; TeX output 2000.07.05:1406[3[͠;3EBˍtffG$$ffȍ"V cmbx10Arc9hiveTforMathematicalLogicman9uscriptNo. 8K`y cmr10(willUUbGeinsertedbytheeditor)F34$ffffffGdK.F.Gauss8 !", cmsy10J.H.PoincarGeNff cmbx12RiemannianZgeometryandHilbs3ertspaceappliedtometamagicalgametheoryandthesurvivalffproblemofSchgodinger'scat9N cmbx12I.Stepstowardsatheoryofalmosteverythingv$o cmr9Receiv9ed:T31March1999/RevisedTversion:14OctobAer1999/ PublishedTonline:20Decem9bAer1999{c% cmsy9 jSpringer-V:erlag20002t : cmbx9Abstract.0W:eRindicateacompletesetofelemen9taryinv|rariantsfortheringofWitt<v9ectorsoverapAerfect eldofprimecharacteristic,wherethisringisequippAedwithTitsuniquem9ultiplicativeTsetofrepresen9tativesTfortheresidue eld.`YffGdӍTheoremsCEofAx,KoGchenandErsovtellusthattheelementarytheoryofa henselianvqaluationringofequalcharacteristic0iscompletelydeterminedby cmmi10pandthemaximalidealisgeneratedbyip.F*oracompletediscretevqaluationringwiththesepropGertiestheana-logueJof` eldofrepresentatives'Jis`multiplicativeJsetofrepresentativesJforthe[residue eld'andisduetoWitt.F*orproGofsofthisandrelatedresultsmentioned&bGelowthatweshallusewerefertoSerre[S62<,Ch.IGI].W*enowproGceedUUtoprecisestatements.FixaprimenumbGerp.LetAbeacompletediscretevqaluationringwithmaximalldideal3%n eufm10m톲=pAldandpGerfectresidue eld6 & cmb10k =통A=m(ofcharacteristicp).Let٢:}ɵA!k ڲbGetheresidueclassmap.Thereisauniquemultiplica- JˉffG K._F.Gauss:Departmen9tofMathematics,UniversityofIllinoisatUrbana- Champaign,Urbana,TIL61801,USA.e-mail:0ߤN cmtt9gauss@math.uiuc.edu J.H.P9oincarXe:iEquipAed'Analyse,T:our46,UniversitXedeParisVI,4PlaceJussieu,F-75230TP9arisCedex05,F:rance.e-mail:poincare@math.bu.eduK.F.GaussE4w9assuppAortedbyagrantoftheSwissNationalScienceF:oundation.J.H.P9oincarXenwaspartiallysuppAortedbyNSFngrantsDMS-98043192andNCR-9850637TatBostonUniv9ersity1j cmti9Key>wordsorphrases:>Riemanniangeometry{HilbAertspace{Gametheory{Surviv|ralT{Sc9hr`odinger'sTcatMathematicsNthatis,[ٲ(f(x))L=x>andf(xy[ٲ)L=f(x)f(y[ٲ)>forx;y޸2Lk #,anduthusf(kG)=S.uThefollowingeasyresultonZ-linearrelationsamongelementsUUofSisdecisive.Let.k=(k1|s;:::;knq~)bGeann-tupleofintegers,andletX=(X1|s;:::;Xnq~)bGeann-tupleofdistinctindeterminates.Givenann-tupleb=(b1|s;:::;bnq~)of+LelementsinanabGelian(additive)groupBq,putk5fϵb:=k1|sb1aB+tE+knq~bn.MLemmaT1.4:': cmti10Ther}'e arepolynomialsR1|s;:::;RN k2ڹFpR[X],dependingonlyonpandk~andnotonA,suchthatforallx=(x1|s;:::;xnq~)2k6^n :퍑Epkw8f(x)=0(UX)R1|s(x)=8ײ=RN(x)=0;wher}'ef(x):=(f(x1|s);:::;f(xnq~):E׍Pr}'oof.UIByUU[S62<,Prop.9,p.47]wehaveforx2k6^n :}fkw8f(x)=1 X sִi=0㋵fcPiTL(xpr 0ncmsy5O \cmmi5i 4)cWpi!>`whereڵPid2FpR[X]depGendsonlyoni,pandkP.TheidealofFp[X]generated byethepGolynomialsPiTL,i2N,eisgeneratedby nitelymanyamongthem,sayR1|s;:::;RN.UUThenR1;:::;RN haveUUthepropGertydescribedinthelemma.