ontherightsideonlytheχ0conditionforT(equivalently,therightnoetherianpropertyforT)ispotentiallysubtletoanalyze.Thehigherχconditionsautomaticallymustfail,asfollows.

Proposition4.2.AssumeHypothesis3.1.ThenTfailsχiontherightforalli≥1.

Proof.WemayassumethatTisrightnoetherian,sinceotherwiserightχ0failsforTandsobyde nitionrightχifailsforalli≥0.Also,weneedonlyshowthatTfailsrightχ1.Forthis,thesameargumentoutlinedin[17,p.424]workshere;sinceitissimplewebrie yrepeatit.Byhypothesis,wehaveSI=I,dimkT/I<∞,anddimkS/I=∞.Sothenaturalmap

T→HomQgr-T(πI,πI)

hasacokernelwhichisnotrightbounded,sinceS Hom

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

10DANIELROGALSKI

NowwewillidealizeleftidealsofSwhicharegeneratedbyacodimension-1subspaceoftheelementsofdegree1.Speci cally,fromnowonwewillconsiderthefollowinghypothesisandnotations.

Hypothesis5.1.Letkbeanalgebraicallyclosedbase eld.Choosesomed≥2,apointc∈Pd,andanautomorphism ∈AutPd.LetφbeagradedautomorphismofU=k[x0,...,xd]suchthat isthecorrespondingautomorphismofprojU=Pd,andde neS=S( )tobetheleftZhangtwistofU=k[x0,...,xd]bytheautomorphismφ.(Althoughtheautomorphismφcorrespondingto isdeterminedonlyuptoscalarmultiple[7,Example7.1.1],itiseasytocheckthatchangingφbyanonzeroscalardoesnotchangetheringSuptoisomorphism.)LetIbetheleftidealofSconsistingofallhomogeneouselementsvanishingatthepointc.De neT=T( ,c)=I(I) S.Alsowritecn= n(c)forn∈Z.

Ingeneral,thepropertiesoftheringT=T( ,c)dependonthepropertiesoftheorbitC={cn}n∈Z.Wearemostinterestedinthe“generic”case,andsowewillusuallyassumeatleastthatCisin nite.Undersuchanassumption,weseenextthattheidealizerringsThavethefollowingbasicproperties.

Lemma5.2.AssumeHypothesis5.1.Ifthepoints{cn}n∈Zarealldistinct,then

(1)T=k+I.

(2)T(n)isnotgeneratedindegree1foranyn≥1.

(3)dimk(S/IS)<∞.

(4)TSis nitelygenerated.

(5)Tisa nitelygeneratedk-algebra.

(6)Hypothesis3.1issatis ed.

Proof.(1)WehaveTn={x∈Sn|Ix I}.Ifφn(I)=I,thensinceIisprimeinU,φn(I) x Iforcesx∈I.Sinceweassumethatchasin niteorderunder ,φn(I)=Iforalln=0andsoTn=Inforn≥1.

(2)IfT(n)weregeneratedindegreeoneforsomen≥1,thenwouldwehaveTnTn=T2n,whichinthecommutativeringUtranslatestoφn(I)n In=I2n.SinceIandφn(I)aredi erenthomogeneousprimeidealsofUwhicharegeneratedindegree1,itiseasytoseethatsuch anequationisimpossible. ∞∞(3)SetJ=IS.WehavethatJ=i=0ISi=i=0φi(I) Ui.Sincethepoints{ci}arealldistinct,itisclearthatthevanishingsetoftheidealJinPdisempty.ThusdimkU/J<∞bythegradedNullstellensatz;equivalently,dimk(S/IS)<∞.

(4)BythegradedNakayamalemma,ak-basisofS/T≥1S=S/ISisaminimalgeneratingsetforTS,so(4)followsimmediatelyfrom(3).

