Generalized tilting modules with finite injective

arXiv:math/0602572v3[math.RA]31Dec2006GeneralizedTiltingModulesWithFiniteInjectiveDimension∗†ZhaoyongHuang‡DepartmentofMathematics,NanjingUniversity,Nanjing210093,People’sRepublicofChinaAbstractLetRbealeftnoetherianring,SarightnoetherianringandRUageneralizedtiltingmodulewithS=End(RU).TheinjectivedimensionsofRUandUSareidenticalprovidedbothofthemarefinite.UndertheassumptionthattheinjectivedimensionsofRUandUSarefinite,wedescribewhenthesubcategory{ExtnS(N,U)|NisafinitelygeneratedrightS-module}issubmodule-closed.Asaconsequence,weobtainanegativeanswertoaquestionposedbyAuslanderin1969.Finally,somepartialanswerstoWakamatsuTiltingConjecturearegiven.1.IntroductionLetRbearing.WeuseModR(resp.ModRop)todenotethecategoryofleft(resp.right)R-modules,andusemodR(resp.modRop)todenotethecategoryoffinitelygeneratedleftR-modules(resp.rightR-modules).Wedefinegen∗(RR)={X∈modR|thereexistsanexactsequence···→Pi→···→P1→P0→X→0inmodRwithPiprojectiveforanyi≥0}(see[W2]).AmoduleRUinmodRiscalledselforthogonalifExtiR(RU,RU)=0foranyi≥1.Definition1.1[W2]AselforthogonalmoduleRUingen∗(RR)iscalledageneralizedtiltingmodule(sometimesitisalsocalledaWakamatsutiltingmodule,see[BR])ifthereexistsanexactsequence:0→RR→U0→U1→···→Ui→···suchthat:(1)Ui∈addRUforanyi≥0,whereaddRUdenotesthefullsubcategoryofmodRconsistingofallmodulesisomorphictodirectsummandsoffinitesumsofcopiesofRU,and(2)afterapplyingthefunctorHomR(,RU)thesequenceisstillexact.∗2000MathematicsSubjectClassification.16E10,16E30.†Keywordsandphrases.generalizedtiltingmodules,injectivedimension,U-limitdimension,submodule-closed,WakamatsuTiltingConjecture.‡E-mailaddress:[e-mail]calledafaithfullybalancedbimoduleifthenaturalmapsR→End(US)andS→End(RU)opareisomorphisms.By[W2]Corollary3.2,wehavethatRUSisfaithfullybalancedandselforthogonalwithRU∈gen∗(RR)andUS∈gen∗(SS)ifandonlyifRUisgeneralizedtiltingwithS=End(RU)ifandonlyifUSisgeneralizedtiltingwithR=End(US).LetRandSbeArtinalgebrasandRUageneralizedtiltingmodulewithS=End(RU).Wakamatsuprovedin[W1]Theoremthattheprojective(resp.injective)dimensionsofRUandUSareidenticalprovidedbothofthemarefinite.TheresultontheprojectivedimensionsalsoholdstruewhenRisaleftnoetherianringandSisarightnoetherianring,byusinganargumentsimilartothatin[W1].Inthiscase,RUSisatiltingbimoduleoffiniteprojectivedimension([M]Proposition1.6).However,becausethereisnodualityavailable,Wakamatsu’sargumentin[W1]doesnotworkontheinjectivedimensionsovernoetherianrings.So,itisnaturaltoaskthefollowingquestions:WhenRisaleftnoetherianringandSisarightnoetherianring,(1)DotheinjectivedimensionsofRUandUScoincideprovidedbothofthemarefinite?(2)IfoneoftheinjectivedimensionsofRUandUSisfinite,istheotheralsofinite?Theanswertothefirstquestionispositiveifoneofthefollowingconditionsissatisfied:(1)RUS=RRR([Z]LemmaA);(2)RandSareArtinalgebras([W1]Theorem);(3)RandSaretwo-sidednoetherianringsandRUisn-Gorensteinforalln([H2]Proposition17.2.6).Inthispaper,weshowinSection2thattheanswertothisquestionisalwayspositive.Bythepositiveanswertothefirstquestion,thesecondquestionisequivalenttothefollowingquestion:AretheinjectivedimensionsofRUandUSidentical?Theaboveresultmeansthattheanswertothisquestionispositiveprovidedthatbothdimensionsarefinite.Ontheotherhand,forArtinalgebras,thepositiveanswertothesecondquestionisequivalenttothevalidityofWakamtsuTiltingConjecture(WTC).Thisconjecturestatesthateverygeneralizedtiltingmodulewithfiniteprojectivedimensionistilting,orequivalently,everygeneralizedtiltingmodulewithfiniteinjectivedimensioniscotilting.Moreover,WTCimpliesthevalidityoftheGorensteinSymmetryConjecture(GSC),whichstatesthattheleftandrightself-injectivedimensionsofRareidentical(see[BR]).InSection4,wegivesomepartialanswerstoquestion(2).LetRandSbetwo-sidedartinianringsandRUageneralizedtiltingmodulewithS=End(RU).WeprovethatiftheinjectivedimensionofUSisequaltonandtheU-limitdimensionofeachofthefirst(n−1)-sttermsisfinite,thentheinjectivedimensionofRUisalsoequalton.ThusittrivialthattheinjectivedimensionofUSisatmost1ifandonlyifthatofRUisatmost1.WeremarkthatforanArtinalgebra2R,itiswellknownthattherightself-injectivedimensionofRisatmost1ifandonlyiftheleftself-injectivedimensionofRisatmost1(see[AR3]p.121).Inaddition,weprovethattheleftandrightinjectivedimensionsofRUandUSareidenticalifRU(orUS)isquasiGorenstein,thatis,WTCholdsforquasiGorensteinmodules.Foran(R−S)-bimoduleRUSandapositiveintegern,wedenoteEn(US)={M∈modR|M=ExtnS(N,U)forsomeN∈modSop}.Foratwo-sidednoetherianringR,Auslandershowedin[A]Proposition3.3thatanydirectsummandofamoduleinE1(RR)isstillinE1(RR).HethenaskedwhetheranysubmoduleofamoduleinE1(RR)isstillinE1(RR).RecallthatafullsubcategoryXofmodRissaidtobesubmodule-closedifanynon-zerosubmoduleofamoduleinXisalsoinX.ThentheaboveAuslander’squestionisequivalenttothefollowingquestion:IsE1(RR)submodule-closed?InSection3,undertheassumptionthatRisaleftnoetherianring,SisarightnoetherianringandRUisageneralizedtiltingmodulewithS=End(RU)andtheinjectivedimensionsofRUandUSbeingfinite,wegivesomenecessaryandsufficientconditionsforEn(