@UTu@UTtThexZ-linearrelationstogetherwiththemultiplicativexrelationss=s1|ss2amongtheelementsofS"generateallpGolynomialrelationsoverZamongelementsUUofS:MLemmaT2.4L}'et7U.RandVPbemultiplicativelyclosedsubsetsof eldsEandFFofOchar}'acteristic0.LetS:U2n!VbeOabijectionsuchthat(u1|su2)S=(u1|s)(u2)forallu1|s;u2n2 U,andsuchthatforallkB2 Z^n andallu 2U^nwe have:k>u=0(UX)k>(u)=0. Thenextendstoanisomorphismfr}'omthesub eldQ(U)ofE'tontothesub eldQ(V8)ofFc.Pr}'oof.UILet]P*=P USiTGciTLX^iF2Z[X]wherethesumisover] nitelymanyi2N^nq~.Then,givenu2U^n,wehavePc(u)=P USiTGciTLu^id=0ifandonlyifP ğici(u^i)=P ;i/ciTL(u)^id=Pc((u))=0,/ibythehypGothesisofthelemma.TheconclusionofUUthelemmafollowseasily*.ut[3[͍ffffGElemen9taryTtheoryofringsofWittvectorsB`3;3EBF*orandc^0wareelementarilyequivalent.HereeG*=v[ٲ(E^S)isthevqaluegroupofthevaluationv onthefraction eldEAof[B.̲withvqaluationringBq,andc^01Ȳ,v[ٟ^0mandE^0!arede nedinthesamewaywithIBq^0jinsteadofBq.ThesevqaluegroupsareconsideredasorderedabGeliangroups.F*ollowing(KoGchen[Ko75!,pp.407{408],theideaoftheproGofistopasstosucientlyjsaturatedmoGdelswherethevqaluationcanbedecomposedintoaUUvqaluationofequalcharacteristic0andacompletediscretevaluation.Pr}'oof.UIOnedirectionisobvious.F*ortheotherdirectionweassumethatBq=m(B)Bq^0N=m(Bq^0)and*c^01Ȳ.T*oshowthatthen(Bq;)(B^0N;^0aƲ),wemay6assumethesetwomoGdelsofTŲare@1|s-saturated.W*efocuson(Bq;),butDthesameanalysiswillapplyto(Bq^0N;^0aƲ).W*ecoarsenvtothevqaluationev ?onME_ڲwithvqaluegroupx卑ݫe :=V=Z&Ӹ1M(where1:=v[ٲ(p)isthesmallestpGositiveelement1ofc)bysettingKyeveײ(a)=v[ٲ(a)j+Z11fora2E^S.1ThevqaluationringofoeUUv gisx卑8"e6˵BA:=Bq[1=p]=fa2EZ:v[ٲ(a)n81UUforsome+˵ng򭍲withr'maximalidealY~ermײ:=m(x卑WeB9)=fa2E5V:v[ٲ(a)n1UUforall ng,r'andresiduev eldKײ:=x卑eB =YWerm !ofcharacteristic0.ThenYterm!isalsoaprimeidealofBq,!andA:=B=YWerm is!avqaluationringofK,withmaximalidealpA.Theresidueclassmap:x卑ieB~]!KmapsㅵBcontoA,andinducesbypassingtoquotientsanisomorphismBq=pBT͍N+3N=䙵A=pAoftheresidue eldsofB oandA.ßW*eputk :=~Bq=pBY=A=pAßbyidentifyingtheseresidue eldsviathisisomorphism.XThuspw=A޲(jBq)Xwherepw:B!kandXӵk:A!karetheAjresidueclassmaps.HenceSZ:=()isamultiplicativelyAjclosedsubsetof4AthatismappGedbijectivelyontok vbytheresidueclassmapA;R!k p.By@1|s-saturationAisacompletediscretevqaluationring,andtherefore(A;S)isalsoamoGdelofTc,byLemma1.W*enowshowhowto\lift"thequotient(K(;A;S)of(x卑WeB9;Bq;)backto(x卑WeB9;Bq;).Thebijection"7!