(5)Tisgeneratedasak-algebrabysomeelementsti∈T≥1ifandonlyifT≥1isgeneratedasleftT-idealbytheti;sotoprove(5)wejustneedtoshowthatTIis nitelygenerated.Sincebypart(4)weknowthatTSis nitelygenerated,wehaveTS≤n=Sforsomen≥0.ThenTS≤nT1=ST1=Iisa nitelygeneratedleftT-module.

(6)SincedimkIn=dimkSn 1foralln≥1,itisclearthatdimkS/I=∞.Theothernecessarypropertiesfollowfrom(1)and(4).

Thenoetherianpropertyontheleftisalsostraightforwardtoanalyze.

Proposition5.3.AssumeHypothesis5.1,andthatthepoints{cn}n∈Zarealldistinct.ThenTisleftnoetherian.

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY11

Proof.WehavethatT=k+IandthatTSis nitelygenerated,byLemma5.2.ThusthehypothesesofTheorem2.1aresatis edandtoshowthatTisleftnoe-therianweneedtoshowthatHom

S(S/I,S/J)is nitedimensional,sowealsomayassumethatJ=U≥1.

Nowwemaymaketheidenti cationofvectorspaces

Hom

S(S/I,S/J)n=0forn 0.ThusHom

iS(S/I,S/J)n～=Ext

dS(S/I,S)～

dn=

(U/I,U)～Ext

=(U/I)[d]easilyfrom

aKoszulresolutionofU/I.SoSfailsχdontheleft.Ontheotherhand,

[14,Proposition8.6(1)]provesthatsince{cn}n∈Ziscriticallydense,wehavedimkExt

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

12DANIELROGALSKI

Similarly,againassumingm≥n,wehave(fS∩T)m=(φm n(f) Um n)∩Im.Ifφm n(f)∈I,thenasIisprime,(φm n(f) Um n)∩Im=φm n(f) Im n=(fT)m.Conversely,ifφm n(f)∈I,then(fS∩T)m=(fS)m=(fT)m.

Nowsince{cn}n∈Zisacriticallydensesetofpoints,everyhomogeneousf∈Ssatis esf∈φn(I)forn 0,whichisequivalenttoφn(f)∈Iforn 0.Weconcludethatforanyhomogeneous0=f∈Tthemodules(fS∩T)/fTandS/(fS+T)are nitedimensional,asrequired.

6.Thestrongnoetherianproperty

WecontinuetostudyidealizerringsTsatisfyingHypothesis5.1,andwemaintainthenotationintroducedintheprevioussection.In[14],theauthorshowedtheexistenceofringswhicharenotstronglynoetherianoneitherside.HerewewillshowthattheidealizerringsTaretypicallystronglynoetherianononesidebutnottheother.

LetAbeanarbitraryk-algebra.WecallaleftA-moduleMstronglynoetherianifM kBisanoetherianleftA kB-moduleforeverycommutativenoetheriank-algebraB.Moregenerally,MisuniversallynoetherianifM kBisnoetherianoverA kBforeverynoetheriank-algebraB.

Proposition6.1.AssumeHypothesis5.1,andassumefurtherthatthesetofpoints{cn}n∈Ziscriticallydense.ThenTisanoetherianringsuchthat

(1)Tisuniversallyleftnoetherian.

(2)Tisnotstronglyrightnoetherian.

Proof.ThatTisnoetherianfollowsfromPropositions5.3and5.4.

(1)WenotethattheringSisuniversallyleftnoetherian,asfollows.Foranynoetheriank-algebraB,theringU kB～=B[x0,...,xd]isnoetherianbytheHilbertbasistheorem.ThensinceS kBisaleftZhangtwistofU kB,itisalsoleftnoetherian[19,Theorem1.3].NowweprovethatTisuniversallynoetherianontheleft.WeknowthatM=T(S/T)is nitelygeneratedbyLemma5.2,andsincedimkMn=1foralln≥1weseethatMmusthaveKrulldimension1.By[1,Theorem4.23],MisauniversallynoetherianleftT-module.SoifBisanynoetheriank-algebra,thenM kB=(S kB)/(T kB)isanoetherianleftT kB-module.Thenby[1,Lemma4.2],sinceS kBisleftnoetherian,T kBisalsoleftnoetherian.

