Shimomura, Katsumi

Tottori University Research Result Repository

J Fac Educ TottO? univ(Nat Sci。),41(1992)1-5Note on the Bousneld Localizatiomwith Respect tO F(7)Katsumi SHIMoMuRA*(re¢θθんθrr ИP/′ど晩 /99フ)§1・  IntrOductionFor a gcncralizcd homology thcoryど*,Bousncld[2]dcancs thc tocalzation LEχ,of a spcctrun1 / with rcspcct to E*, and discuss about convcrgcncc of a gcnerahzcdAdams spccttal scqucncc bascd on E*. Considcr thc Johnson―Wilson spccttum E(η)([3])for each positive integcr tt and a prilnc p, whose cocfncicnt ring is E(猾)*=ZID[υl,中●,υ“,υ「1].ThiS Spectrum induces a generalized homology thcoryE(■)*.  ThCn thcrc cxists thc gcncralizcS Adams spectral sequencc convcrging to ahomotopy group π*(LE仰)χ)Of E(乃)-10Calization of a spcctrum/with E2~tCrm Eゴ=Ext≧岳)*(E(И)(E(つ*,E(η)*(χ))WhiCh WC dcnote E(■)をι(χ)・ On the othcr hand,wcalso havc the Adams―Novikov spectral sequcnce converging to a hom6topy groupπ*(ズ)Ofa p‐local conncctcd spcctrumズwith E2~tCrm Ey=Extと,ネ(BP)(BP*,BP*(χ))which wc dcnotc BPシr(ズ〉Lct ズ bc a p‐local conncctcd spcctrum and ηχ:二て―→LE(.)/ bc thc localizationmap, that is, thc homology theory E(■)*(―) induCes an isomorphismE●)*(ηχ):』(η)*(χ)→E(η)*(LE仲)χ〉 ThCn this g?cs risc to a map ηχ*:BP彦ど(/)→E(ηttt(/〉WC also havc a map φ*:BPゾ(ズ)→E(η)ンr(ズ)induccd by thc Thommap Φ:BP→E(η).HCrC WC havcTHEOREM.ηtt Tんοw ttη φ:】P→E(Ю)肋7T/じ9∫ チカど んじα′″αr力η ttη ηχ:ズ→ LE(″)ズ υ力 r力ど 即 ″αと,zι″ノ4置aη夕∫ ∫pゼじどrα′ ∫ゼ?クθ刀じθ∫・This thcorcm mcans that the maps bet、veen c E2~tCrms given abovc arc thcsamc,  It sccms that it is a forklore but thcre docs not sccnl to appcar anywhcrc.§2.  Bousneld localizationThroughout this papcr wc work on the hottotopy catcgory of spectra.WC bC8in、v th thc dcinition of thc Bousncld iocalization([2],scc alSO[4]〉LctE*(―)dCnOtc a gcncralizcd homology thcory.Wc cal1/E*‐んどα′if[C,ズ]*=O fOrany C with E*((D=0.ThS dcnnition implics immcdiatcly             、COROLLARY 2.1.LすLう?E*―んじα′dP“ど′ク陶・ T/2ど乃 9αじ力 麗?pデ:ズ→ yψ′チカ* Dcpartinent oF Mathematics, Facuhy of Education, T9ttori Univcrsity, Tottori, 680,JAPANSHIMoMuRA,KE*(ア)なοttοrp力ね“カカじ9∫αヵねο“θrplis1/mデ*:[ⅥL]*堅Eズ,L]*.PROOF COnSidcr thc conbcring χ=二r→c.Thcn thc assumption on/impliCSE*(C)=0,Which shows EC,L]*=O sincc L is E*-local,Now thc corollary followsfro■l thc cxact scqucncc induccd froHl thc conbcring abovc`                 q.c.d.COROLLARY 2.2.Sttψο∫?チカαどうοチカ げ Ψ?じ″α χ αηプ y αr?E*イοじα友 丁 α ttrvP′ ズ → y肋力 じ奮 α狩 ねοttοΨれisれE*(デ),肋留 デた ,力οttοチ9Pノゼ?