MODULIOFpDIVISIBLEGROUPS3withrespecttomultiplicationbypThenGisafunctorfromRalgebrastoQpvectorspaceswhichwecalltheuniversalcoverofGAnimportantobservationisthatifSRisasurjectionwithnilpotentkern ID: 340093
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2MODULIOFP-DIVISIBLEGROUPSReferences601.IntroductionInthisarticleweproveseveralresultsaboutp-divisiblegroupsandtheirmoduli.Ourworktouchesuponvariousthemesknowntoariseinthecontextofp-divisiblegroups,includingthecrystallineDieudonnemodule([Mes72],[BBM82]),therigid-analyticmodulispacesofRapoport-Zinkandtheirassociatedperiodmaps([RZ96]),andthemorerecentworkofFarguesandFontaineonthefundamentalcurveofp-adicHodgetheory([FF11]).Thetheoryofperfectoidspaces([Sch12])arisesrepeatedlyinourconstructionsandarguments,andinparticularitprovidestheappropriatecontextforstudyingRapoport-Zinkspacesatinnitelevel.OurrstresultconcernsthefullfaithfulnessofthecrystallineDieudonnemodulefunctor.GivenaringRinwhichpisnilpotent,andap-divisiblegroupGoverR,letM(G)denoteits(covariant)Dieudonnecrystal,asdenedin[Mes72].ThequestionofthefullfaithfulnessofMhasalonghistory(see[dJM99]forasurvey),andisknowntohaveanarmativeanswerinanumberofsituations,forinstancewhenRisanexcellentringincharacteristicpwhichisalocallycompleteintersectionring(loc.cit.).Incontrast,ourresultappliestoringsincharacteristicpwhicharetypicallynon-noetherian.WesayaringRincharacteristicpissemiperfectiftheFrobeniusmap:R!Rissurjective.Sucharingiscalledf-semiperfectifthetopologicalringlim �Rhasanitelygeneratedidealofdenition.Forinstance,thequotientofaperfectringbyanitelygeneratedidealisf-semiperfect.AnimportantexampleisOC=p,whereCisanyalgebraicallyclosedextensionofQp.TheoremA.LetRbeanf-semiperfectring.ThentheDieudonnemodulefunctoronp-divisiblegroupsuptoisogenyisfullyfaithful.IfmoreoverRisthequotientR=S=JofaperfectringSbyaregularidealJS,thentheDieudonnemodulefunctoronp-divisiblegroupsisfullyfaithful.Remark1.0.1.Inthiscase,aDieudonnemodulecanbemadeveryexplicit:Ingeneral,ifRisasemiperfectring,wehavetheFontaineringsAcris(R)andB+cris(R),andthefunctorM7!M=M(Acris(R))(resp.M7!M=M(Acris(R))[p�1])inducesanequivalenceofcategoriesbetweenthecategoryofDieudonnecrystalsoverR(resp.,uptoisogeny)andthecategoryofniteprojectiveAcris(R)(resp.B+cris(R))-modulesequippedwithFrobeniusandVerschiebungmaps.Weremarkthatthecaseofperfecteldsisclassical,andthatthecaseofperfectringsisanunpublishedresultduetoGabber,relyingonaresultofBerthelotforthecaseofperfectvaluationrings,[Ber80].Recently,thiswasreprovedbyLau,[Lau13].Inalloftheseresults,oneknowseventhattheDieudonnemodulefunctorisessentiallysurjective,andonealsohasadescriptionofnitelocallyfreegroupschemes.Ourmethodofproofisentirelydi erent.IthandlesrstthecaseofmorphismsQp=Zp!p1byanexplicitcomputationinvolvingintegralityoftheArtin-Hasseexponential,andreducestothiscasebyatrickwelearnedfromapaperofdeJongandMessing,[dJM99].Wenotethatthisreductionsteponlyworksformorphismsofp-divisiblegroups(andnotfornitelocallyfreegroupschemes),andevenifoneisonlyinterestedintheresultforperfectrings,thisreductionstepwillintroduceratherarbitraryf-semiperfectrings.Inthisreductionstep,acertaintechnicalresultisnecessary,forwhichwegiveaproofthatreliesonFaltings'salmostpuritytheoremintheformgivenin[Sch12].Asexplainedbelow,TheoremAhasdirectconsequencesinp-adicHodgetheory,sothatthisstateofa airsmaybereasonable.OfparticularinterestarethemorphismsQp=Zp!G.ObservethatifSisanyR-algebra,thentheQp-vectorspaceHomS(Qp=Zp;G)[1=p]isidentiedwiththeinverselimit~G(S)=lim �G(S)(taken MODULIOFp-DIVISIBLEGROUPS3withrespecttomultiplicationbyp).Then~GisafunctorfromR-algebrastoQp-vectorspaceswhichwecalltheuniversalcoverofG.AnimportantobservationisthatifS!Risasurjectionwithnilpotentkernel,andifGSisaliftofGtoS,then~GS(S)iscanonicallyisomorphicto~G(R)(andsodoesnotdependonthechoiceofGS).Inotherwords,~GisacrystalontheinnitesimalsiteofR.Usingthisconstruction,wecanexplainhowTheoremAiscloselyrelatedtop-adicHodgetheory.Infact,takeR=OC=p,whereCisanalgebraicallyclosedcompleteextensionofQp,andletGbeanyp-divisiblegroupoverOC,withreductionG0toOC=p.LetM(G)betheniteprojectiveB+cris-modulegivenbyevaluatingtheDieudonnecrystalofGonAcris=Acris(OC=p),andinvertingp.Then,usingTheoremAformorphismsQp=Zp!G0uptoisogeny,onendsthatM(G)'=p=Hom(Qp=Zp;G0)[p�1]=~G(OC=p)=~G(OC):Ontheotherhand,thereisanexactsequence(1.0.1)0!T(G)[p�1]!~G(OC)!LieG C!0;wherethelattermapisthelogarithmmap,andT(G)denotestheTatemoduleofG.Thistranslatesintotheexactsequence0!T(G)[p�1]!M(G)'=p!LieG C!0relatingtheetaleandcrystallinehomologyofG.WeremarkthatTheoremAalsohandlestherelativecaseofthisresult,andalsotheintegralversion.Oursecondmainresultisaclassicationofp-divisiblegroupsoverOC,whereCisstillanalgebraicallyclosedcompleteextensionofQp.WerecallthatonehasaHodge-Tatesequence0!LieG C(1)!T(G) C!(LieG_)_ C!0;where(1)denotesaTatetwist.TheoremB.Thereisanequivalenceofcategoriesbetweenthecategoryofp-divisiblegroupsoverOCandthecategoryoffreeZp-modulesTofniteranktogetherwithaC-subvectorspaceWofT C(�1).TheequivalencecarriesGtothepair(T;W),whereTistheTatemoduleofGandW=LieG C.Inparticular,theresultisaclassicationintermsoflinearalgebra,insteadof-linearalgebraasusually,and(relatedtothat)isintermsofitsetalecohomology,insteadintermsofitscrystallinecohomology.WenotethatTheoremBcanberegardedasap-adicanalogueofRiemann'sclassicationofcomplexabelianvarieties,whichareclassiedbytheirsingularhomologytogetherwiththeHodgeltration.ResultsinthedirectionofTheoremBhavepreviouslybeenobtainedbyFarguesin[Far12].Inparticular,thefullyfaithfulnesspartisprovedalreadythere.WeproveessentialsurjectivityrstinthecasethatCissphericallycomplete,withsurjectivenormmapC!R0.Inthatcase,theargumentisverydirect;thekeyideaistolookrstattherigid-analyticgenericbreofthe(connected,say)p-divisiblegroup(consideredasaformalscheme),whichisanopenballandcanbeconstructeddirectlyfrom(T;W).Fortheconverse,onehastoseethatacertainrigid-analyticvarietyisanopenball.Ingeneral,onecanshowthatitisanincreasingunionofclosedballs;undertheassumptionsonC,onecanconcludethatitisanopenball.Afterwards,wemakea(somewhatindirect)descentargumentinvolvingRapoport-Zinkspaces(andTheoremC)todeducethegeneralcase.ItwillbecrucialtorelatetheexactsequenceinEq.(1.0.1)tovectorbundlesontheFargues-FontainecurveX,whichistheobjectinvestigatedin[FF11].ThisisdenedasX=ProjP,wherePisthegradedQp-algebraP=Md0(B+cris)'=pd; 4MODULIOFP-DIVISIBLEGROUPSandB+cris=B+cris(OC=p).Thereisaspecialpoint12Xcorrespondingtothehomomorphism:B+cris!C;wewritei1:f1g!Xfortheinclusion.ForeveryisocrystalMoverFp,thereisacorrespondingvectorbundleonOC,givenbythevectorbundleassociatedtoLd0(M B+cris)'=pd,andFarguesandFontaineshowthatallthevectorbundlesonXariseinthisway.InparticularthereisabijectionH7!E(H)betweenisogenyclassesofp-divisiblegroupsoverFpandvectorbundlesonXwhoseslopesarebetween0and1.Notethati1E(H)=M(H) C.Usingthisclassicationofvectorbundles,weshowinTheorem5.1.4thateveryp-divisiblegroupGoverOCisisotrivialinthesensethatthereexistsap-divisiblegroupHoverFpandaquasi-isogeny:H FpOC=p!G OCOC=p:Thus(G;)isadeformationofHinthesenseofRapoport-Zink.AppealingtoTheoremA,wecanidentify~G(OC)=~H(OC=p)=(M(H) B+cris)'=pwiththespaceofglobalsectionsofE=E(H).TheexactsequenceinEq.(1.0.1)appearswhenonetakesglobalsectionsintheexactsequenceofcoherentsheavesonX,(1.0.2)0!F!E!i1(LieG C)!0;whereF=T ZpOX,seeProposition5.1.6.Tosummarizethesituation,ap-divisiblegroupHgivesavectorbundleEonX,whileadeformationofHtoOCgivesamodicationofE(inthesensethatFandEareisomorphicawayfrom1).WeshouldnotethatvectorbundlesoverXareequivalentto'-modulesovertheRobbaring,andthatouruseofXcouldbereplacedbyKedlaya'stheoryof'-modulesovertheRobbaring,[Ked04].WenowturntoRapoport-Zinkspacesandtheirassociatedperiodmaps,asin[RZ96].LetHbeap-divisiblegroupofdimensiondandheighthoveraperfecteldkofcharacteristicp.LetMbetheassociatedRapoport-Zinkspace.Thisisaformalschemeparametrizingdeformations(G;)ofH,seeDenition6.1.1.Passingtothegenericbre,wegetanadicspaceMad.Notethat[RZ96]usesrigidspaces,butweworkwithadicspaces,asitisimportantforustoconsidernon-classicalpoints(inparticulartotalkabouttheimageoftheperiodmorphismlater).1Weremarkthatif(R;R+)isacompleteanoid(W(k)[1=p];W(k))-algebra,thenamorphismSpa(R;R+)!MadcorrespondstoacoveringofSpa(R;R+)byopenanoidsSpa(Ri;R+i),togetherwithadeformation(Gi;i)ofHtoanopenboundedsubringofeachR+i.Wewillsimplyrefertosuchadatumasapair(G;).If(K;K+)isananoideld,thenMad(K;K+)isthesetofdeformationsofHtoK+.Thereisa(Grothendieck-Messing)periodmorphism:Mad!F`,whereMadistheadicgenericbreofMandF`istheGrassmannianvarietyofd-dimensionalquotientsofM(H) Qp(consideredasanadicspace).Ifxisa(C;OC)-pointofMadcorrespondingtoadeformation(G;)ofHtoOC,then(x)2F`(C;OC)correspondstothequotientM(H) C!WgivenbyGrothendieck-Messingtheory.Ourthirdmaintheoremprovidesadescriptionoftheimageof.A(C;OC)-valuedpointxofF`correspondstoad-dimensionalquotientM(H) C!W.LetE=E(H)bethevectorbundleoverXassociatedtoH,sothati1E=M(H) C.Themapi1E!WinducesamapE!i1i1E!i1W.LetFbethekernel,sothatwehaveamodicationofEasinEq.