The discrete module category for the ring of K-the

arXiv:math/0603045v1[math.AT]2Mar2006THEDISCRETEMODULECATEGORYFORTHERINGOFK-THEORYOPERATIONSFRANCISCLARKE,MARTINCROSSLEY,ANDSARAHWHITEHOUSEAbstract.Westudythecategoryofdiscretemodulesovertheringofdegreezerostableoperationsinp-localcomplexK-theory.WeshowthattheK(p)-homologyofanyspaceorspectrumissuchamodule,andthatthiscategoryisisomorphictoacategorydefinedbyBousfieldandusedinhisworkontheK(p)-localstablehomo-topycategory[2].Wealsoprovideanalternativecharacterisationofdiscretemodulesaslocallyfinitelygeneratedmodules.1.IntroductionAnexplicitdescriptionofthetopologicalringAofdegreezerosta-bleoperationsinp-localcomplexK-theorywasgivenin[5].Hereweconsiderthecategoryof‘discretemodules’overthisring.WefocusattentionondiscretemodulesbecausetheK(p)-homologyofanyspaceorspectrumissuchamodule;seeProposition2.6.InSection2werecallsomeresultsfrom[5]abouttheringA,definediscreteA-modules,andshowhowthecategoryDAofsuchmodulesiseasilyseentobeacocompleteabeliancategory.OurfirstmainresultcomesinSection3,whereweprovideaninter-estingalternativecharacterisationofdiscreteA-modules:anA-moduleisdiscreteifandonlyifitislocallyfinitelygenerated.WealsoshowthatthecategoryDAisisomorphictothecategoryofcomodulesoverthecoalgebraK0(K)(p)ofwhichAisthedual.ThecategoryDAaroseindisguisedformin[2].ThereBousfieldintroducedacertaincategoryA(p)asthefirststepinhisinvestigationoftheK(p)-localstablehomotopycategory.InSection4werecallBousfield’s(ratherelaborate)definition,andweprovethathiscategoryisisomorphictothecategoryofdiscreteA-modules.ThisallowsustosimplifyandclarifyinSection5someconstruc-tionsinBousfield’swork.Inparticular,thereisarightadjointtotheforgetfulfunctorfromA(p)tothecategoryofZ(p)-modules.ForBousfield,thisfunctorhastobeconstructedinanadhocfashion,sep-aratingcases.Inourcontext,itisrevealedassimplyacontinuousHomDate:27thFebruary2006.2000MathematicsSubjectClassification.Primary:55S25;Secondary:19L64,11B65.Keywordsandphrases.K-theoryoperations,K-theorymodules,K-localspectra.12F.CLARKE,M.D.CROSSLEY,ANDS.WHITEHOUSEfunctor.Wegiveaconstructioninthislanguageofafour-termexactsequenceinvolvingtherightadjoint.Bousfield’saimwastogiveanalgebraicdescriptionoftheK(p)-localstablehomotopycategory.Hesucceededatthelevelofobjects,andinSection6wetranslatehismainresultintoourlanguageofdiscreteA-modules.Ofcourse,p-localK-theorysplitsasasumofcopiesoftheAdamssummand.Wehavechosentowritethemainbodyofthispaperinthenon-splitcontext,butverysimilarresultsholdinthesplitsetting.Werecordtheseinanappendix.WenotethatafullalgebraicdescriptionoftheK(p)-localstablehomotopycategoryhasbeengivenbyFranke[6].Theinterestedreadermaywishtoconsult[12,8].Inadifferentdirection,wenotethefurtherworkofBousfield,buildingaunifiedversionofK-theoryinordertocombineinformationfromdifferentprimes[3].Throughoutthispaperpwilldenoteanoddprime.WewouldliketothankPeterKrophollerforpointingoutaresultwhichweneedintheproofofProposition3.9.Thethirdauthorac-knowledgesthesupportofaScheme4grantfromtheLondonMathe-maticalSociety.2.DiscreteA-modulesLetpbeanoddprimeandletA=K0(p)(K(p))betheringofdegreezerostableoperationsinp-localcomplexK-theory.Thisringcanbedescribedasfollows;see[5].Chooseanintegerqthatisprimitivemodulop2,andletΨq∈AbethecorrespondingAdamsoperation.Let(2.1)qi=q(−1)i⌊i/2⌋,anddefinepolynomialsΘn(X),foreachintegern0,byΘn(X)=Qni=1(X−qi).ThenlettheoperationΦn∈AbegivenbyΦn=Θn(Ψq).Forexample,Φ4=(Ψq−1)(Ψq−q)(Ψq−q−1)(Ψq−q2).TheseoperationshavebeenchosensothatanyinfinitesumPn0anΦn,withcoefficientsaninthep-localintegersZ(p),converges.Thefollowingtheoremsaysthatanyoperationcanbewrittenuniquelyinthisform.Theorem2.2.[5,Theorem6.2]TheelementsofAcanbeexpresseduniquelyasinfinitesumsPn0anΦn,wherean∈Z(p).Foreachm≥0,wedefineAm=XnmanΦn:an∈Z(p)⊆A,sothatAmistheidealofoperationswhichannihilatethecoefficientgroupsπ2i(K(p))for−m/2i(m+1)/2,andthusitdoesnotTHEDISCRETEMODULECATEGORYFORK-THEORYOPERATIONS3dependonthechoiceofprimitiveelementq.WeobtainadecreasingfiltrationA=A0⊃A1⊃···⊃Am⊃Am+1⊃···.WeusethisfiltrationtogivetheringAatopologyinthestan-dardway(see,forexample,Chapter9of[11]):theopensetsareunionsofsetsoftheformy+Am,wherey∈Aandm0.WenotethatAiscompletewithrespecttothistopology.Indeed,an-otherwaytoviewthetopologicalringAisasthecompletionofthepolynomialringZ(p)[Ψq]withrespecttothefiltrationbytheprincipalidealsΦmZ(p)[Ψq]=Z(p)[Ψq]∩Am.(Bywayofwarning,thisfiltrationisnotmultiplicativeinthesenseof[11],and,sinceAisnotNoether-ian[5,Theorem6.10],itisnotthecompletionofapolynomialringwithrespecttoanymultiplicativefiltration.)Definition2.3.AdiscreteA-moduleMisanA-modulesuchthattheactionmapA×M→MiscontinuouswithrespecttothediscretetopologyonMandthere-sultingproducttopologyonA×M.Inpracticeweusethefollowingcriteriontorecognisediscretemod-ules.Lemma2.4.AnA-moduleMisdiscreteifandonlyifforeachx∈M,thereissomensuchthatAnx=0.Proof.Fixingx∈M,themapsendingα∈Ato(α,x)∈A×Miscontinuous.ThusifMisdiscrete,themapα7→αx∈Miscontinuous,whichimpliesthatitskernelcontainsAnforsomen.Supposenowthatforeachx∈M,thereexistsnsuchthatAnx=0.Ifx,y∈Maresuchthatαx=y,andAnx=0,then(α+An)x=ysothat(α+An)×