(2)TheproofwhichwenowpresentthatTisnotstronglynoetherianontherightisquiteanalogoustotheproofin[14,§7]thattheringRstudiedinthatpaperisnotstronglynoetherian.Letus rstmakeafewcommentsaboutnotation.Weusesubscriptstoindicateextensionofscalars,forexampleUB=U kB.TheautomorphismφofUnaturallyextendstoanautomorphismofUBsuchthatSBisagaintheleftZhangtwistofUBbyφ.Weextendalsoournotationalconvention,sothatjuxtapositionmeansmultiplicationinSBand meansthecommutativemultiplicationinUB.Fixonceandforallsomeparticularchoiceofhomogeneouscoordinatesforeachofthepointsin{cn}n∈Z Pdk.Thenforf∈UB,theexpressionf(cn)denotespolynomialevaluationatthe xedcoordinatesforcn,givingawell-de nedvalueintheringB.

Becausebyassumptionthepointset{cn}n∈Ziscriticallydense,thesameproofasin[14,Theorem7.4]showsthatthereexistsanoetheriancommutativek-algebra

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY13

Bwhichisauniquefactorizationdomain,constructedasanin nitea neblowupofa nespace,andcontainingelementsf,g∈(UB)1withthefollowingproperties:

(1)g(ci)= if(ci)forsome i∈B,foralli≤0.

(2)Foralli 0,f(ci)isnotaunitinB.

(3)gcd(f,g)=1inUB.

Notethatahomogeneouselementf∈UBisin(TB)≥1=I kBifandonlyiff(c0)=0.Nowforeachn≥1wemaychoosesomeelementθn∈(SB)n\(TB)nwithcoe cientsink.Puttingtn=( nf g)θn,wehaveintermsofthecommutativemultiplicationinUBthattn=φn( nf g) θn,andsinceφn( nf g)(c0)= ( nf g)(c n)=0weseethattn∈(TB)n+1.Supposeforsomenthattn+1=n

i=1tiriwithri∈(TB)n i+1.Then

φn+1( n 1f g) θn+1=n

i=1φn+1( if g) φn i+1(θi) ri.

Rewritingthisequationintheformh1 φn+1(f)=h2 φn+1(g),andusingthatgcd(f,g)=1,wemayconcludethatφn+1(g)dividesh1,where

h1= n 1θn+1 n

i=1 iφn i+1(θi) ri.

Then(φn+1(g))(c0)=g(c n 1)dividesh1(c0).Eachri∈(TB)≥1andsori(c0)=0,andbyassumptionθn+1∈TBandsoθn+1(c0)∈k×.Thusg(c n 1)divides n 1,whichimpliesthatf(c n 1)isaunitinB.This ncontradictsproperty(2)above forn 0.Thusforn 0wemusthavetn+1∈i=1tiTB.WeconcludethattiTBisanin nitelygeneratedrightidealofTB,soT kBisnotrightnoetherianandTisnotstronglyrightnoetherian.

7.TensorProductsofalgebras

InProposition6.1weshowedexplicitlythatTisnotstronglyrightnoetherianbyexhibitingacommutativenoetheriank-algebraBsuchthatT kBisnotrightnoetherian.Necessarily,suchaBisnota nitelygeneratedcommutativealgebra.Bycontrast,ifweallowourselvestotensorbynoncommutativeringsthenwemay nda nitelygeneratednoetheriank-algebraB′suchthatT kB′isnotrightnoetherian.Infact,wewillseeinthenexttheoremthatonemaytakeB′tobeTitself.

InordertostaywithintheclassofN-gradedalgebras,inadditiontotensorproductsitwillbeusefulalsotoconsiderSegreproducts,de nedasfollows.IfAands ∞BaretwoN-gradedalgebrasweletA kBbetheN-gradedalgebran=0An kBn.Thefollowinglemmaisthenelementary.

