Class and rank of differential modules

arXiv:math/0602344v3[math.AC]25Jan2007CLASSANDRANKOFDIFFERENTIALMODULESLUCHEZARL.AVRAMOV,RAGNAR-OLAFBUCHWEITZ,ANDSRIKANTHIYENGARAbstract.Adifferentialmoduleisamoduleequippedwithasquare-zeroendomorphism.Thisstructureunderpinscomplexesofmodulesoverrings,aswellasdifferentialgradedmodulesovergradedrings.Weestablishlowerboundsontheclass—asubstituteforthelengthofafreecomplex—andontherankofadifferentialmoduleintermsofinvariantsofitshomology.Thesere-sultsspecializetobasictheoremsincommutativealgebraandalgebraictopol-ogy.OneinstanceisacommongeneralizationoftheequicharacteristiccaseoftheNewIntersectionTheoremofHochster,Peskine,P.Roberts,andSzpiro,concerningcomplexesovercommutativenoetherianrings,andofatheoremofG.Carlssonondifferentialgradedmodulesovergradedpolynomialrings.ContentsIntroduction11.Differentialmodules42.Differentialflags93.Classinequality.I144.Classinequality.II195.Rankinequalities206.Square-zeromatrices24AppendixA.Ranksofmatrices25Acknowledgements27References27IntroductionThispaperhasitsrootsinaconfluenceofideasfromcommutativealgebraandalgebraictopology.SimilaritiesbetweentwoseriesofresultsandconjecturesinthesefieldswerediscoveredandefficientlyexploitedbyGunnarCarlssonmorethantwentyyearsago.OnthetopologicalsidetheydealtwithfiniteCWcomplexesadmittingfreetorusactions;onthealgebraicone,withfinitefreecomplexeswithhomologyoffinitelength.However,nosinglestatement—letalonecommonproof—coverseventhebasiccaseofmodulesoverpolynomialrings.Inthispaperweexplorethecommonalityoftheearlierresultsandprovethatbroadgeneralizationsholdforallcommutativealgebrasoverfields.TheyincludeDate:February2,2008.2000MathematicsSubjectClassification.Unclassified.Keywordsandphrases.differentialmodules,finitefreeresolutions.ResearchpartlysupportedbyNSFgrantDMS0201904(L.L.A.),NSERCgrant3-642-114-80(R.O.B.),andNSFgrantDMS0442242(S.I.).12L.L.AVRAMOV,R.-O.BUCHWEITZ,ANDS.IYENGARbothCarlsson’stheoremsondifferentialgradedmodulesovergradedpolynomialringsandtheNewIntersectionTheoremforlocalalgebras,duetoHochster,Peskine,P.Roberts,andSzpiro.Theyalsosuggestprecisestatementsaboutmatricesovercommutativerings,thatimplyconjecturesonfreeresolutions,duetoBuchsbaum,Eisenbud,andHorrocks,andconjecturesonthestructureofcomplexeswithalmostfreetorusactions,duetoCarlssonandHalperin.Theseconjecturesareamongthefundamentalopenquestionsonbothsidesofthisnarrative.Thefocushereisonasimpleconstruct:amoduleoveranassociativeringR,equippedwithanR-linearendomorphismofsquarezero.WecallthesedataadifferentialR-module.Theyarepartofthestructureunderlyingthefamiliarandubiquitousnotionsofcomplexordifferentialgradedmodule.DifferentialmodulesassuchappearedalreadyfivedecadesagoinCartanandEilenberg’streatise[10],wheretheyareassignedmostlydidacticfunctions.Ourgoalistoestablishthatthesebasicobjectsareofconsiderableinterestintheirownright.Toillustratethedirectionandscopeofthegeneralitysogained,takeacomplexP=0−→Pl∂l−−→Pl−1−→···−→P1∂1−−−→P0−→0offinitefreemodulesoveraringR.ThemoduleP=LnPnwithendomorphismδ=Ln∂nisadifferentialR-modulePΔ.Withrespecttoanobviouschoiceofbasisfortheunderlyingfreemodule,δisrepresentedbyablocktriangularmatrixA=0A010...0000A12...00000...00..................000...0Al−1,l000...00withA2=0,Resultsonfinitefreecomplexesareequivalenttostatementsaboutsuchmatrices.Thekeycontention,supportedbyourresults,isthatsuchstatementsshouldextendinsuitableformtoanystrictlyuppertriangularmatrixA=0A01A02...A0l−1A0l00A12...A1l−1A1l000...A2l−1A2l..................000...0Al−1l000...00withA2=0,Matricesofthistypearisefromsequencesofsubmodules{Fn}=0⊆F0⊆F1⊆···⊆Fl=DinadifferentialR-moduleDwithendomorphismδ,satisfyingforeverynthecon-ditions:Fn/Fn−1isfreeoffiniterankandδ(Fn)⊆Fn−1.Wesaythat{Fn}isafreedifferentialflagwith(l+1)foldsinD.WhenDadmitssuchaflagwesaythatitsfreeclassisatmostl,andwritefreeclassRD≤l;else,wesetfreeclassRD=∞.TheprojectiveclassofDisdefinedanalogously,andisdenotedprojclassRD.NotethatifDhasfinitefree(respectively,projective)class,thenitisnecessarilyfinitelygeneratedandfree(respectively,projective).DIFFERENTIALMODULES3ThehomologyofDistheR-moduleH(D)=Ker(δ)/Im(δ).Acentralresultofthispaperlinksthesizeofitsannihilator,AnnRH(D),totheclassofDbyaClassInequality.LetRbeanoetheriancommutativealgebraoverafieldandDafinitelygenerateddifferentialR-module.OnethenhasprojclassRD≥heightIwhereI=AnnRH(D).TheexampleD=KΔ,whereKistheKoszulcomplexondelementsgeneratinganidealofheightd,showsthattheinequalitycannotbestrengthenedingeneral.ForthedifferentialmodulePΔdefinedaboveonehasl≥projclassRPΔandH(PΔ)=MnHn(P),sotheNewIntersectionTheoremfollowsfromtheClassInequality.ThehypothesisthatRcontainafieldisduetotheuseinourproofofHochster’sbigCohen-Macaulaymodules[18].Theconclusionholdswheneversuchmodulesexist,inparticular,whendimR≤3,see[19],orwhenRisCohen-Macaulay.P.RobertsprovedthattheIntersectionTheoremholdsforallnoetheriancommutativeringsR,see[21],andweconjecturethatsodoestheClassInequality.TheClassInequality,aresultaboutcommutativeringsingeneral,hasitsorigininthestudyoffreeactionsofthegroup(Z/2Z)donaCWcomplexX.Carlsso