(1.0.2).TheoremC.ThepointxisintheimageofifandonlyifF=OhX.Inordertoprovethistheorem,itisenoughtoconsiderthecasewhereCissphericallycompleteandthenormmapC!R0issurjective.Inthatcase,TheoremBisavailabletoconstructthedesiredp-divisiblegroupG.TheconstructionoftheisogenyoverOC=preliesonTheoremA.On 1Asinourpreviouswork,wepreferadicspacesoverBerkovichspaces. 6MODULIOFP-DIVISIBLEGROUPSNotethatanypoint(G;; )ofM1givesamapZhp!T(G)ad~Gad=~Had;wheretherstmapis ,andtheisomorphismisinducedfrom.Inotherwords,allp-divisiblegroupsGparametrizedbyMhavethesameuniversalcover~H,andareoftheform~H=,where~HisaZp-lattice.Thefollowingtheoremanswersthequestionforwhich~Honecanformthequotient~H=.Thedescriptionusesthequasi-logarithmmapqlog:~Had!M(H) Ga.Weremarkthatforanydeformation(G;)ofH,wehaveacommutativediagram~Hadqlog// M(H) Ga Gadlog// LieG GaInparticular,=ker(~Had!Gad)mapstothekernelofM(H)!LieG,whichexplainstheconditionontherankinthefollowingtheorem.TheoremD.ThemoduliproblemM1isrepresentablebyanadicspace,anditisisomorphictothefunctorwhichassignstoacompleteanoid(W(k)[1=p];W(k))-algebra(R;R+)thesetofh-tupless1;:::;sh2~Had(R;R+)forwhichthefollowingconditionsaresatised.(i)ThecokernelWofthemapRh(qlog(s1);:::;qlog(sh))�������������!M(H) RisaprojectiveR-moduleofrankd.(ii)Forallgeometricpointsx=Spa(C;OC)!Spa(R;R+),thesequence0!Qhp(s1;:::;sh)������!~Had(C;OC)!W RC!0isexact.Moreover,M1(~Had)hisalocallyclosedsubspace,thespaceM1ispreperfectoid,andM1lim �nMn.ThefactthatwecandescribeM1explicitlyindependentlyofp-divisiblegroupsreliesonTheoremC.3ThelastpartofthetheoremsaysmoreexplicitlythatforanyperfectoideldextensionKofW(k)[p�1],thebase-changeM1;KofM1toSpa(K;OK)hasanaturalcompletion^M1;K,whichisaperfectoidspaceoverK,andonehasanequivalenceofetaletopoi^M1;K;etlim �nMn;K;et(atleastafterbase-changetoaquasicompactopensubsetofsomeMn;K,soastoworkwithqcqstopoi,whereprojectivelimitsarewell-dened).Weremarkthattakinginverselimitsinthecategoryofadicspacesisnotcanonical,correspondingtothephenomenonthatonadirectlimitlim�!AiofBanachalgebrasAi,onecanputinequivalentnorms(whichareequivalentwhenrestrictedtoanindividualAi,butinequivalentinthedirectlimit).ThespaceM1hasanicemodulidescription,butitisnotcleartowhichnormonlim�!Aiitcorresponds.Ontheotherhand,^M1;Kischaracterizedasgivinglim�!Aithenormmakinglim�!Aithenorm-1-subalgebra,whereAiAiisthesetofpower-boundedelements.(Thiscorrespondstotheweakestpossibletopologyonlim�!Ai.)In[Fal10], 3WenotethatusingTheoremA,onecanmakeexplicitwhat~Had(R;R+)isintermsofDieudonnetheory,atleastwhen(R;R+)isaperfectoidanoid(K;OK)-algebraforsomeperfectoideldK.ThisgivesadescriptionofM1purelyintermsofp-adicHodgetheory. MODULIOFp-DIVISIBLEGROUPS9isbounded.WriteAforthesubringofpower-boundedelementsinA.Also,wedenotebyAAtheidealoftopologicallynilpotentelements.Finally,ananoidringisapair(A;A+),withAanf-adicringandA+AanopensubringwhichisintegrallyclosedandcontainedinA.Morphismsbetweenanoidrings(A;A+)and(B;B+)arecontinuousringhomomorphismsA!BsendingA+intoB+.Forinstance,ifaringAisgiventhediscretetopologythenAisbothadic(0servesasanidealofdenition)andf-adic,and(A;A)isananoidring.Foralesstrivialexample,letKbeacompletenonarchimedeaneldwithringofintegersOKandmaximalidealmK.ThenOKisanadicringand(K;OK)isananoidring.TheintegralTatealgebraA+=OKhXiii2Iinanynumberofvariablesisanadicring.If$2mK,then$A+servesasanidealofdenitionforA+,sothatA+isf-adic.TheTatealgebraA=KhXiii2I=A+[$�1]isanf-adicringbecauseitcontainsA+asanopensubring.InthiscasewehaveA=A+,sothat(A;A+)isananoidring.Itshouldbenotedthat(A+;A+)isananoidringaswell.WenotethatitisnotalwaysthecasethatA+Aisbounded.Examplesarisebytakingnonreducedringsappearinginrigid-analyticgeometry,suchasA=K[X]=X2,inwhichcaseonecanchooseA+=A=OK+KX.Denition2.1.2.LetAbeatopologicalring.AcontinuousvaluationofAisamultiplicativemapjjfromAto�[f0g,where�isalinearlyorderedabeliangroup(writtenmultiplicatively)suchthatj0j=0,j1j=1,jx+yjmax(jxj;jyj),andforall 2�,thesetfx2A:jxj gisopen.Twovaluationsjji:A!�i[f0g(i=1;2)areequivalentiftherearesubgroups�0i�icontainingtheimageofjjifori=1;2andanisomorphismoforderedgroups:�01!�02forwhich(jfj1)=jfj2,allf2A.Denition2.1.3.Let(A;A+)beananoidring.TheadicspectrumX=Spatop(A;A+)isthesetofequivalenceclassesofcontinuousvaluationsonAwhichsatisfyjA+j1.Forx2Xandf2A,wewritejf(x)jfortheimageoffunderthecontinuousvaluationjjcorrespondingtox.Letf1;:::;fn2Abeelementssuchthat(f1;:::;fn)AisopeninA,andletg2A.DenethesubsetXf1;:::;fn;gbyXf1;:::;fn;g=x2Xjfi(x)jjg(x)j6=0;i=1;:::;nFiniteintersectionsofsuchsubsetsarecalledrational.WegiveXthetopologygeneratedbyitsrationalsubsets.Asanexample,ifKisanonarchimedeaneld,thenSpatop(K;OK)consistsofasingleequivalenceclassofcontinuousvaluations,namelythevaluationwhichdenesthetopologyonK.ThespaceSpatop(OK;OK)isatraitcontainingandaspecialpointsdenedbyjf(s)j=(1;f62mK0;f2mK:Foranyanoidring(A;A+),HuberdenesapresheafofcompletetopologicalringsOXonX=Spatop(A;A+),togetherwithasubpresheafO+X.Theyhavethefollowinguniversalproperty.Denition2.1.4.Let(A;A+)beananoidring,X=Spatop(A;A+),andUXbearationalsub-set.Thenthereisacompleteanoidring(OX(U);O+X(U))withamap(A;A+)!(OX(U);O+X(U))suchthatSpatop(OX(U);O+X(U))!Spatop(A;A+) MODULIOFp-DIVISIBLEGROUPS11ThecategoryofadicspacesisdenotedAdic.Welistsomerstproperties.Proposition2.1.6.(i)Let(A;A+),(B;B+)beanoidrings,andassumethat(A;A+)issheafyandcomplete.ThenHom(Spa(A;A+);Spa(B;B+))=Hom((B;B+);(A;A+)):Inparticular,thefunctor(A;A+)7!Spa(A;A+)isfullyfaithfulonthefullsubcategoryofsheafycompleteanoidrings.(ii)Thefunctor(A;A+)7!Spanaive(A;A+)factorsover(A;A+)7!Spa(A;A+).(iii)ThefunctorSpa(A;A+)7!Spanaive(A;A+)comingfrompart(ii)extendstoafunctorX7!Xnaive:Adic!(V).Inparticular,anyadicspaceXhasanassociatedtopologicalspacejXj,givenbytherstcomponentofXnaive.(iv)ThefullsubcategoryofadicspacesthatarecoveredbySpa(A;A+)with(A;A+)sheafyisequiva-lenttoAdichunderX7!Xnaive.Underthisidentication,weconsiderAdichasafullsubcategoryofAdic.(v)LetXbeanhonestadicspaceandlet(B;B+)beananoidring.ThenHom(X;Spa(B;B+))=Hom((B;B+);(OX(X);O+X(X))):Here,thelatteristhesetofcontinuousringhomomorphismsB!OX(X)whichmapB+intoO+X(X).Proof.Easyandlefttothereader.AbasicexampleofananoidadicspaceistheclosedadicdiscSpa(A;A+),whereA=KhXiandA+=OKhXi.See[Sch12],Example2.20foradiscussionofthevespeciesofpointsofSpatop(A;A+).Anexampleofanonanoidadicspaceistheopenadicdisc,equaltotheopensubsetofSpa(OKJXK;OKJXK)denedbytheconditionj$j6=0,where$2mK.AcoveringoftheopendiscbyanoidadicspacesoverKisgivenbythecollectionSpa(KhX;Xn=$i;OKhX;Xn=$i)forn1.Inthefollowing,wewilloftenwriteSpa(A;A+)foreitherofSpa(A;A+)orSpatop(A;A+).Asbydenition,theiropensubsetsagree,wehopethatthiswillnotresultinanyconfusion.However,wewillnevertalkaboutSpanaive(A;A+)inthefollowing.2.2.Formalschemesandtheirgenericbres.Inthissubsection,werecallthedenitionofformalschemesthatwewilluse,andrelatethemtothecategoryofadicspaces.FixacompletenonarchimedeaneldKwithringofintegersO=OKandxsome$2O,j$j1.ConsiderthecategoryNilpOofO-algebrasRonwhich$isnilpotent.ItsoppositeNilpopOhasthestructureofasite,usingZariskicovers.Wegettheassociatedtopos(NilpopO).AnyadicO-algebraAwithidealofdenitionIcontaining$givesrisetothesheaf(SpfA)(R)=lim�!nHom(A=In;R)onNilpopO.Asbefore,onedenesopenembeddingsin(NilpopO),andhenceformalschemesoverOasthosesheavesonNilpopOwhichadmitanopencoverbySpfAforAasabove,cf.[RZ96],rstpageofChapter2.Proposition2.2.1.ThefunctorSpfA7!Spa(A;A)extendstoafullyfaithfulfunctorM7!MadfromformalschemesoverOwhichlocallyadmitanitelygeneratedidealofdenitiontoadicspacesoverSpa(O;O). 