Lemma7.1.LetAandBbeN-gradedalgebras.IfA kBisleft(right)noetherian,thenA kBisleft(right)noetherian.

Proof.SinceanyhomogeneousleftidealIofA Bsatis es(A B)I∩(A B)=I,aproperascendingchainofhomogeneousleftidealsofA BinducesaproperascendingchainofleftidealsofA B.

WethankJamesZhangforpointingouttousthefollowingusefulfact.ssss

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

14DANIELROGALSKI

Lemma7.2.LetAbeconnectedN-gradedandnoetherian.ThenAis nitelypresented.

Proof.Let

···→r1

i=1r0 i=1A[ d1i]→A[ d0i]→A→k→0

beagradedfreeresolutionofAkbyfreemodulesof niterank.ThenonemaycheckthatAhasapresentationwithr0generatorsandr1relations.

Thefollowingtheoremshowsthatitispossibleto ndtwoconnectedgradednoetherianringswhosetensorproductisnoetherianononesideonly,aswellapairofconnectedgradednoetherianringswhosetensorproductisnoetherianonneitherside.

Theorem7.3.AssumeHypothesis5.1,andinadditionthat{cn}n∈Ziscriticallydense.LetT′=T kTop.Then

(1)TandT′arenoetherian nitelypresentedconnectedgradedk-algebras.

(2)T kTisleftnoetherian,butnotrightnoetherian.

(3)T′ kT′～=T′ k(T′)opisneitherleftnorrightnoetherian.

Proof.(1)TheringTisnoetherianbyPropositions5.3and5.4.Infact,byPropo-sition6.1Tisuniversallyleftnoetherian.ItfollowsimmediatelythatTopisuni-versallyrightnoetherian.ThusT kTopisbothleftandrightnoetherian.ByLemma7.1,T′isnoetherian.ThenbyLemma7.2,bothTandT′are nitelypresented.

(2)Aswesawinpart(1),Tisuniversallyleftnoetherian,sothatT kTisleftnoetherian.NowwewillprovethatT Tisnotrightnoetherian.ByLemma7.1,sitisenoughtoprovethatT Tisnotrightnoetherian.sForagradedringAwewillusetheabbreviationAs=A A.NowletX=projUs～=Pd×Pd.ThegradedringUshastheautomorphismφ φwithcorrespondingautomorphism × ofX.ThegradedringSsmaybethoughtofastheleftZhangtwistofUsbyφ φ,andweidentifytheunderlyingvectorspaces.Inparticular,anyhomogeneouselementofSsde nesavanishinglocusinX.Nowlet Xbethediagonalsubscheme,andletJbetheleftidealofSsconsistingofthoseelementswhichvanishalong .Since( × )( )= ,itfollowseasilythatJisatwo-sidedidealofSs.WritingK=I kI,aleftidealofSs,wehaveTs=k⊕K.ThentoprovethatTsisnotrightnoetherian,byLemma2.2itwillbeenoughtoshowthat(J∩K)/JKisnot nite-dimensionaloverk.

Let indicatemultiplicationinthecommutativeringUs.SinceJisinvariantunderφ φ,wehaveJ K=JK,andsoitwillbeequivalenttoprovethatM=(J∩K)/(J K)isnotatorsionUs-module.Toshowthis,weconsiderthe onX,looklocallyatthepointp=(c,c),andprovethatcorrespondingsheafM p=0.M

Chooselocala necoordinatesu1,...udforaprincipalopensetAd Pdsuchthatthepointccorrespondstotheorigin.Letv1,...vdbethesamecoordinatesfortheequivalentopensetAdinthesecondcopyofPd,sothatu1,...ud,v1,...vdarelocalcoordinatesforana neneighborhoodA2dofpinXsuchthatpistheorigininthesecoordinates.NowletpbethehomogeneousprimeidealofUscorrespondingss

