Graded Rings and Equivariant Sheaves on Toric Vari

GradedRingsandEquivariantSheavesonToricVarietiesMarkusPerlingDepartmentofMathematicsUniversityofKaiserslautern,Germanyemail:[e-mail]2001AbstractInthisnotewederiveaformalismfordescribingequivariantsheavesovertoricvarieties.ThisformalismisageneralizationofacorrespondenceduetoKlyachko,whichstatesthatequivariantvectorbundlesontoricvarietiesareequivalenttocertainsetsofltrationsofvectorspaces.WesystematicallyconstructthetheoryfromthepointofviewofgradedringtheoryandthiswayweconsiderablyclarifyearlierconstructionsofKaneyamaandKlyachko.Wealsoconnecttheformalismtothetheoryofne-gradedmodulesoverCox’homogeneouscoordinateringofatoricvariety.Asanapplicationweconstructminimalresolutionsofequivariantvectorbundlesofranktwoontoricsurfaces.1IntroductionIn[Kly90]Klyachkoshowedthatequivariantvectorbundlesonatoricvarietyareequiva-lenttocertainsetsofltrationsofvectorspaces.Inthepreprint[Kly91]thisequivalencewasextendedtoequivarianttorsionfreesheavesonsmoothtoricvarietiesandappliedtomoduliproblemsforbundlesoverP2.Itwasclaimedtherethatresultsof[Kly90]concerningvectorbundlesareinasimilarfashionalsovalidfortorsionfreesheaves,butunfortunatelynoproofsweregiven.Theaimofthisnoteistopickupsomeoftheideasof[Kly91]andtodeliverproperdenitionsoftheltration-formalismforequivarianttor-sionfreesheavesandtogeneralizeittoarbitraryequivariantcoherentsheavesonnotnecessarilysmoothtoricvarietes.OnebasicideaisthatifoneconsidersanequivariantquasicoherentsheafEoverthetoricvarietyXassociatedsomefan(seesection2fornotation),thentheactionofthetorusTonitsaneT-invariantopensubsetsU,2,inducesisotypicaldecompositionsofthemodulesofsectionsofEintoT-eigenspaces:(U;E)=Mm2M(U;E)m;1whereM=ZdimTdenotesthecharactergroupofthetorus.IncaseEiscoherent,theeigenspaces(U;E)marenite-dimensionalvectorspaces.Foreachm;m02Mthemodulestructureoverthecoordinateringk[M]ofUinducesmaps(U;E)m!(U;E)m0e7!(m0m)ebymultiplicationwiththecharacter(m0m).Thismultiplicationmapexistsifandonlyifm0misanelementofthesemigroupM=\M.SothesemigroupMinducesinanaturalwayapreorderonMbysettingmm0im0m2M.Thesetofvectorspaces(U;E)mandcharacters(m)formsadirectedfamilyofvectorspaceswithrespecttothispreorder,andwewilldenotesuchdataa-family.Weobtainforeach2suchafamily,andviceversa,givenasetof-familiesforall2,weobtainasystemofsheavesEovereachUwhich,ifcertaincompatibilityconditionsarefullled,gluetoaglobalsheafoverthetoricvarietyX.Suchacompatiblesetof-familieswillbecalleda-family.Ultimately,wewillarriveatanequivalenceofcategoriesbetweenequivariantquasicoherentsheavesoverXand-families.IncasethatEistorsionfree,allthemapsinitsassociated-familyareinjectiveandonecanformulatetheresultintermsofmultiltrationsofacertainlimitvectorspaceE0.ThisisessentiallytheformulationwhichKlyachkogavein[Kly91]andwhichgeneralizedtheearlierltrationsforequivariantvectorbundlesof[Kly90].Weshowthattheconstructionsof[Kly90]and[Kly91]simplifyconsiderablyiftheysystematicallyarederivedfromthepointofviewofgradedringtheory.ThispointofviewalsohasbeenadoptedearlierbyKaneyamain[Kan75,Kan88]forclassicationofequivariantvectorbundlesonsmoothtoricvarietesandespeciallyonPn.Moreover,wewillenhancetheseconstructionsbyshowinghownegradedmodulesoverthehomogeneouscoordinateringofatoricvariety([Cox95])canbeincorporated.AsanapplicationweconstructminimalresolutionsofequivariantvectorbundlesofranktwoonsmoothtoricsurfacesintermsofequivariantEuler-likeshortexactsequences.Inaforthcomingpaperwewillextendthisapplicationtoafullclassicationofsuchvectorbundles.Planofthepaper:Insection2,wewillrecallsomebasicnotionsconcerningtoricvarieties.Section3collectsgeneralfactsofgradedringtheorywhichcannoteasilybefoundintheliterature.Insection4webrieydesribetheconstructionofhomogeneouscoordinateringsovertoricvarietiesduetoCox.Themainpartofthisnoteissection5,wherewedevelopourformalismforequivariantsheaves,andweconcludewithanapplicationinsection6.Acknowledgements:IwouldliketothankProf.G.TrautmannandDr.J.Zintlformanydiscussionsonthesubject.2BasicFactsforToricVarietiesByanalgebraicvarietyoveranalgebraicallyclosedeldk,weunderstandaseparatedschemeofnitetypeoverspec(k),whichweassumetobereduced,ifnotstatedotherwise.AtoricvarietyXisanormalvarietywhichcontainsanalgebraictorusTasanopendense2subsetsuchthatthetorusmultiplicationextendstoanactionofthealgebraicgroupTonX.Forbackgroundontoricvarietieswerefertostandardliteraturesuchas[MO78],[Oda88]and[Ful93].Werecallfrom[MO78],xx5.3,5.4and[Ful93]somebasicfactswhichinthesequelwillfrequentlybeused.AtoricvarietyXisdenedbyafancontainedintherealvectorspaceNR=NZRofalatticeN=ZnandisdenotedX=X.LetMbethelatticedualtoNandleth;i:MN!ZbethenaturalpairingwhichextendstothescalarextensionsMR:=MZRandNR.ElementsofMaredenotedbym,m0,etc.ifwrittenadditively,andby(m),(m0),etc.ifwrittenmultiplicatively,i.e.(m+m0)=(m)(m0).ThelatticeMisthenaturalgroupofcharactersofthetorusT=Hom(M;k)=(k)n.AconeofthefanisaconvexrationalpolyhedralconecontainedinNR.Fortheseconesthefollowingstandardn