12MODULIOFP-DIVISIBLEGROUPSProof.LetusexplainhowonecanreconstructSpfAfromSpa(A;A).First,notethatSpfASpa(A;A)isthesubsetofvaluationswithvaluegroupf0;1g,recoveringthetopologicalspace.Torecoverthestructuresheaf,itisenoughtoshowthattheglobalsectionsofthesheacationO]Spa(A;A)ofOSpa(A;A)aregivenbyAitself.NotethatthereisacontinuousmapSpa(A;A)!SpfA:x7!x,givenbysendinganycontinuousvaluationxonAtothevaluationf7!jf(x)j=0jf(x)j11jf(x)j=1:Moreover,everyx2Spa(A;A)specializestox2Spf(A)Spa(A;A).ItfollowsthatanycoveringofSpa(A;A)isrenedbyacoveringcomingaspullbackviathespecializationfromacoverofSpfA,soweonlyhavetocheckthesheafpropertyforsuchcoverings.ButthisreducestothesheafpropertyofthestructuresheafofSpfA.NotethatifAisanadicO-algebrawithidealofdenitionIcontaining$,andF2(NilpopO),thenHom(SpfA;F)=lim �nF(A=In):Weabbreviatetheleft-handsidetoF(A),therebyextendingthefunctorFtothelargercategoryofadicO-algebraswithidealofdenitioncontaining$.ThefollowingpropositiongivesamodulidescriptionofthegenericbreMad=MadSpa(O;O)Spa(K;O):Proposition2.2.2.Let(R;R+)beacompleteanoid(K;O)-algebra.(i)TheringR+isthelteredunionofitsopenandboundedO-subalgebrasR0R+.(ii)ThefunctorMad:CA (K;O)!Setsisthesheacationof(R;R+)7!lim�!R0R+M(R0)=lim�!R0R+lim �nM(R0=$n):Proof.(i)First,thereissomeopenandboundedsubringR0R+.Next,wecheckthatifR1;R2R+areboundedsubrings,thensoistheimageofR1 R2!R+.Indeed,Ri$�niR0forcertainni2Z,andthentheimageofR1 R2iscontainedin$�n1�n2R0.Inparticular,thereareopenandboundedO-subalgebras,andthesetofsuchisltered.Finally,ifR0R+isanopenandboundedO-subalgebraandx2R+,thenxispower-bounded,sothatallpowersofxarecontainedin$�nR0forsomen,andhenceR0[x]$�nR0isstillopenandbounded.(ii)ItsucestocheckinthecaseM=Spf(A;A),whereAissomeadicO-algebrawithnitelygeneratedidealIcontaining$.ThenitfollowsfromHom((A;A);(R;R+))=Hom(A;R+)=lim�!R0R+Hom(A;R0)=lim�!R0R+lim �nHom(A;R0=$n):Indeed,anycontinuousmapA!R+isalsocontinuousforthe$-adictopologyonA;thenA[1 $]!RisacontinuousmapofK-Banachspaces.Inparticular,theimageofAinR+isbounded,andhencecontainedinsomeopenandboundedO-subalgebraR0.AsR0is$-adicallycomplete,onegetstheidenticationHom(A;R0)=lim �nHom(A;R0=$n): 14MODULIOFP-DIVISIBLEGROUPSWecall(^Ai;^A+i)thestrongcompletionof(Ai;A+i).Remark2.3.5.Asanexample,thenonreducedrigid-analyticpointX=Spa(K[X]=X2;OK+KX)ispreperfectoid.Notethatinthisexample,theanoidalgebra(K[X]=X2;OK+KX)isalreadycompleteforitsnaturaltopology.However,articiallyenforcingthep-adictopologyonOK+KXmakesthecompletionsimply(K;OK),whichiscertainlyaperfectoidanoid(K;OK)-algebra.Proposition2.3.6.LetXbeapreperfectoidadicspaceoverSpa(K;OK).Thenthereexistsaperfectoidspace^XoverSpa(K;OK)withthesameunderlyingtopologicalspace,andsuchthatforanyopensubsetU=Spa(A;A+)Xforwhich(^A;^A+)isperfectoid,thecorrespondingopensubsetof^Xisgivenby^U=Spa(^A;^A+).Thespace^XisuniversalformorphismsfromperfectoidspacesoverSpa(K;OK)toX.Proof.Easyandlefttothereader.Wecall^XthestrongcompletionofX.Wehavethefollowingbehaviourwithrespecttopassagetolocallyclosedsubspaces.Proposition2.3.7.LetXbeapreperfectoidadicspaceoverSpa(K;OK),andletYXbealocallyclosedsubspace.ThenYispreperfectoid.Remark2.3.8.Again,thisisallonecanexpectasYmaybenonreduced.Proof.Thestatementisclearforopenembeddings,soletusassumethatYXisaclosedsubspace.WemayassumethatX=Spa(A;A+)isanoidwith(^A;^A+)perfectoid.ThenY=Spa(B;B+)isgivenbyaclosedidealI,sothatB=A=IandB+theintegralclosureoftheimageofA+inB.Weclaimthat^BisaperfectoidK-algebra,whichimpliestheresult.NotethatYisthelteredintersectionofitsrationalneighborhoodsUiX;indeed,foranyf1;:::;fk2I,onecanconsiderthesubsetswherejf1j;:::;jfkjp�n,andthenYistheirintersection.Then^B+=\lim�!YUiO+X(Ui):Indeed,anyf2Ibecomesinnitelyp-divisibleinthedirectlimit,andthusgetskilledinthecompletion.ButeachOX(Ui)isperfectoid,hencesois^B.Finally,wewillneedonemorenotion.Denition2.3.9.LetXbeanadicspaceoverSpa(Qp;Zp).ThenXispreperfectoidifforanyperfectoideldKofcharacteristic0,thebase-changeXK=XSpa(Qp;Zp)Spa(K;OK)ispreperfec-toid.Proposition2.3.10.LetKbeaperfectoideld,andletXbeanadicspaceoverSpa(K;OK).ThenXispreperfectoidasanadicspaceoverSpa(K;OK)ifXispreperfectoidasanadicspaceoverSpa(Qp;Zp).Proof.Byassumption,XKispreperfectoidoverSpa(K;OK).ButthereisaclosedembeddingX!XK,whichshowsthatXispreperfectoidoverSpa(K;OK)byProposition2.3.7.Proposition2.3.11.LetXbeapreperfectoidadicspaceoverSpa(Qp;Zp),andletYXbealocallyclosedsubspace.ThenYispreperfectoid.Proof.ThisfollowsdirectlyfromProposition2.3.11.Example2.3.12.LetA=ZpJX1=p11;:::;X1=p1nKbethecompletionoftheZp-algebralim�!Zp[X1=pm1;:::;X1=pmn]withrespecttotheI=(p;X1;:::;Xn)-adictopology. MODULIOFp-DIVISIBLEGROUPS15Proposition2.3.13.LetRbeanadicZp-algebrawithidealofdenitioncontainingp.Then(SpfA)(R)=(lim �x7!xpR)n=(lim �x7!xpR=p)n=(R[)n;whereR[=lim �x7!xpR=p,equippedwiththeinverselimittopology.Proof.Easyandlefttothereader.LetX=(SpfA)adbethegenericbreofSpa(A;A)!Spa(Zp;Zp).ThenXisanadicspaceoverSpa(Qp;Zp),equaltotheunionofanoidsSpa(Am;A+m)denedbyjX1jm;:::;jXnjmjpj;m1:ItiseasytoseethatXisapreperfectoidspaceoverSpa(Qp;Zp).Lemma2.3.14.TheadicspaceXoverSpa(Qp;Zp)isthesheafassociatedto(R;R+)7!lim�!R0R+(R[0)n:Proof.ThisfollowsdirectlyfromProposition2.2.2.2.4.Inverselimitsinthecategoryofadicspaces.Inverselimitsrarelyexistinthecategoryofadicspaces,evenwhenthetransitionmapsareane.Theproblemisthatif(Ai;A+i)isadirectsystemofanoidrings,thenthedirectlimittopologyoflim�!Aiwillnotbef-adic,andthereisnocanonicalchoiceforthetopology.(Anotherproblemarisesbecauseofnon-honestadicspaces.)Wewillusethefollowingdenition.Denition2.4.1.LetXibealteredinversesystemofadicspaceswithquasicompactandqua-siseparatedtransitionmaps,letXbeanadicspace,andletfi:X!Xibeacompatiblefamilyofmorphisms.WewriteXlim �XiifthemapofunderlyingtopologicalspacesjXj!lim �jXijisahomeomorphism,andifthereisanopencoverofXbyanoidSpa(A;A+)X,suchthatthemaplim�!Spa(Ai;A+i)XiAi!Ahasdenseimage,wherethedirectlimitrunsoverallopenanoidSpa(Ai;A+i)XioverwhichSpa(A;A+)X!Xifactors.IftheinverselimitconsistsofasinglespaceX0,thenwewriteXX0insteadofXlim �X0.Proposition2.4.2.Let(Ai;A+i)beadirectsystemofcompleteanoidrings.AssumethatthereareringsofdenitionAi;0Aicompatibleforalli,andnitelygeneratedidealsofdenitionIiAi;0suchthatIj=IiAj;0Aj;0forji.Let(A;A+)bethedirectlimitofthe(Ai;A+i),equippedwiththetopologymakingA0=lim�!iAi;0anopenadicsubringwithidealofdenitionI=lim�!iIi.ThenSpa(A;A+)lim �Spa(Ai;A+i):Proof.Itisclearthat(A;A+)withthegiventopologyisananoidring.Moreover,givingavaluationon(A;A+)isequivalenttogivingacompatiblesystemofvaluationsonall(Ai;A+i).ThecontinuityconditionisgivenbyaskingthatjInj!0asn!1,checkingthatalsocontinuousvaluationscorrespond,thusjSpa(A;A+)j=lim �jSpa(Ai;A+i)j:Theconditiononringsissatisedbydenition,nishingtheproof. 16MODULIOFP-DIVISIBLEGROUPSProposition2.4.3.InthesituationofDenition2.4.1,letYi!Xibeanopenimmersionofadicspaces,andletYj=YiXiXj!Xjforjibethepullback,aswellasY=YiXiX!X.ThenYlim �jiYj.Proof.SeeRemark2.4.3in[Hub96].Proposition2.4.4.LetXbeapreperfectoidspaceoverSpa(K;OK),whereKisaperfectoideld.Let^XbethestrongcompletionofX.Then^XX.Proof.Immediate.Ofcourse,thisshowsthatonemayhaveXX0,whileX6=X0.Letushoweverrecordthefollowingresult.Proposition2.4.5.LetKbeaperfectoideld,letXibeaninversesystemofadicspacesoverSpa(K;OK)withqcqstransitionmaps,andassumethatthereisaperfectoidspaceXoverSpa(K;OK)suchthatXlim �Xi.Thenforanyperfectoidanoid(K;OK)-algebra(B;B+),wehaveX(B;B+)=lim �Xi(B;B+):Inparticular,ifYisaperfectoidspaceoverSpa(K;OK)withacompatiblesystemofmapsY!Xi,thenYfactorsoverXuniquely,makingXuniqueuptouniqueisomorphism.Proof.WemayassumethatX=Spa(A;A+)isanoidsuchthat(A;A+)isaperfectoidanoid(K;OK)-algebra.ThenX(B;B+)=Hom((A;A+);(B;B+)):Aslim�!im(X)Spa(Ai;A+i)XiAi!Ahasdenseimage,itfollowsthatthemapX(B;B+)!lim �Xi(B;B+)isinjective.Conversely,givenacompatiblesystemofmapsSpa(B;B+)!Xi,itfollowsthatSpa(B;B+)!XifactorsovertheimageofXinXi.Wegetamaplim�!im(X)Spa(Ai;A+i)XiA+i!B+:Passingtop-adiccompletions,theleft-handsidebecomesequaltoA+,as(A;A+)isstronglycomplete,andtheleft-handsideisdenseinA+.ThisgivesthedesiredmapA+!B+,i.e.(A;A+)!