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY15

tothepointp=(c,c).SettingU′=(Us)(p)=OX,p,J′=J(p),andK′=K(p),wehave p=M(p)～M=(J′∩K′)/(J′K′)

wherewereverttotheuseofjuxtapositiontoindicatemultiplicationinthecommu-tativelocalringU′.Explicitly,U′isthepolynomialringk[u1,...ud,v1,...vd]lo-calizedatthemaximalidealm=(u1,...,ud,v1,...,vd),J′=(u1 v1,...,ud vd),andK′=(u1,u2,...,ud)(v1,v2,...vd).Nowitisclearthatw=u1v2 u2v1∈ p=0,asweneededtoshow.J′∩K′,butw∈J′K′sincew∈m3 J′K′.ThusMs(3)Notethat(T′)op～=Top T～=T′.ThefactthatT′ T′isneitherleftnorrightnoetherianfollowsimmediatelyfrompart(2). Remark7.4.AssumingthesetupofHypothesis5.1,theringR=k I1 Swhichisgeneratedbythedegree1pieceofTisagradedringofthetypestudiedinthearticle[14].Incasethepoints{cn}n∈Zarecriticallydense,thisringRhassimilarlystrangepropertiesundertensorproducts.Forexample,asimilarbutslightlymorecomplicatedversionoftheargumentinTheorem7.3(2)abovewouldshowthatR kRisneitherleftnorrightnoetherian.

8.proofofthemaintheorem

Inthe nalsection,werecapitulateallofourprecedingresultstoproveTheo-rem1.1,whichwerestateasTheorem8.2below.TheonlythingwehavelefttoshowisthatgiventhesetupofHypothesis5.1,thereexistsaplentifulsupplyofchoicesofapointc∈Pdandanautomorphism ∈AutPdsuchthatC={cn}n∈Ziscriticallydense.Thissituationhasalreadybeenstudiedinthepaper[14];werepeattheresultforthereader’sreferenceasthenextproposition.

WecallasubsetofavarietyXgenericifitscomplementiscontainedinacountableunionofclosedsubvarietiesZ X.Notethataslongasthebase eldkisuncountable,anygenericsubsetisintuitively“almostall”ofX,inparticularitisnonempty.Thusthe rstpartofthefollowingpropositionshowsthatifthebase eldkisuncountable,thenanysuitablygeneralpair( ,c)willleadtoacriticallydensesetC.Thesecondpartshowsthatincasechark=0wemayeasilywritedownmanyexplicitexamplesofpairs( ,c)forwhichCiscriticallydense.

Proposition8.1.[14,Theorem12.4,Example12.8]AssumeHypothesis5.1andsetC={cn}n∈Z.

(1)Letkbeuncountable.Foranygivenc∈Pd,thereisagenericsubset

Y AutPd=PGL(k,d)suchthatif ∈YthenCiscriticallydense.

(2)Ifchark=0,c=(1:1:···:1),and isde nedby

(a0:a1:···:ad)→(a0:p1a1:p2a2:···:pdad),

thenCiscriticallydenseifandonlyifp1,...,pdgenerateamultiplicative

subgroupofk×whichisisomorphictoZd.

Finally,wesummarizeallofthepropertiesthattheringThasincasethesetofpointsCiscriticallydense.

Theorem8.2.AssumeHypothesis5.1.Letkbeuncountableandassumethatthepair( ,c)ischosensothatC={cn}n∈Ziscriticallydense.ThentheidealizerringT=I(I)=T( ,c)isanoetherianconnected nitelypresentedgradedringwiththefollowingproperties:

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

16DANIELROGALSKI

(1)Tisleftuniversallynoetherian,butnotstronglyrightnoetherian.

(2)T kTisleftnoetherianbutnotrightnoetherian.TheSegreproductT′=

T kTopisalsoa nitelypresentedconnectedgradednoetherianring,butT′ kT′isnoetherianonneitherside.

Proj-TandT-Projhavethesameunderlyingcategorybutnon-isomorphic

distinguishedobjects;speci cally,Proj-T～=(QchPd,OPd)andT-Proj～=dd(QchP,I),whereIisthesheafofidealscorrespondingtothepointc∈P.Tsatis esleftχd 1butnotleftχd,andTfailsχ1ontheright.

cd(Proj-T)=cd(T-Proj)=d.