([ٲ):F!lSݲisRPmultiplicative,sobythesecondlemmamapstheringZ[]isomorphicallyontoZ[S]"K.Thusthefraction eldQ()"E]Oof¹Z[] r[3[͍4ffffGK.F.Gauss,TJ.H.P9oincarXe;3EBis=Ractuallycontainedinx卑VeB S,andmapsQ()isomorphicallyontoQ(S). ?SincewBxishenselian,soisitsloGcalizationx卑ΫeB .Theresidue eldKofx卑ΫeB'being ofcharacteristic0,itfollowsthatthereisa eldLwithQ()͸Lx卑$eBsuch thatmapsLisomorphicallyontoallofK.Then(L;BO˸\ZL;) isthedesiredliftingof(K(;A;S),thatis,(L;B\7L;)(x卑WeB9;Bq;)andrestrictstoanisomorphism(L;B¸\QL;)T͍3+33=K(K(;A;S).W*enowshiftourattentionfrom;(Bq;)(anexpansionofthemixedcharacteristicvqaluationringB)to(x卑WeB9;L;BSٸ\hL;)"whichweviewastheequalcharacteristicvqaluationringx卑;eBequippGeduwithaliftingofitsexpandedresidue eld(K(;A;S).uNotethatBisde nablerin(x卑WeB9;L;B\L;)rasfollows:BG=fx2x卑oeB i:xy"2Yoerm TforUUsome6#ʵy2BQ\8Lg.W*eVnowcarryoutthesameconstructionwith(Bq^0N;^0aƲ),introGducingŵev[ٟ^0 L,x卑Ӑec^0 X,x卑eBq^0 ,K^0U,k^0 ,A^09,S^0#]andL^0вinthesamewayweobtainedthecorrespGondingunaccentedob8jectsfrom(Bq;).AsweindicatedabGoveitnowsucestoshowAthat(x卑WeB9;L;B6\ŵL;)(x卑WeBq^0 r;L^09;Bq^0`o\L^09;^0aƲ).AConsidertheringsWc(kG)andZWc(kG^0W)ofWittvectorsoverk Ҳandk x^0 ,andforpGerfectsub eldsF?W(OG;L)(O0V;L0 GtR)(UX)*c0ůand1L_L0 ctRo:5Z[3[͍ffffGElemen9taryTtheoryofringsofWittvectorsB`5;3EBThe}followingvqariantofthetheoremcanbGeobtainedinthesameway*, byappGealingtoacorrespondingvqariantofLemma3(see[KuPr89"?]).W*eletk6andk^01denotetheresidue eldsofthevqaluationringsBandBq^0N,andletandUUc^0bGetheirvqaluegroupsasinthetheorem.PropQosition.B)L}'et㎲(Bq;)and(B^0N;^0aƲ)b}'emodelsofTGsuchthat(Bq;)Wd(Bq^0N;^0aƲ)(sother}'earenaturalinclusionsk k6^0pVand*c^01Ȳ).Then+Gf(Bq;)(B0N;0aƲ)(UX)k Nk60pVand!e*c01ȵ:R}'emark.)(InGLemma1wedescribGedtheZ-linearrelationsamongtheele-mentsrofSZA.Anotherwaytodothis,insomerespGectsmoreilluminating,isUUasfollows.First,anyroGotofunityinAbGelongstoSandanytuple=(1|s;:::;nq~)(n>0)YofroGotsofunityid2Asatis esnon-trivialZ-linearrelations.Theserelations#proGduceincertainobviouswaysfurtherrelations,forexample,foranyUUs2Smn8f0gthetuples8satis esthesameZ-linearrelationsas.Secondly*,anelementa2AԲbGelongstoSaifandonlyifFc(a)=a^pR,whereFK4isthecanonicalliftingoftheF*robGeniusmaptoanautomorphismofA(seeUU[S62<]).Usinglothislastfactonecanshow,following[H],thatallZ-linearrelationsamongIelementsofSsֲarisefromtheZ-linearrelationsamongtheroGotsofunityUUinA.ThiswaspGointedouttomebyHrushovski.Acknowxledgements.RwTheauthorswishtothankH.Mink9owskiandD.HilbAertfor stim9ulatingTdiscussionsandencouragement.References[A73],3Ax,J.:Ametamathematicalapproac9htosomeproblemsinnumbAer,3theory:.TAMSSympAosium(1973)161{190[H],3Hrusho9vski,E.:TheManin-MumfordconjectureandthemoAdeltheory,3ofTdi erence elds.Preprin9t[Ko75],3KoAc9hen,S.:Themodeltheoryoflocal elds.In:LogicConference,Kiel,31974(ProAceedings),LectureNotesinMathematics499,Berlin1975:,3Springer,Tpp.384{425[KuPr89],3Kuhlmann,AF.-V.andPrestel,A.:Onplacesofalgebraicfunction elds.,3J.Treineangew.Math.400,185{202(1989)[S62],3Serre,TJ.-P:.:CorpsN cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Y&Y cmex10Z#