(B;B+).Itseemsreasonabletoexpectthatthereisagooddenitionofanetalesiteforanyadicspace,andthatthefollowingresultistrue.Conjecture2.4.6.AssumethatXiisaninversesystemofquasicompactandquasiseparatedadicspaces,andthatXlim �Xi.ThentheetaletoposofXisequivalenttotheprojectivelimitoftheetaletopoioftheXi(consideredasabredtoposintheobviousway).WewillonlyneedthecaseswhereallXiandXareeitherstronglynoetherianadicspaces,orperfectoidspaces.Inthatcase,thereisadenitionoftheiretaletopoi,andwehavethefollowingresult. MODULIOFp-DIVISIBLEGROUPS17Theorem2.4.7([Sch12],Thm.7.17).LetKbeaperfectoideld,andassumethatallXiarestronglynoetherianadicspaces(inparticularquasicompactandquasiseparated)overSpa(K;OK),andthatXisaperfectoidspaceoverSpa(K;OK),Xlim �Xi.ThentheetaletoposofXistheprojectivelimitoftheetaletopoioftheXi.Weremarkthatthenotionintroducedhereisstrongerthanthenotion0introducedin[Sch12],whichonlytalksabouttheresidueeldsinsteadofthesectionsofthestructuresheafinanopenneighborhood.Asexplainedin[Hub96,Section2.4],oneshouldexpectthattheweakernotion0issucientintheconjectureifallXiareanalyticadicspaces,asisthecaseintheprevioustheorem.3.Preparationsonp-divisiblegroupsInthissection,weworkoveraringRwhichisp-torsion,andconsiderp-divisiblegroupsGoverR.LetNilpopRdenotethecategoryoppositeofthecategoryofR-algebrasonwhichpisnilpotent.Ofcourse,thelastconditionisvacuous,butlaterwewillpasstop-adicallycompleteZp-algebrasR.3.1.Theuniversalcoverofap-divisiblegroup.WewillconsiderGasthesheafonNilpopR,sendinganyR-algebraStolim�!G[pn](S).Lemma3.1.1.AssumethatGisisogenoustoanextensionofanetalep-divisiblegroupbyaconnectedp-divisiblegroup.ThenthefunctorGisrepresentablebyaformalscheme(stilldenotedG),whichlocallyadmitsanitelygeneratedidealofdenition.Proof.Itisclearthatisogeniesinducerelativelyrepresentablemorphisms,whichdonotchangetheidealofdenition,sowemayassumethatGisanextensionofanetalebyaconnectedp-divisiblegroup.Thenonereducestotheconnectedcase,inwhichcaseGisaformalLiegroup,givingtheresult.Onecanalsoconsiderthecompletion^GofG,whichwedeneasthefunctoronNilpopRsendinganyR-algebraSonwhichpisnilpotenttothesubsetofthosex2G(S)forwhichthereissomenilpotentidealISsuchthatx2G(S=I)isthezerosection.Werecallthatthisisalwaysrepresentable.Lemma3.1.2.Thefunctor^GisaformalLievariety,andinparticularrepresentablebyananeformalschemewithnitelygeneratedidealofdenition.IfGisconnected,thenG=^G.Moreover,ifLieGisfree,then^G=SpfRJX1;:::;XdK:Proof.Cf.[Mes72].Considerthesheaf~G(S)=lim �p:G!GG(S);onNilpopR,whichwecalltheuniversalcoverofG.Clearly,~GisasheafofQp-vectorspaces.Proposition3.1.3.(i)Let:G1!G2beanisogeny.Thentheinducedmorphism~:~G1!~G2isanisomorphism.(ii)LetS!Rbeasurjectionwithnilpotentkernel.Thenthecategoriesofp-divisiblegroupsoverRandSuptoisogenyareequivalent.Inparticular,foranyp-divisiblegroupGoverR,theuniversalcover~Gliftscanonicallyto~GSoverS.Moreover,~GS(S)=~G(R).(iii)AssumethatGisisogenoustoanextensionofanetalebyaconnectedp-divisiblegroup.Then~Gisrepresentablebyaformalscheme(stilldenoted~G),whichlocallyadmitsanitelygenerated MODULIOFp-DIVISIBLEGROUPS19Lemma3.1.6.Thesheaf~GG^GisrepresentablebyananeformalschemeSpfSwithnitelygeneratedidealofdenitionISgivenbythepreimageoftheidealdeningtheunitsectionin^G.Moreover,S,IandS=Iare atoverR,andifRisofcharacteristicp,thenSisrelativelyperfectoverR,i.e.therelativeFrobeniusmorphismS=R:S R;R!Sisanisomorphism.Proof.AsGp!Gisrelativelyrepresentable(andnitelocallyfree),themapGG^G!^Gisrelativelyrepresentable,andthusGG^Gisrepresentable,wherethetransitionmapG!Gismultiplicationbyap-power.Moreover,theidealofdenitiondoesnotchange:Onecantakethepreimageoftheidealdeningtheunitsectionin^G.Uponpassingtotheinverselimit(i.e.,thedirectlimitonthelevelofrings),wegetrepresentabilityof~GG^G:The atnessassertionsareeasytocheck,as^GisaformalLievarietyandthetransitionmorphismsareniteandlocallyfree.TocheckthatSisrelativelyperfectoverR,onehastocheckthatthepullbackof^G(p)alongF:G!G(p)is^G,whichisobvious.Remark3.1.7.Notethat~GG^Gisstillacrystalontheinnitesimalsite,ascheckingwhetherasectionofGonathickeningS!RofRliesin^G(S)canbecheckedafterrestrictingtoR,bydenitionof^G.3.2.Theuniversalvectorextension,andtheuniversalcover.LetEGbetheuniversalvectorextensionofG:0!V!EG!G!0:Deneamap(3.2.1)sG:~G(R)!EG(R)asfollows.Representanelementx2~G(R)byasequencexn2G(R).Liftxntoanelementyn2EG(R),andformthelimitlimn!1pnyn.Thelimitexistsbecausethedi erencespm+nym+n�pnynlieinpnV(R).Adi erentchoiceofliftschangesthevalueofybyanelementoftheformlimn!1pnvnwithvn2V(R),butthislimitis0.SetsG(x)=y.ItiseasytoseethattheformationofsGisfunctorialinG.WenotethatifG=Qp=Zp,thentheuniversalvectorextensionofGis0!Ga!GaQp Zp!Qp=Zp!0;andthemorphismsG:~G!EGisexactlytheinclusionofQpinto(RQp)=Zp.Lemma3.2.1.ThemorphismsGextendsfunctoriallytoamorphismofcrystals~G!EGonthenilpotentcrystallinesite(i.e.,thickeningswithanilpotentPDstructure).Proof.LetS!RbeanilpotentPDthickening.Asonecanalwaysliftp-divisiblegroupsfromRtoS,itisenoughtoprovethefollowingresult.Lemma3.2.2.LetG,Hbetwop-divisiblegroupsoverS,withreductionsG0,H0toR.Letf0:G0!H0beanymorphism.Thenthediagram~G(S)sG// ~f0 EG(S)Ef0 ~H(S)sH// EH(S) 20MODULIOFP-DIVISIBLEGROUPScommutes.Proof.Somemultiplepnf0off0liftstoamorphismf:G!HoverS.Itisclearthatthediagram~G(S)sG// ~f EG(S)Ef ~H(S)sH// EH(S)commutes,aseverythingisdenedoverS.Letg2Hom(~G(S);EH(S))denotethedi erenceofthetwomorphismsinthediagramofthelemma.Thenwejustprovedthatpng=0.Ontheotherhand,~G(S)isaQp-vectorspace.Itfollowsthatg=0.NowassumethatSisatopologicalZp-algebra,equippedwithasurjectionS!R,whosekernelistopologicallynilpotentandhasaPDstructure.WedonotrequirethePDstructuretobe(topologically)nilpotent.Forexample,weallowthesurjectionZ2!F2.Inthefollowing,wewriteM(G)forthecrystalLieEG.Notethatbythetheoryof[BBM82],itcanbemadeintoacrystalonthecrystallinesite,notjustonthenilpotentcrystallinesite.Denition3.2.3.Thequasi-logarithmmapisdenedasthemapqlogG:~G(S)!M(G)(S)[p�1]givenasthecomposite~G(S)sG�!EG(S)logEG�!M(G)(S)[p�1]:Remark3.2.4.Inthisdenition,wewanttoevaluateEGonS.Thismaynotbepossible,asthethickeningS!Risnotrequiredtohavea(topologically)nilpotentPDstructure.Theeasiestwayaroundthisisthefollowing.ConsiderR0=S=I2,whereI=ker(S!R).ThenS!R0isatopologicallynilpotentPDthickening.Uptoisogeny,workingoverRandR0isequivalent.Asthequasi-logarithmonlyinvolvesobjectsuptoisogeny,wecanusethedenitionforS!R0togettheresultforS!R.Lemma3.2.5.LetG0bealiftofGtoS.Thediagram~G(S)qlogG// M(G)(S)[p�1] G0(S)logG0// (LieG0)[p�1]commutes.Remark3.2.6.Thislemmasaysthatontheuniversalcover~G(S),whichisindependentofthelift,onehasquasi-logarithmmap(whichisalsoindependentofthelift),andwhichspecializestothelogarithmmapofanychosenliftG0uponprojectiontothecorrespondingquotientM(G)(S)!LieG0.Proof.ReducetothecaseofatopologicallynilpotentPDthickeningasinthepreviousremark.Thenthemapsfrom~G(S)=~G0(S)bothfactoroversG0:~G0(S)!EG0(S).Theremainingdiagramisjustthefunctorialityoflog. 22MODULIOFP-DIVISIBLEGROUPS3.4.Thelogarithm.TheLiealgebraLieGisalocallyfreeR-module.Wecanassociatetoitanadicspace.WegivetheconstructionforanylocallyfreeR-module.Proposition3.4.1.LetMbealocallyfreeR-module.ThesheacationofthefunctoronCA Spa(R;R)sending(S;S+)toM RSisrepresentablebyanadicspaceoverSpa(R;R),whichwedenoteM Ga.Proof.OneimmediatelyreducestothecasewhereMisfree,andhencewhereM=R.Now,wecanassumethatR=Zp,asthegeneralcasefollowsfromthisbybase-change.Wehavetoshowthatthesheacationofthefunctor(S;S+)7!Soncompleteanoid(Qp;Zp)-algebrasisrepresentablebyanadicspaceGaoverQp.NotethatasS=S+[p�1],itcanbewrittenasthedirectlimitofthefunctor(S;S+)7!S+alongthemultiplicationbypmap.ThelatterfunctorisrepresentablebySpa(QphXi;ZphXi),henceGa=A1isrepresentablebytheincreasingunionoftheballsSpa(QphpmXi;ZphpmXi),m0.Theobjectsintroducedsofarsitinanexactsequence.Proposition3.4.2.(i)ThereisanaturalZp-linearlogarithmmaplogG:Gad!LieG GaofsheavesoverCA Spa(R;R).(ii)ThefunctorGadisrepresentablebyanadicspaceoverSpa(R;R).(iii)Thesequence0!Gad[p1]!Gad!LieG GaofsheavesofZp-modulesoverCA Spa(R;R)isexact.(iv)Thefunctor~GadisrepresentablebyanadicspaceoverSpa(R;R).(v)Thesequence0!V(G)ad!~Gad!LieG GaofsheavesofQp-vectorspacesoverCA Spa(R;R)isexact.Proof.Functoriallyin(S;S+),wehavetodeneaZp-linearmaplogG:Gad(S;S+)!