AlthoughnoVeroneseringofTisgeneratedindegree1,onehasisomor-phismsT-Proj～=T(n)-ProjandProj-T～=Proj-T(n)foralln≥1.s(3)(4)(5)(6)

Proof.NotethatbyProposition8.1,wemayindeed ndapair( ,c)sothatCiscriticallydense.ThenTisnoetherianbyPropositions5.4(2)and5.3,andTis nitelypresentedbyLemma7.2.

Now(1)followsfromProposition6.1,and(2)fromTheorem7.3.

For(3),notethatsinceSisaleftZhangtwistofU,wehaveS-Gr U-GrandsoiteasilyfollowsthatS-Proj～=U-Proj.NowtheoppositeringSopofSisisomorphictotheleftZhangtwistofUbyφ 1;thismaybecheckeddirectly,orseetheproofof[14,Lemma4.2(1)].ThuswealsohaveanisomorphismProj-S～=Proj-U.BySerre’stheorem,wealsohaveanequivalenceofcategoriesU-Qgr QchPd,whereQchPdisthecategoryofquasi-coherentsheavesonPd.

NowusingProposition3.2,itfollowsthat

T-Proj～=(S-Qgr,πI)～=(QchPd,I)

Proj-T～=(QchPd,OPd).=(Qgr-S,πS)～

Sinced≥2,theidealsheafIwhichde nestheclosedpointcisnotlocallyfree,soinparticularwehaveI～OPdand(3)isproved.=

Next,result(4)isacombinationofPropositions4.2and5.4(1).SinceS-Proj～=U-ProjandProj-S～=Proj-U,itfollowseasilythatcd(S-Proj)=gd(S-Qgr)=cd(Proj-S)=gd(Qgr-S)=d,andso(5)isaconsequenceofProposition3.5.Finally,(6)followsfromProposition3.4andLemma5.2(2).

WeclosewithafewremarksconcerningTheorem8.2.

Remark8.3.Theorem8.2(2)showsthatthetensorproductoftwonoetherian nitelypresentedconnectedgradedalgebras(overanalgebraicallyclosed eld)canfailtobenoetherian.Thisanswers[6,Appendix,OpenQuestion16′].

Remark8.4.SupposethatAisaconnectedgradednoetherianringsatisfyingleftχ1suchthatA-Proj～=(QchX,OX)forsomeproperschemeX.KeelershowedthatinthiscaseAmustbeequalinlargedegreetoatwistedhomogeneouscoordinateringB(X,L,σ)whereLisσ-ample[8,Theorem7.17].Inparticular,Amustbeuniversallynoetherianandmustsatisfyχonbothsides.

NowconsiderinsteadconnectedgradednoetherianringsAwithleftχ1suchthatA-Proj～=(QchX,F)forsomeproperschemeX,butwhereFisnotassumedtobethestructuresheaf.ThenA=T,whereTsatis estheconclusionsofTheorem8.2,isanexampleshowingthatringswithmuchmoreunusualbehaviormayoccurinthiscase.and

We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $\chi$-conditions of Artin and Zhang and the strong noetherian pr

IDEALIZERRINGSANDNONCOMMUTATIVEPROJECTIVEGEOMETRY17

Acknowledgments

MuchoftheresearchforthisarticlewascompletedattheUniversityofMichiganandtheUniversityofWashington;theauthorthanksbothinstitutions.TheauthoralsothanksJamesZhangandPaulSmithforhelpfulconversations,andespeciallyTobySta ordforhismanyusefulsuggestionswhichimprovedthisarticle.

References

[1]M.Artin,L.W.Small,andJ.J.Zhang,Generic atnessforstronglyNoetherianalgebras,

J.Algebra221(1999),no.2,579–610.

[2]M.ArtinandJ.J.Zhang,Noncommutativeprojectiveschemes,Adv.Math.109(1994),

no.2,228–287.[3]

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