A pairing in homology and the category of linear t

Apairinginhomologyandthecategoryoflineartiltingcomplexesforaquasi-hereditaryalgebraVolodymyrMazorchukandSergeOvsienkoAbstractWeshowthatthereexistsanaturalnon-degeneratepairingofthehomomorphismspacebetweentwoneighborstandardmodulesoveraquasi-hereditaryalgebrawiththefirstextensionspacebetweenthecorrespondingcostandardmodulesandviseversa.Investigationofthisphenomenonleadstoafamilyofpairingsinvolvingstan-dard,costandardandtiltingmodules.Inthegradedcase,undersome”Koszul-like”assumptions(whichweprovearesatisfiedforexamplefortheblocksofthecategoryO),weobtainanon-degeneratepairingbetweencertaingradedhomomorphismandgradedextensionspaces.Thismotivatesthestudyofthecategoryoflineartiltingcomplexesforgradedquasi-hereditaryalgebras.Weshowthat,underassumptions,similartothosementionedabove,thiscategoryrealizesthemodulecategoryfortheKoszuldualoftheRingeldualoftheoriginalalgebra.AsacorollaryweobtainthatundertheseassumptionstheRingelandKoszuldualitiescommute.1IntroductionanddescriptionoftheresultsLet|beanalgebraicallyclosedfield.Iftheoppositeisnotemphasized,inthispaperbyamodulewemeanaleftmoduleandwedenotebyRad(M)theradicalofamodule,M.Fora|-vectorspace,V,wedenotethedualspacebyV∗.LetAbeabasic|-algebra,whichisquasi-hereditarywithrespecttothenaturalor-derontheindexingset{1,2,...,n}ofpairwise-orthogonalprimitiveidempotentsei(see[CPS,DR1,DR2]fordetails).LetP(i),Δ(i),∇(i),L(i),andT(i)denotetheprojective,standard,costandard,simpleandtiltingA-modules,associatedtoei,i=1,...,n,respec-tively.SetP=⊕ni=1P(i),Δ=⊕ni=1Δ(i),∇=⊕ni=1∇(i),L=⊕ni=1L(i),T=⊕ni=1T(i).Weremarkthat,evenifthestandardA-modulesarefixed,thelinearorderontheindexingsetofprimitiveidempotents,withrespecttowhichthealgebraAisquasi-hereditary,isnotuniqueingeneral.WedenotebyR(A)andE(A)theRingelandKoszuldualsofArespectively.Agradedalgebra,B=⊕i∈ZBi,willbecalledpositivelygradedprovidedthatBi=0foralli0andRad(B)=⊕i0Bi.ThispaperhasstartedfromanattempttogiveaconceptualexplanationfortheequalitydimHomA(Δ(i−1),Δ(i))=dimExt1A(∇(i),∇(i−1)),(1)1whichisprovedatthebeginningofSection2.Ourfirstmainresult,provedalsoinSection2,isthefollowingstatement:Theorem1.(1)Leti,j∈{1,...,n}andji.Thenthereexistsabilinearpairing,h·,·i:HomA(Δ(j),Δ(i))×Ext1A(∇(i),∇(j))→|.(2)Ifj=i−1,thenh·,·iisnon-degenerate.Theorem1explainstheoriginsof(1)andmotivatesthestudyofh·,·i.Ithappensthatinthegeneralcase,thatisforji−1,theanalogueofTheorem1(2)isnolongertrue.WegiveanexampleatthebeginningofSection3.Inthesamesectionwepresentsomespecialresultsandamodificationofh·,·iinthegeneralcase.AnattempttolifttheaboveresultstohigherExt’snaturallyledustothedefinitionofadifferentpairing,whichusesaminimaltiltingresolutionofthecostandardmodule.InSection4weconstructandinvestigateapairingbetweenExtlA(∇,∇)andHomA(Δ,Tl),whereTlisthel-thcomponentofaminimaltiltingresolutionof∇.Inthecasel=1thisnewpairinginducestheonewehaveconstructedinSection2.Thenewpairingisrarelynon-degenerate.Inanattempttofindsomeconditions,whichwouldensurethisproperty,wenaturallycametothegradedcase.InSection5weshowthatinthegradedcaseournewpairinginducesanon-degeneratepairingbetweenthegradedhomomorphismandthegradedfirstextensionspacesundertheconditionthatthecostandardmodulesadmitlineartiltingresolutions.Herethelinearityoftheresolutionmeansthefollowing:weshowthatforapositivelygradedquasi-hereditaryalgebraalltiltingmodulesaregradableandthuswecanfixtheirgradedliftsputtingtheir”middles”indegree0;thelinearityoftheresolutionnowmeansthatthei-thtermoftheresolutionconsistsonlyoftiltingmodules,whose”middles”areexactlyindegreei.Thisobservationbringsthelinearcomplexesoftiltingmodulesintothepictureandservesasabridgetothesecondpartofthepaper,inwhichwestudythecategoryofallsuchlinearcomplexes.Theabovementionedconditionoftheexistenceofalineartiltingresolutionforcostan-dardA-modulesimmediatelyresemblestheconditions,whichappearedin[ADL]duringthestudyofthefollowingquestion:whentheKoszuldualofaquasi-hereditaryalgebraisquasi-hereditarywithrespecttotheoppositeorder?In[ADL,Theorem3]itwasshownthatthisisthecaseifandonlyifboththestandardandcostandardA-modulesadmitalinearprojectiveandinjective(co)resolutionrespectively(algebras,satisfyingthesecon-ditions,werecalledstandardKoszulin[ADL]).ThisresemblancemotivatedustotakeacloserlookatthecategoryoflinearcomplexesoftiltingA-modules.Themoststrikingpropertyofthiscategoryisthefactthatitcombinestwoobjectsofcompletelydifferentnatures:tiltingmodulesforaquasi-hereditaryalgebra,whichgiverisetotheso-calledRingelduality;andlinearresolutions,whicharethesourceofacompletelydifferentdual-ity,namelytheKoszulduality.Undersomenaturalassumptions,whichroughlymeanthatallobjectsweconsiderarewell-definedandwell-coordinatedwitheachother,inSection6weproveoursecondmainresult:2Theorem2.AssumethatAisapositivelygradedquasi-hereditaryalgebra,suchthat(i)thegradingonR(A),inducedfromthecategoryofgradedA-modules,ispositive;(ii)standardA-modulesadmitalineartiltingcoresolution,(iii)costandardA-modulesadmitalineartiltingresolution.TheaboveconditionsimplythatthequadraticdualR(A)!ofR(A)isquasi-hereditary(withrespe