(LieG Ga)(S):RecallthatGadisthesheafassociatedto(S;S+)7!lim�!S0S+G(S0);andLieG Gaisthesheafassociatedto(S;S+)7!LieG RS.Changingnotationslightly,weseethatitisenoughtodescribeafunctorialmaplogG:G(R)!LieG[p�1]foranyp-adicallycompleteandseparated atZp-algebraR,andp-divisiblegroupGoverR.ThenLemma2.2.5of[Mes72]givesaZp-linearbijectionlogG:ker(G(R)!G(R=p2))=!p2LieG:AsmultiplicationbypistopologicallynilpotentonG(R),anysectionx2G(R)admitssomeintegern0suchthat[pn]G(x)2ker(G(R)!G(R=p2)).ThisallowsonetoextendthemorphismtologG:G(R)!LieG[p�1];asdesired.ItalsoshowsthatthekernelispreciselyG(R)[p1],whichprovespart(iii).Forpart(ii),notethatwehaveprovedthatthereisanopensubsetUoftheidentityofGadonwhichlogGisan MODULIOFp-DIVISIBLEGROUPS25ThenR[=FpJX1=p1;Y1=p1K,andwegetcanonicalTeichmullerlitsX;Y2W(R[).Let p(x)denotethep-thdividedpowerofx.Thenwecanconsidertheelement = p(X2) p(Y2)� p(XY)2:Itiseasytoseethatp =0,andalso'( )=0.However,onecancheckthat 6=0.Forthis,lookatthethickeningS=Fp[X1=p1;Y1=p1;U;V]=(X2+;X2Y;X1+Y;XY1+;XY2;Y2+;XU;YU;XV;YV;U2;V2)ofR.Hererunsthrough1=pn,foralln0.NotethatthekernelKofS!RhasanFp-basis(X2;XY;Y2;U;V;UV).OnecandenedividedpowersonKinauniquewayextending p(aX2+bXY+cY2+dU+eV+fUV)=apU+cpV:OnegetsaninducedmapAcris(R)!Ssending toUV6=0.Notethat canbeinterpretedasamorphismofDieudonnecrystalsfromQp=Zptop1,whichcannotcomefromamorphismofp-divisiblegroups.Inthisrstexample,Risf-semiperfect.Anotherpathologyoccursforsemiperfectringswhicharenotf-semiperfect.ConsidertheringR=Fp[X1=p11;X1=p12;:::]=I;whereI=(X1;X2;:::;8n:X1=pn1=(X2X3)1=pn=(X4X5X6)1=pn=:::):Inthisexample,R[=FpJX1=p11;X1=p12;:::K=(8n:X1=pn1=(X2X3)1=pn=(X4X5X6)1=pn=:::):Thisringiscompleteforthetopologymakingtheidealsk(J)=(Xpk1;Xpk2;:::),k0,abasisofopenneighborhoodsof0.Thentheelement[X1]pliesinthekernelofW(R[)!Acris(R).Indeed,itisenoughtoshowthatitisdivisiblebypkforallk.ButwecanwriteX1asaproductofkelementsofJ,X1=Y1Yk.(Forexample,ifk=3,takeY1=X4,Y2=X5,Y3=X6.)Then[X1]p=[Y1]p[Yk]p=(p!)k p(Y1) p(Yk);asclaimed.However,forf-semiperfectrings,thissecondpathologycannotoccur.Lemma4.1.7.LetRbeanf-semiperfectring.ThenthecanonicalmapW(R[)!Acris(R)isinjective.Proof.Fortheproof,wewillconstructcertainexplicitPDthickenings.LetJ=ker(R[!R);wemayassumethat(J)=Jp.Thisimpliesthatthesubset[J]:=fXi0[ri]pi2W(R[)jri2JgW(R[)isanideal.Moreover,W(R[)iscompleteforthe[J]-adictopology.LetWPDW(R[)[p�1]denotethesubringgeneratedbyalldividedpowersofelementsof[J].ItisthequotientofthePDhullofW(R[)!Rbyitsp-torsion.NowconsiderWPD;n=WPD=(WPD\'n([J][p�1])):ThenelementsofWPD;ncanbewrittenuniquelyasasumPi2Z[ri]pi,whereri2R[=n(J),andzeroforisucientlynegative.ItiseasytoseethatthePDstructureonWPD(R[)passesuniquelytothequotientWPD;n.Moreover,WPD;nisp-adicallycomplete.Onegetsaninducedmap MODULIOFp-DIVISIBLEGROUPS27withan2W(R[),convergingto0,denesap-torsionelementinAcris(R).ThenthereexistsXn0bn n(T)2W(R[)hTiwith(X�T)Xn0bn n(T)=pXn0an n(T):Thismeanspa0=Xb0,pa1=Xb1�Tb0,andingeneralpan=Xbn�nTbn�1.UsingthatXisnottorsioninR[,onechecksinductivelythatbnisdivisiblebyp,whichthenshowsthatPn0an n(T)mapsto0inAcris(R),asdesired.Remark4.1.10.ThenalcomputationshowingthatAcris(R)isp-torsionfreeifJ=ker(R[!R)isprincipalandgeneratedbyanon-torsionelement,isthesameastheclassicalcomputationshowingthatFontaine'sringAcris=Acris(OC=p)isp-torsionfree(andgeneralizesit),whereC=Qpisanalgebraicallyclosedcompleteextension.Moregenerally,wehavethefollowingproposition.Wewillnotneedthefollowingresultsintherestofthepaper,andincludethemonlyforcompleteness.Proposition4.1.11.LetRbeanf-semiperfectringsuchthatR=S=J,whereSisaperfectring,andJ=(s1;:::;sn)isanidealgeneratedbyaregularsequences1;:::;sn2S.ThenAcris(R)isp-torsionfree.Moreover,:R!RadmitsthestructureofaPDthickening.Proof.Easyandlefttothereader.WenotethatthePDstructureon:R!Rdependsonthechoiceoftheregularsequence(s1;:::;sn).Corollary4.1.12.LetRbeanf-semiperfectringsuchthatR=S=J,whereSisaperfectring,andJisanidealgeneratedbyaregularsequence.AssumeTheorem4.1.4.ThentheDieudonnemodulefunctoronp-divisiblegroupsoverRisfullyfaithful.Proof.WefollowtheproofofLemma11in[Fal10].LetG;Hbetwop-divisiblegroupsoverR.WehavetocheckthatHom(G;H)!Hom(M(G);M(H))isanisomorphism.AsAcris(R)isp-torsionfree,bothgroupsare atoverZp.Asthekernelandcokernelarep-torsion,weseethatthekernelistrivial;wehavetoshowthatthecokernelistrivialaswell.Forthis,wehavetocheckthatiff2Hom(G;H)suchthatM(f)isdivisiblebyp,thenfisdivisiblebyp.Forthis,evaluateM(f)atthePDthickening:R!R.Byassumption,thisevaluationis0,asRisp-torsion.FixingliftsofG0andH0to:R!R,wendthatfliftstof0:G0!H0,andthatthemapinducedbyf0onLiealgebrasoftheuniversalvectorextensionistrivial.Weclaimthatf0factorsasacompositeG0F!G0(1)!H0(1)V!H0;whereG0(1);H0(1)arethepullbacksofG0;H0via:R!R.Byduality,itsucestocheckfactor-izationoverG0(1).ButG0!H0istrivialonLiealgebras,whichmeansthatthemapbecomestrivialonthekernelofFrobenius,i.e.thedesiredfactorization.Butthenwegetamapg(1):G0(1)!H0(1),whichisthesamethingasamapg:G!HoverR.Onechecksdirectlythatpg=f,asdesired.4.2.ThecaseQp=Zp!p1.Inthissection,wewillprovethefullfaithfulnessresultsuptoisogenyforthecaseofhomomorphismsfromQp=Zptop1.WenotethattheDieudonnemoduleM(Qp=Zp)(resp.,M(p1))isafreeAcris(R)-moduleofrank1,withFactingonabasiselementasp(resp.,as1). 28MODULIOFP-DIVISIBLEGROUPSLemma4.2.1.ThereisacanonicalidenticationHomR(Qp=Zp;p1)=1+JR[;underwhichthecanonicalmapHomR(Qp=Zp;p1)!Hom(M(Qp=Zp);M(p1))[p�1]=B+cris(R)'=pisidentiedwiththemap1+J!B+cris(R)'=p:r7!log([r]):Proof.ThisfollowsfromLemma3.5.1above.Fortherstpart,notethatanelementofHomR(Qp=Zp;p1)isgivenbyasequencer0;r1;:::2Rsuchthatri=rpi+1,andr0=1.Thisgivesrisetoanelementofr2R[suchthatrmapsto1inR,i.e.r21+J.LetI=ker(W(R[)!R).Thefollowinglemmaimpliesthat1+J!B+cris(R)'=pisinjective.Lemma4.2.2.Letw21+I.Iflogw=0,thenw=1.Ifp6=2,theninfactw=1.Proof.Assumerstp6=2.Notethatwp21+pA.Wehavelogwp=0,sothat0=logwp=(wp�1)1�wp�1 2+(wp�1)2 3�::::Thefactorontherightliesin1+pA2A,whencewp=1(inAcris(R),henceinW(R[)byLemma4.1.7).ButasR[isperfect,therearenonontrivialp-throotsinW(R[),hencew=1.Ifp=2,arguesimilarlywithw421+4A,showingthatw4=1.Forp=2,theonlynontrivial4-throotsofunityinW(R[)are1,givingtheresult.Inthefollowing,weuseProposition4.1.5and4.1.2toreplaceRbyanf-semiperfectringforwhichJ=ker(R[!R)satises(J)=Jp.Fork0,letI(k)W(R[)betheidealI(k)=[Jpk]W(R[)+p[Jpk�1]W(R[)+:::+pk[J]W(R[)+pk+1W(R[);sothatI(0)=I.IntermsofthepresentationofWittvectorsassequences,wehaveI(k)=f(a0;a1;:::)jai2Jpkfori=0;1;:::;kg:ThusI(k)isthekernelofthehomomorphismfromW(R[)ontoWk+1(R[=Jpk),whereWiisthefunctoroftruncatedWittvectorsoflengthi.SinceW(R[)=lim �Wk(R[),andR[=lim �R[=Jpk,thefollowinglemmaisimmediate.Lemma4.2.3.TheringW(R[)iscompletewithrespecttothelineartopologyinducedbytheidealsI(k),i.e.,W(R[)=lim �W(R[)=I(k).Moreover,foranyi0,(1+I)pi1+I(i).RecallthequotientAcris(R)!W(R[)=[J]ofAcris(R)fromtheproofofLemma4.1.7.WeletNAcris(R)bethekernel.NotethatNisgeneratedby n([x]),forx2Jandn1.Lemma4.2.4.Themap'1:Icris(R)!Acris(R)preservesNIcris(R).Therestriction'1jNistopologicallynilpotent.Proof.RecallthatNisgeneratedasAcris(R)-modulebyz= n([x]),wherex2Jandn1.Wecomputefromthedenitionof'1,'1(z)='1( n([x]))=(np)! pn! np([x])2N:Moreover,thisisdivisibleby(n!)p�1,fromwhichoneeasilydeducesthat'1istopologicallynilpotent. 34MODULIOFP-DIVISIBLEGROUPSYisanFp-vectorspace.ThusY FpOC=pisfree.Ifitiskilledbyp1�,itfollowsthatY=0,asdesired.Werecallthatwewanttoshowthatthecokernelof :�TG(T+=p) Zp(T+) W(R)T+=pk!M(G) W(R)T+=pkiskilledbyp3�forsome�0.Takeanyt2TG(T+=p).Write:TG(T+=p)!M(G) W(R)T+=pkforthecanonicalmap.Weclaimthatp(t)liesintheimageof .Thiswillgivetheresult,astheimageof isthenaT+=pk-submoduleofM(G) W(R)T+=pkcontainingp(TG(T+=p)),givingtheresultby(4.3.1).Nowmodulopk,(t)2M(G) W(R)T+=pkwilllieinM(G) W(R)S+i=pkforilargeenough,whereweusethealgebrasS+iconstructedinLemma4.3.10.NotethatthemapsconstructedinLemma4.3.10giverisetoanelementx2(T+) W(R)S+i.Thent x2TG(T+=p) Zp(T+) W(R)S+i�!(TG(T+=p) Zp(T+)) W(R)T+mapsunder totheimageof(t) x2�M(G) W(R)S+i=pk Zp�(T+) W(R)S+iundertheevaluationmap�M(G) W(R)S+i=pk Zp�(T+) W(R)S+i!M(G) W(R)S+i=pkcontractingthesecondandthirdfactor.ButwehavechosenxsothatitactsthroughmultiplicationbyponS+i.Thismeansthattheresultwillbep(t),asdesired.WecannowcompletetheproofofTheorem4.3.1.RecallthatSistheR-algebrawhichrepresentsTG.WewanttoprovethatthecokernelofAcris(S)!M(G)iskilledbyp2.Itisenoughtoprovethisresultmodulopkforanyk3.Letm2p2M(G).ByLemma4.3.11,xhasapreimageinTG(T+=p) Zp(T+)modulopk.WewritethispreimageasPni=1ai i,withai2TG(T+=p),i2(T+).Itisenoughtoprovethattheimageofai iinM(G)isalsointheimageofAcris(S)!M(G).Butai2TG(T+=p)givesrisetoamapS!T+=pcarryingtheuniversalelementofTG(S)toai2TG(T+=p).WegetamorphismAcris(S)!Acris(T+=p)!T+;usingtheusualmap:Acris(T+=p)!T+,comingfromthefactthatT+!T+=pisaPDthickening.Byduality,thisgivesamap(T+)!Acris(S),throughwhichwecanconsideriasanelement0i2Acris(S).Itisnowaneasydiagramchasetocheckthat0imapstotheimageofai iinM(G).ThiscompletestheproofofTheorem4.3.1.4.4.Thegeneralcase.Inthissection,wewillnishtheproofofTheorem4.1.4.ThisfollowsanideawelearntfromapaperofdeJongandMessing,[dJM99](seetheproofofProp.1.2),namelyinordertoprovefullfaithfulnessformorphismsG!H,oneusesbase-changetotheringwithuniversalhomomorphismsQp=Zp!GandH!p1,overwhichoneappliestheresultforthespecialcaseofmorphismsQp=Zp!p1.WeneedabasiclemmaonHopfalgebras.LetGandHbenitelocallyfreegroupschemesoveranyringR,withcoordinateringsAG,AH.ThecoordinateringofGH_isHomR(AH;AG)(R-modulehomomorphisms),sothatthelatterhasthestructureofanR-algebra. 36MODULIOFP-DIVISIBLEGROUPSProof.Notethatiff=0,thenfisp-torsion,soitisenoughtoprovethatiff=0,thenp4f=0.Atthispoint,weneedTheorem4.3.1.Assumethatf=0andconsiderthediagramM(Qp=Zp)(Acris(S))f=0ww M(G)(Acris(AH_))fAH_ // M(G)(Acris(S))fS M(H)(Acris(AH_))// h M(H)(Acris(S))hS M(p1)(Acris(AH_))// M(p1)(Acris(S))Let betheimageof12Acris(S)=M(Qp=Zp)(Acris(S))inM(G)(Acris(S)).Takeanyv2M(G)(Acris(AH_)).LetAcris(S)betheAcris(AH_)-lineardualofAcris(S).ByTheorem4.3.1,thereexists2Acris(S)suchthat( )=p2v2M(G)(Acris(AH_)):TheelementinducesacommutativediagramM(Qp=Zp)(Acris(S))f=0ww M(G)(Acris(AH_))fAH_ M(G)(Acris(S))oo fS M(H)(Acris(AH_))h M(H)(Acris(S))oo hS M(p1)(Acris(AH_))M(p1)(Acris(S))oo whichshowsthat(hfAH_)(p2v)=0,i.e.p2(hfAH_)=0.Now,lookatthecommutativediagramM(G)(Acris(R))p2f// M(H)(Acris(R)) g**TTTTTTTTTTTTTTTTM(G)(Acris(AH_))=044 p2fAH_// M(H)(Acris(AH_))h// M(p1)(Acris(AH_)):Weclaimthatthekernelofg:M(H)(Acris(R))!M(p1)(Acris(AH_))=Acris(AH_)iskilledbyp2.Indeed,useTheorem4.3.1forH_toseethatHomAcris(R)(Acris(AH_);Acris(R))!HomAcris(R)(M(H)(Acris(R));Acris(R)) 38MODULIOFP-DIVISIBLEGROUPSSimilarly,thebasechangetoS3oftheuniversalmorphisms(H1)S1!p1and(H2)S2!p1tintoacommutativediagram(H1)S3$$IIIIIIIII( H)S3// (H2)S3zzuuuuuuuuup1:Fori=1;2,fiinducesamorphismfi2HomSi(Qp=Zp;p1)[p�1].Weclaimthatthesemor-phismsagreeuponbasechangetoS3.ConsiderthediagramM(Qp=Zp)(Acris(S3))f1jS3'' =// M(Qp=Zp)(Acris(S3))f2jS3ww M(G1)(Acris(S3))// M(G2)(Acris(S3)) M(H1)(Acris(S3))// M(H2)(Acris(S3)) M(p1)(Acris(S3))=// M(p1)(Acris(S3)):Thecentersquareinthisdiagramcommutesbyhypothesis.ThetopandbottomsquaresinthediagramcommutebyapplyingMtothecommutativetrianglesinLemma4.4.5.Thusf1jS3=f2jS3.Butthisimpliesthatf1jS3=f2jS3.ThemorphismsfideterminethesequencesofR-linearmapsrfi;n:A0Hi!AGi.RecallthatS3=HomR;cont(A0H2;AG1).ThefactthattheficoincideuponbasechangetoS3meansexactlythatthedesireddiagramcommutes.Lemma4.4.6.Letf:M(G)!M(H)beamorphismofcrystalsoverR.(i)Letff:M(GG)!M(HH)betheobviousmorphism.Wehaverff;n=rf;n rf;nasR-linearmapsA0H^ A0H!AG AG.(ii)Wehave(rf;n)_=rf_;nasR-linearmapsA0G_!AH_.Proof.Easyandlefttothereader.Finally,wecanshowthatrf;n:A0H!AGisamorphismofHopfalgebras.ConsiderthediagonalmapsG:G!GGandH:H!HH.ThenM(G):M(G)!M(GG)=M(G) M(G)isalsothediagonalmap,andsimilarlyforM(H).ThediagramM(G)M(G)// f M(GG)ff M(H)M(H)// M(HH) MODULIOFp-DIVISIBLEGROUPS39commutes.ByLemma4.4.4,thediagramA0H^ A0Hrf;n^ rf;n 0H// A0Hrf;n AG AGG// AGalsocommutes.ButG:AGAG!AGissimplythemultiplicationmap,andsimilarlyfor0H.Thusrf;nisanR-algebrahomomorphism.BydualizingandusingthesameresultforM(H_)!M(G_),oneseesthatrf;nisalsocompatiblewithcomultiplication,asdesired.Wegetthatrf;nliesinHomR-Hopf;cont(A0H;AG)=lim�!mHomR-Hopf(AH[pm];AG[pm])=lim�!mHom(G[pm];H[pm]);wherethelastHomisinthecategoryofgroupschemesoverR.Thusform�nlargeenough,rf;nisinducedbyamorphism 0n:G[pm]!H[pm].Letusassumethatf2HomS(Qp=Zp;p1)isintegral,whichwecanaftermultiplicationbyapowerofp.Thenthemorphismsrf;ncomefromelementssn2S,n0,satisfyingspn+1=sn,s0=1.Inparticular,spnn=1.ByLemma4.4.1,thismeansthatpn 0n=0,sothat 0nfactorsthroughamap n:G[pn]!H[pn].Nowtheconditionspn+1=snforalln0meansthatthemorphisms ncombinetoamorphism :G!Hofp-divisiblegroups.Insummary,wehaveconstructedamapHomR(M(G);M(H))[p�1]!HomR(G;H)[p�1];thatisinjectivebyProposition4.4.2.Ontheotherhand,thecompositionHomR(G;H)[p�1]!HomR(M(G);M(H))[p�1]!HomR(G;H)[p�1]istheidentitybyconstruction,nishingtheproofofTheorem4.1.4.5.Onp-divisiblegroupsoverOCInthissection,wexacompletealgebraicallyclosedextensionCofQp,andwillclassifyp-divisiblegroupsoverOC.FixamapFp!OC=p.Also,inthissectionwewillwriteB+cris=B+cris(OC=p);whichcomeswithanaturalmap:B+cris!C.5.1.Fromp-divisiblegroupstovectorbundles.First,wewillrelatesomeoftheconstructionsthatoccuredsofartothetheoryofvectorbundlesovertheFargues-Fontainecurve.LetPbethegradedQp-algebraP=Md0(B+cris)'=pd;andletX=ProjP.ThenXisacurve,inthesensethatitisaconnected,separated,regularNoetherianschemeofdimension1,cf.[FF11],Theoreme10.2.Moreover,Xisalsocompleteinthesensethatthereisahomomorphismdeg:Div(X)!Zwhichissurjective,nonnegativeone ectivedivisors,andzeroonprincipaldivisors.Fromhere,onedenestherankanddegreeofanycoherentsheafonX,andonegetsthefollowingresult. MODULIOFp-DIVISIBLEGROUPS41Proposition5.1.6.LetGbeap-divisiblegroupoverOC.LetE=E(G0)forG0=G OCOC=pandF=OX ZpTbeassociatedvectorbundlesoverX.Hereandinthefollowing,wewriteT=T(G)(OC).(i)ThereisanaturalexactsequenceofcoherentsheavesoverX,0!F!E!i1(LieG C)!0:Theglobalsectionsofthismapgivethelogarithmsequence0!T[p�1]!~G(OC)!LieG C!0:(ii)Undertheidenticationi1E=M(G) OCC;theadjunctionmorphismE!i1i1Einducesonglobalsectionsthequasi-logarithmmorphism~G(OC)!M(G) C:WhenrestrictedtoT[1 p],itinducesasurjectivemap G:T C!(LieG_ C)_M(G) C:(iii)Thesequence0!LieG C(1) _G_(1)�!T C G�!(LieG_ C)_!0isexact.Inotherwords,ap-divisiblegroupofheighthoverOCgivesrisetoamodicationEofthetrivialvectorbundleF=OhXalong12X.Wenotethat Gcanbedescribedmoreelementaryasfollows.OnehasthemapQp=Zp ZpT!G;whichisdenedasamapofp-divisiblegroupsoverOC,andinducesbyCartierdualityamapG_!p1 ZpT_:OnLiealgebras,thisgivesamapLieG_ C!T_ C.Dualizingagain,wegetthemapT C G!(LieG_ C)_:Proof.Firstofall,allidenticationsofmapsareimmediatefromourpreparationsonp-divisiblegroups.Inparticular,thenaturalmapF!EinducedfromT!M'=pfactorsthroughF0=ker(E!i1(LieG C)):Moreover,theinducedmapF!F0isinjective.Indeed,itsucestocheckthisforF!E.ConsiderthediagramF// E // i1(LieG C) 0// F// E0// i1(T C(�1))// 0Onthelowerline,E0=T ZpOX(1),andthesequencecomesfromthep-divisiblegroupT(�1) Zpp1.Byadirectcomputationforp1,thelowerlineisexact.Inparticular,F,!E.ButFandF0arevectorbundlesofthesamerankhanddegree0,thusF=F0byProposition5.1.1. 42MODULIOFP-DIVISIBLEGROUPSPart(iii)istakenfrom[FGL08],AppendixCtoChapter2.Onecouldalsoreproveitusingthemachineryemployedhere.Infact,thesurjectivityof Gfollowsdirectlyfrompart(i)afteri1.Forpart(iii),itonlyremainstoseethatthecomposite G _G_(1)is0.WehaveacompositemapfG:E_G_!F_G_=FG!EG;whereweusesubscriptstodenotethep-divisiblegroupwithrespecttowhichweconstructthevectorbundles.Also,weuseatrivializationZp(1)=ZptoidentifythevectorbundlesF_G_andFG.Wehavetoseethati1fGis0.Forthis,itisenoughtoshowthatfG:E_G_!EG=E_G_ OXOX(1)ismultiplicationbyt2H0(X;OX(1)),wheret=log([])comesfromtheelement2O[CinducedfromthechosentrivializationZp(1)=Zp.Forthis,onecanreplaceGbyT ZpQp=Zp,whereitisimmediate.5.2.Classicationofp-divisiblegroups.Themaintheoremofthissectionisthefollowingresult.Theorem5.2.1.Thereisanequivalenceofcategoriesbetweenthecategoryofp-divisiblegroupsoverOCandthecategoryoffreeZp-modulesTofniteranktogetherwithaC-subvectorspaceWofT C(�1).Thefunctorisdenedinthefollowingway.ToG,oneassociatestheTatemoduleT=T(G),togetherwithW=LieG C,embeddedintoT Cvia _G_:LieG C!T C(�1):Proof.Westartbyprovingthatthefunctorisfullyfaithful.ThiswasalreadyobservedbyFargues,[Far12].LetG0=T(G)(�1) Zpp1beamultiplicativep-divisiblegroupequippedwithamapG!G0inducinganisomorphismT(G)=T(G0).WealsogetamapLieG!LieG0=T(G)(�1) OC.Itiseasytocheckthatthismapisidentiedwith _G_(afterinvertingp).Theresultingadicspacestintoacommutativediagram0// Gad[p1]= // Gad // LieG Ga 0// G0ad[p1]// G0ad// LieG0 GaWeclaimthattherightsquareiscartesian.Indeed,ifx2G0ad(R;R+)mapstoLieG Ga,thenfornsucientlylarge,pnx2Gad(R;R+),asonasmallneighborhoodof0,logG0isanisomorphism.Onegetsx2Gad(R;R+),asGadpn// Gad G0adpn// G0adiscartesian.ItfollowsthatwecanreconstructGadfrom(T;W)=(T(G);LieG C).ButthenwecanalsoreconstructG=GYGadSpfH0(Y;O+Y); MODULIOFp-DIVISIBLEGROUPS43whereYrunsthroughtheconnectedcomponentsofGad.Itremainstoprovethatthefunctorisessentiallysurjective.Forthis,letusrstassumethatCissphericallycomplete,andthatthenormmapjj:C!R0issurjective.So,assumegiven(T;W).WedeneG0asbefore,andthenGadG0adasthebreproduct.WehavetoseethatGYGadSpfH0(Y;O+Y)denesap-divisiblegroup.Onechecksdirectlythat(therigid-analyticspacecorrespondingto)Gadisap-divisiblerigid-analyticgroupinthesenseof[Far12]:Denition5.2.2.Ap-divisiblerigid-analyticgroupoverCisacommutativesmoothrigid-analyticgroupGoverCsuchthatp:G!Gisnitelocallyfreeandfaithfully at,andp:G!Gistopologicallynilpotent.Abasicfactisthatthesedecomposeintoconnectedcomponents.Lemma5.2.3.[Far12,Corollaire9]LetGbeap-divisiblerigid-analyticgroupoverC.ThenG=G00(G),whereG0isaconnectedp-divisiblerigid-analyticgroupoverC,and0(G)=(Qp=Zp)rforsomeintegerr.ByTheoreme15of[Far12],itisenoughtoshowthattheconnectedcomponent(Gad)0oftheidentityisisomorphictoad-dimensionalopenunitballBd:Thattheoremveriesthatif(Gad)0=Bd,thentheformalgroupinducedonSpfH0((Gad)0;O+(Gad)0)=SpfOCJT1;:::;TdKisp-divisible,i.e.thekernelofmultiplicationbypisniteandlocallyfree.Moreover,weknowthefollowing.Proposition5.2.4.[Far12,Proposition14,Lemma13]Onecanwrite(Gad)0asanincreasingunionofconnectedanoidsubgroupsUn(Gad)0,suchthatUn=Bdforalln0.Proof.ThesubgroupsUncanbedenedasthe(connectedcomponentof)theintersectionof(Gad)0withincreasingclosedballsinG0ad=Bh.Fortheconvenienceofthereader,werecallthecrucialargumentinshowingthatUn=Bd.WriteUn=Spa(A;A+).Using[BGR84],weknowthatA+istopologicallyofnitetypeoverOC,andithasthestructureofatopologicalHopfalgebra.Let A=A+ OCk,wherekistheresidueeldofOC.ThenSpec Aisaconnectedcommutativeanereducedgroupschemeoverksuchthatmultiplicationbypisnilpotent.Asitisreducedandagroupscheme,itissmooth;itfollowsthatitisanextensionofatorusbyaunipotentgroup.Asmultiplicationbypisnilpotent,thetorusparthastobetrivial.Itfollowsthat A=k[T1;:::;Td],andhenceA+=OChT1;:::;Tdi,andthusUn=Bd.Nowwenishtheproofbyusingthefollowingproposition(whichistheonlyplacewhereweuseourassumptionsonC).Proposition5.2.5.AssumethatKisasphericallycompletenonarchimedeaneldforwhichthenormmapjj:K!R0issurjective.LetXbeanadicspaceoverSpa(K;OK)withapoint02XwhichcanbewrittenasanincreasingunionX=SXnofquasicompactopensubsets02X0X1:::.Moreover,weassumethefollowingconditions.(i)Foralln0,theinclusionXnXn+1isastrictopenembedding. MODULIOFp-DIVISIBLEGROUPS45Remark5.2.9.Theconverseto(ii)isfalse.Indeed,letG1beQp=Zp,letG2beconnectedofdimension1andheight2,andletG3bep1.ThentherearemapsG1!G2!G3whosecompositionis0,andsuchthatonTatemodules0!T1!T2!T3!0andLiealgebras0!W1!W2!W3!0,onegetsexactsequences.However,thesequence0!G1!G2!G3!0isnotexact,asonecheckseasilyonthespecialbre.Proof.For(i),usetheHodge{Tatesequence.Part(ii)isclear.6.Rapoport{Zinkspaces6.1.Thespaceoflevel0,andtheperiodmorphism.LetHbeap-divisiblegroupoveraperfecteldkofcharacteristicp,ofheighthanddimensiond.Denition6.1.1.LetR2NilpW(k).AdeformationofHtoRisapair(G;),whereGisap-divisiblegroupoverRand:H kR=p!G RR=pisaquasi-isogeny.LetDefHbetheassociatedfunctoronNilpW(k),takingRtothesetofisomor-phismclassesofdeformations(G;)ofHtoR.RecallthefollowingtheoremofRapoport{Zink,[RZ96].Theorem6.1.2.ThefunctorDefHisrepresentablebyaformalschemeMoverSpfW(k),whichlocallyadmitsanitelygeneratedidealofdenition.Moreover,everyirreduciblecomponentofthereducedsubschemeisproper.LetM(H)denotetheDieudonnemoduleofH,afreeW(k)-moduleofrankh.AssumenowthatRisap-adicallycompleteW(k)-algebraequippedwiththep-adictopology.ThenviaGrothendieck{Messingtheory,(G;)givesrisetoasurjectionoflocallyfreeR[1 p]-modulesM(H) R[1 p]!LieG[1 p],whichdependson(G;)onlyuptoisogeny.Wegettheinducedperiodmap:Mad!F`;whichwecallthecrystallineperiodmap.Here,F`isthe agvarietyparametrizingd-dimensionalquotientsoftheh-dimensionalW(k)[1 p]-vectorspaceM(H)[1 p];weconsiderF`asanadicspaceoverSpa(W(k)[1 p];W(k)).Moreover,wewillconsiderthefollowingvariantofDefH.ConsiderthefunctorDefisogHthatassociatestoap-adicallycomplete atW(k)-algebraRequippedwiththep-adictopologythesetofdeformations(G;)ofHtoR,moduloquasi-isogenyoverR.Observethatgivingaquasi-isogenyoverRisstrictlystrongerthangivingacompatiblesystemofquasi-isogeniesoverthequotientsR=pn,sothatDefisogHcannotbedenedonNilpW(k).Usingtheusualprocedure,onegetsasheaf(DefisogH)adonthecategoryofcompleteanoid(W(k)[1 p];W(k))-algebras.Wenotethatfactorsoveramap,stilldenoted,:(DefisogH)ad!F`:Ourrstgoalistoprovethefollowingresult,whichisessentiallycontainedin[dJ95].Proposition6.1.3.Thesheaf(DefisogH)adisrepresentablebyanadicspace,whichisidentiedwithanopensubspaceUF`under.Weremarkthatifd=1,i.e.weareworkingintheLubin-Tatecase,then(DefisogH)ad=F`,themapiscalledtheGross{Hopkinsperiodmap,andtheresultfollowseasilyfromthetheoryin[GH94]. 50MODULIOFP-DIVISIBLEGROUPS(i)Thematrix(qlog(s1);:::;qlog(sh))2(M(H) R)hisofrankexactlyh�d;letM(H) R!Wbetheinducedniteprojectivequotientofrankd.(ii)Forallgeometricpointsx=Spa(C;OC)!Spa(R;R+),thesequence0!Qhp(s1;:::;sh)������!~Had(C;OC)!W RC!0isexact.Weremarkthattheconditionontherankissayingmorepreciselythatallminorsofsizeh�d+1vanish,andthattheminorsofsizeh�dgeneratetheunitideal.Lemma6.3.6.ThereisanaturalisomorphismoffunctorsM1=M01.Moreover,M01(~Had)hisalocallyclosedsubfunctor.Proof.WestartbyconstructingamapM1!M01.So,let(R;R+)beacompleteanoid(W(k)[1 p];W(k))-algebra,and(G;; )2M1(R;R+),where(G;)isdenedoversomeopenandboundedsubringR0R+.Using,wegetanidentication~HR0=~G.Therefore,wegetamap :Zhp!~Gad(R;R+)=~Had(R;R+);i.e.hsectionss1;:::;sh2~Had(R;R+).Wehavetocheckthattheysatisfyconditions(i)and(ii)above.Condition(ii)isclearlysatised,whenW=LieG R.Forcondition(i),notethatwehaveacommutativediagram~Had(R;R+)// qlog Gad(R;R+)log M(H) R// LieG R:Thisimpliesthatqlog(si)2ker(M(H) R!LieG R),whichisanh�d-dimensionalprojectiveR-submodule.Therefore,allminorsofsizeh�d+1vanish.Tocheckthattheminorsofsizeh�dgeneratetheunitideal,itsucestocheckforallx=Spa(K;K+)2Spa(R;R+),thesectionsqlog(s1);:::;qlog(sh)2M(H) Kgenerateanh�d-dimensionalsubspace.WecanassumethatK=Cisalgebraicallyclosed,andthatK+=OC.Inthatcase,thisfollowsfromthesurjectivityofT(G) C!(LieG_)_ C;cf.Proposition5.1.6(iii).NotethattheseconsiderationsalsoshowthatthequotientofM(H) Rbythesubmodulegeneratedbyqlog(s1);:::;qlog(sh)isexactlyW=LieG R.NowwewillconstructtheinversefunctorM01!M1.First,observethatM01!F`factorsoverUF`,whereUdenotestheimageoftheperiodmorphism.AsUF`ispartiallyproper,itisenoughtocheckthisongeometricpointsSpa(C;OC),i.e.thoseofrank1.WehavethevectorbundleE=E(H)ontheFargues{FontainecurveX.LetF=ker(E!i1W):ThenH0(X;F)=Qhpbycondition(ii).WeclaimthatF=OhX.AsFisofrankhanddegree0,ifF6=OhX,thenthereexistssome�0suchthatOX(),!F.ButdimQpH0(X;OX())=1for�0,contradiction.ByTheorem6.2.1,thisimpliesthedesiredfactorization.Next,considerthefunctorFdenedsimilarlytoM01,butwithcondition(ii)weakenedtotheconditionthatthequotientWofM(H) RdenesapointofUF`.ThenFisclearlylocallyclosedin(~Had)h.Moreover,thereisanaturalmorphismF!U,sothatlocallyonSpa(R;R+)F,onehasadeformation(G0;0)ofHtoanopenandboundedW(k)-subalgebraR0R+.However,(G0;0)isonlywell-deneduptoquasi-isogenyoverR0;i.e.wehaveasection(G0;0)2(DefisogH)ad(F): 52MODULIOFP-DIVISIBLEGROUPSThepropositionconstructsadeformation(G;)ofHtoF0.Moreover,wehaves1;:::;sh2V(G)ad(F0):Infact,theylieinthesubsetT(G)ad(F0).AsT(G)adV(G)adisanopenembedding,thiscanbecheckedonpoints,whereitisclear.Thismeansthats1;:::;shgiveamap :Zhp!T(G)ad(F0);whichisclearlyanisomorphismateverypoint.ThisgivestheinversefunctorM01!F0!M1,asdesired.Infact,M01=F0=M1.NowwecannishtheproofthatM1ispreperfectoid.Indeed,M1(~Had)hislocallyclosed,and~Hadispreperfectoid.Thus,byProposition2.3.11,M1ispreperfectoid.Finally,wegiveadescriptionofM1purelyintermsofp-adicHodgetheory,onthecategoryofperfectoidalgebras.So,letusxaperfectoideldKofcharacteristic0.Proposition6.3.9.ThefunctorM1onperfectoidanoid(K;OK)-algebrasisthesheacationofthefunctorsending(R;R+)tothesetofh-tuplesp1;:::;ph2(M(H) W(k)B+cris(R+=p))'=p;forwhichthefollowingconditionsaresatised.(i)Thematrix((p1);:::;(ph))2(M(H) R)hisofrankexactlyh�d;letM(H) R!Wbetheinducedquotient.(ii)Forallgeometricpointsx=Spa(C;OC)!Spa(R;R+),thesequence0!Qhp(p1(x);:::;ph(x))����������!(M(H) W(k)B+cris(OC=p))'=p!W RC!0isexact.Proof.Wehave~Had(R;R+)=~H(R+)=~H(R+=p)=(M(H) W(k)B+cris(R+=p))'=p;byTheorem4.1.4.Itfollowsthatthedatumofp1;:::;phisequivalenttothedatumofhsectionss1;:::;sh2~Had(R;R+);andtheconditions(i)and(ii)clearlycorrespond.Corollary6.3.10.LetCbeanalgebraicallyclosedcompletenonarchimedeanextensionofQp,andletXbetheassociatedFargues{Fontainecurve.LetF=OhXandE=E(H)beassociatedvectorbundlesonX.ThenM1(C;OC)isgivenbythesetofmorphismsf:F!Ethatgiverisetoamodication0!Ff!E!i1W!0;whereWissomeC-vectorspace. MODULIOFp-DIVISIBLEGROUPS536.4.TheLubin-Tatespaceatinnitelevel.Inthissection,Hisaconnectedp-divisiblegroupoverkofdimension1andheighth.Weassumethatkisalgebraicallyclosed,sothatHonlydependsonh.LetMh1betheassociatedinnite-levelRapoport-Zinkspace.WeshowherethatMh1iscutoutfrom(~Had)hbyasingledeterminantcondition.Letusbeginwiththecaseofh=1,sothatH=p1;k.ThenHliftsuniquelytothep-divisiblegroupp1overW(k).WehaveM(p1;k)=Liep1=W(k).Thequasi-logarithmmapreducestotheusuallogarithmmap,whichsitsinsidetheexactsequence0// V(p1)ad// ~adp1;log// GaofsheavesofQp-vectorspacesonCA Spa(W(k)[1=p];W(k)).Nowlet(R;R+)beacompleteanoid(W(k)[1 p];W(k))-algebra.ByLemma6.3.6,M11(R;R+)isthesetofsectionss2~adp1(R;R+)whichsatisfybothlog(s)=0andtheconditionthats(x)6=0forallpointsx=Spa(K;K+)2Spa(R;R+).ThismeansthatM11=V(p1)adnf0g:Eitherfromthisdescription,orfromthedescriptionofM11asclassifyingtriples(G;; ),itfollowsthatM11=Gn2ZSpa(L;OL);whereListhecompletionofW(k)[1 p](p1).Indeed,thedecompositionbyn2Zcomesfromtheheightof.Assumethatisofheight0;thenG=p1canonically,byrigidityofmultiplicativep-divisiblegroups.Now :Zp!Tp1amountstothechoiceofacompatiblesystemofp-powerrootsofunity,whencetheresult.Nowreturntothegeneralcase,sothatHhasdimension1andheighth.Weconstructadeterminantmorphismfromtheh-foldproductof~Hadinto~adp1;.Let^hM(H)bethetopexteriorpoweroftheW(k)-moduleM(H);then^hM(H)isfreeofrank1withFrobeniusslope1,sothat^hM(H)=M(p1;k).Write~HfortheuniversalcoverofH,consideredoverthebaseW(k).ByCorollary3.1.5,~HhisrepresentablebyaformalschemeSpfR,whereR=W(k)JX1=p11;:::;X1=p1hK.InparticularR=pisaninverselimitoff-semiperfectrings.Theprojectionmaps~Hh!~Hgivehcanonicalelementss1;:::;sh2~H(R=p).AppealingtoTheorem4.1.4,theDieudonnemodulefunctorgivesusisomorphisms H:~H(S)!(M(H) W(k)B+cris(S)) =1and p1:~p1(S)!(M(p1;k) B+cris(S)) =1foranyf-semiperfectringS.Wegetanelement �1p1( H(s1)^^ H(sh))of~p1(S).AsR=pisaninverselimitoff-semiperfectrings,wegetamorphismofformalschemesdet:~Hh!~p1overSpeck,andthenbyrigidityalsooverSpfW(k).Passingtogenericbres,wegetamorphismofadicspacesdet:(~Had)h!~adp1; 54MODULIOFP-DIVISIBLEGROUPSwhichisQp-alternatingwhenconsideredasamapbetweensheavesofQp-vectorspacesonCA Spa(W(k)[1=p];W(k)).Thismorphismmakesthediagram(~Had)hdet// qlog ~adp1;log (M(H) Ga)hdet// M(p1;k) Gacommute.Theorem6.4.1.ThereisacartesiandiagramMh1det// M11 (~Had)hdet// ~adp1;:Proof.First,wehavetoprovethatdet:Mh1!~adp1;factorsoverM11.Forthis,supposethattheh-tuples1;:::;shrepresentsasectionofMh1overananoidalgebra(R;R+).Sincethematrix(qlog(s1);:::;qlog(sh))hasrankh�1,wehavedet(qlog(s1);:::;qlog(sh))=0,i.e.log(det(s1;:::;sh))=0.Itremainstoshowthatdet(s1;:::;sh)(x)6=0atallpointsx2Spa(R;R+).Forthis,itisenoughtocheckongeometricpointsSpa(C;OC),whereitisreadilydeducedfromCorollary6.3.10,notingthatVhF,!VhE.Itremainstoseethatthesquareiscartesian.Forthis,lets1;:::;shrepresentasectionofthebreproductover(R;R+).Thismeansthatdet(s1;:::;sh)isasectionofV(p1)adnf0g.Wehavedet(qlog(s1);:::;qlog(sh))=log(det(s1;:::;sh))=0;meaningthatthematrix(qlog(s1);:::;qlog(sh))hasrankatmosth�1.Thereforetoshowthats1;:::;shliesinMh1,weonlyneedtoshowthatthismatrixhasrankexactlyh�1,andthatcondition(ii)inDenition6.3.5issatised.ThiscanbecheckedongeometricpointsSpa(C;OC).Thetuples1;:::;shcorrespondstoamapF!EofvectorbundlesontheFargues-FontainecurveX,withF=OhXandE=E(H).Sincethesections1^^shofE(p1)vanishesonlyat1,wehaveanexactsequence0!F!E!i1W!0;whereWisaC-vectorspaceequaltothequotientofM(H) W(k)Cbythespanofs1;:::;sh.Countingdegrees,wegetdimW=1,verifyingtheconditionontherank.Takingglobalsections,onegetstheexactsequence0!Qhp(s1;:::;sh)������!~Had(C;OC)!W RC!0;verifyingcondition(ii)inDenition6.3.5.Remark6.4.2.ThespaceM11=FZSpa(L;OL)hasanobviousintegralmodel,namely^M11=FZSpfOL.Thissuggestsdeninganintegralmodel^Mh1asthebreproductof~Hhand^M11over~p1.Ontheotherhandwehavetheintegralmodels^Mhnofnitelevel,duetoDrinfeld.Itisprovedin[Wei12]that^Mistheinverselimitlim �^MninthecategoryofformalschemesoverSpfZp. MODULIOFp-DIVISIBLEGROUPS556.5.ELstructures.Inthissection,wegeneralizethepreviousresultstoRapoport{ZinkspacesofELtype.ItwouldnotbeproblematictoconsidercasesofPELtype.However,thegrouptheorybecomesmoreinvolved,andthepresenttheorymakesitpossibletoconsideranaloguesofRapoport{ZinkspaceswhicharenotofPELtype,sothatwewillleavethisdiscussiontofuturework.LetusxasemisimpleQp-algebraB,anitenondegenerateB-moduleV,andletG=GLB(V).FixamaximalorderOBBandanOB-stablelatticeV.Further,xaconjugacyclassofcocharacters:Gm!GQp,suchthatinthecorrespondingdecompositionofVQpintoweightspaces,onlyweights0and1occur;writeVQp=V0V1forthecorrespondingweightdecomposition,andsetd=dimV0,h=dimV,sothath�d=dimV1.Moreover,xap-divisiblegroupHofdimensiondandheighthoverkwithactionOB!End(H),suchthatM(H) W(k)W(k)[p�1]=V QpW(k)[p�1]asB QpW(k)[p�1]-modules.Inthefollowing,wewriteD=(B;V;~H;)fortherationaldata,andDint=(OB;;H;)fortheintegraldata.Moreover,letEbethere exeld,whichistheeldofdenitionoftheconjugacyclassofcocharacters.SetE=EW(k).Denition6.5.1.TheRapoport{ZinkspaceMDintofELtypeassociatedtoDintisthefunctoronNilpOEsendingRtothesetofisomorphismclassesof(G;),whereG=Risap-divisiblegroupwithactionofOBsatisfyingthedeterminantcondition,cf.[RZ96,3.23a)],and:H kR=p!G RR=pisanOB-linearquasi-isogeny.ThenRapoport{ZinkproverepresentabilityofMDint.Theorem6.5.2.ThefunctorMDintisrepresentablebyaformalscheme,whichlocallyadmitsanitelygeneratedidealofdenition.Onthegenericbre,wehavetheGrothendieck{MessingperiodmorphismGM:(MDint)ad!F`GM;wherethe agvarietyF`GMparametrizesB-equivariantquotientsWofM(H) W(k)RwhichareniteprojectiveR-modules,andlocallyonRisomorphictoV0 QpRasB QpR-modules.Foranyn1,onealsodenesthecoverMDint;nofitsgenericbre(MDint)ad,asparametrizingOB-linearmaps=pn!G[pn]adwhichareisomorphismsateverypoint.WehavethefollowingdenitionofMDint;1.Denition6.5.3.ConsiderthefunctorMDint;1oncompleteanoid(E;OE)-algebras,sending(R;R+)tothesetoftriples(G;; ),where(G;)2(MDint)ad(R;R+),and :!T(G)ad(R;R+)isamorphismofOB-modulessuchthatforallx=Spa(K;K+)2Spa(R;R+),theinducedmap (x):!T(G)ad(K;K+)isanisomorphism.Ourargumentsimmediatelygeneralizetothefollowingresult. MODULIOFp-DIVISIBLEGROUPS59Theorem7.2.3.ThereisanaturalG(Qp)G(Qp)-equivariantisomorphismMD;1=MD;1;underwhichGM:MD;1!F`GMgetsidentiedwithHT:MD;1!F`HT,andviceversa.Proof.Let(R;R+)beanycompleteanoid(E;OE)-algebra.ThenrecallthatMD;1istheshea-cationofthefunctorsending(R;R+)tothesetofB-linearmapss:V!~Had(R;R+)forwhichtheinducedquotientM(H) R!WbytheimageofV RislocallyoftheformV0 R,andwhichgiveexactsequencesateverygeometricpoint.Similarly,MD;1isthesheacationofthefunctorsending(R;R+)tothesetofEndB(V)-linearmapss:EndB(H)!V B~Had(R;R+)forwhichtheinducedquotientM(H) W(k)R=EndB(V) QpR!WbytheimageofEndOB(H) RislocallyoftheformEndB(V0;V) R,andwhichgiveexactsequencesateverygeometricpoint.Letusrstcheckthatsandscorrespond.Indeed,dene(s(f))(v)=f(s(v));wheref2EndB(H)andv2V.Onereadilychecksthatthisdenesabijectivecorrespondencebetweensands.NowconsiderthemapEndB(H) R!M(H) W(k)R=EndB(V) QpRinducedbys.WeclaimthatitisthesameasHomB R(�;V R)appliedtothemapV QpR!M(H) Rinducedfroms;thisisadirectverication.Fromhere,onededucesthatWislocallyoftheformV0 RifandonlyifWislocallyoftheformHomB(V0;V) R,aswellasthecompatibilitywithperiodmaps.Moreover,theexactnessconditionsatgeometricpointscorrespond.7.3.EquivariantcoversofDrinfeld'supperhalf-space.FixaniteextensionFofQp,andlet Pn�1denoteDrinfeld'supperhalf-space,whereweconsidereverythingasanadicspaceoverSpa(C;OC),whereoureldofscalarsCisacompletealgebraicallyclosedextensionofF.AsanapplicationofthedualityofRapoport{Zinkspaces,weprovethefollowingresult.Theorem7.3.1.Let~ ! beaniteetaleGLn(F)-equivariantmorphismofadicspaces.Thereexistsanm0suchthatMDrm! factorsthrough~ .Proof.Let^MLT1be(thestrongcompletionofthebase-changetoCof)theLubin-TatespaceofheightnforFatinnitelevel,andlet^MDr1be(thestrongcompletionofthebase-changetoCof)theDrinfeldspaceofheightnforFatinnitelevel.Thenboth^MLT1and^MDr1areperfectoidspacesoverSpa(C;OC).Byduality,wehavetheGLn(F)D-equivariantisomorphism^MLT1=^MDr1,whereDisthedivisionalgebraoverFofinvariant1 n.(WenotethattheGLn(F)-actionwhichcomesfromthegeneraldualitystatementdi ersfromthestandardGLn(F)-actionontheDrinfeldtowerbyg7!(g�1)t.Thisdoesnotchangeanyofthestatementstofollow.)Let~^MDr1bethebreproduct~ ^MDr1.Then~^MDr1!^MDr1isaGLn(F)-equivariantniteetalemorphismofperfectoidspacesoverSpa(C;OC).Bythedualityisomorphism,itgivesrisetoaGLn(F)-equivariantniteetalemorphism~^MLT1!^MLT1. 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