クテυβ肋 どぞ,PROOF.ThC Corollary abovc shows an isomorphismデ*:[Lズ]*冬[χ,ズ]*anddetne a map σ:7→ズby σ=(デ*)~1(lx).ThCnクデ=デ*(σ)=1ズ・Silnilarly thc map σ givcs rise to a mapデ′denncd byデ′=σ~1(ly)using thcisomorphism σ~1:[χ,7]*笙[Иy]*ShOWn by Corollary 2.1.Thcrcforc wc haveデ′σ=lY・Notc thatデ=lY=′σデ=デ′lχ=デ′,and WC Scc thatriS a homotopy cquivalcncc.q.crd.Wc call a E-local spcctrum LEズthcんどαrデzα√ヵヵof a spcctrum/with rcspcct toE*(―)if thCrc exists a map ηx:ズ→LEχぅwhich is callcd thcんθα′彦,ど力η tη,suchthati(i)thC induccd map E*(ηχ)iS an iSOmorphism,and(ii)if thCrc is a map′ズ→7 With 7 E-local such that』*(デ)iS an iSOmOrphism,thcn therc cxists uniqucly a■lap σ:r→IPEズsuch that,デ=ηx.THEOREM 2.3([23)どυθr/力οttοん2どル οり E*(二)力体 力∫んじα′彦αr力η LEズげ αν ゼじrr"駒/.戸力rど力9′μοrg LEね虎 刀何ο/力′,By the dcinition of thc locdization map ηx,wc B?c』*(LEズ) hC Samc structurcaS E*(ズ)by ηx ThCrcforc wc assume that E*(ηメ)iS thC idCntity l:E*(ズ)=E*(LEχ〉 ThC 10Calization has the following clcmcntary propcrticsiProposition 2.4.  For α ρゞαθど′クη, 'Y,(i)LEχなクη″ク?ル冴げ'ηθ濃(ii)LELE=LE`(lti)ア励ぞrぞ ねα ttηデ:χ→ y With♂力θ乃√ルrg傷ねぬ ,麗リ カ:LEズ→ y∫ク肋v7?力αυ?rE功じα′∫クどヵチカαどE*げ)なαη約 陶θり脱灘 ,ど力tyど力ηχ=デPROOF.(i)SuppoSe that thcrc exists anothcr E*■ocalization上名χ f a Spcctrumズ.Thcn wc havc an E*―cquivalcncc η後:ズ →ーL名/. The SecOnd cOndition of thcdcanition indicatcs the cxistcncc of thc map 9:L修‐>LE SuCh thatση隻=ηァ. Sinccboth ηtt and ηx arc E*―equ?alcnces,σ is also an g*_cquivalencc. Thcrcfore thc lllap σturns out to bc a homotopy cquivalent by Corollary 2.2(fi)BOth LEズand LELEX arC E*―local and the map ηLEX:LEズ→LELEX is anBousncld localization w.rt E(/tlE*―cquivalcncc. This casc again follows fl・om Coronary 2.2.(1li)COnsidcr thc abcr FEX Ofthc map ηx:ズ→ LEズ,and wc have thc long cxactscqucnce…→[LEズ,7]*」4[x,7]*→EFEズ,7]*→_ Notc that E*(FEズ)=0sincc ηtt is E*―cquivalcnt.Thus[FL・ズ,y]*=O by thc dcanition of thc E*■ocalspcctrum,which shows that ηtt iS an epimOrphism.Hcncc wc obtain thc dcsired map力. q.e.d.PROPOSITION 2.5.S2ppο∫"力αどχ → r→zねα じ9″gr∫θ?クゼηε?・ 丁 α77/″0げχ ,rα乃″z prゼE*―んどαL∫οたど力ιοチカgr.PR00F.  Let フ/be any E*―acychc spcctrum and supposc that X and y arc E*¨local.Then[レ:ズ]*=0=[力:y]*.Furthcrmorc the conbcring induccs thc exactsequcncc[町7]*→[町Z]*→[町/]*・Thercforc wc scc mat[町z]*=o,WhiChshows that Z is E*-local.                                         q.c.d.PROPOSITION 2.6.r/7→χ → yねα じり うθ′∫9?クθ刀じθ,rル刀 ∫θ ね LE〃→ 比 X→ LE IPROOF. Lctデd nOte thc map 7→ズ and C dcnotc the conbcr of LEデ:LEフ7→ LE/・  ThCn thc proposition 2.5 shows that C is E*-local.  Considcr thc diagramLE″堂4 LEズ~~→C‖ ‖ |馬V二豊蠍LEズ=空生LE Lin which,stands for thc mapズ→ r Wc alsO scc ttat LEσLEデ=O SinCC,デ=O andLE iS functorial.Hcnce we havc a map力in thc diagra  abovc.Applying E*(―)tOthc diagranl,we obtain a commutativc diagrana with t、vo exact rows,which 8iVCS us anisomorphism E*(力)by the avc lcmma,sincc E*(LE?)=E*(?),WhCrc“?"may bcsubstitutcd by a spcctruln or a map.  Thereforc Coro1lary 2.2 11nplics that 力 is ahomotopy cquivalcncc. q.e.d.§3. Generalized Adams specttal sequenceNcxt rccall[1]thC gCncralizcd Adams spcctral scqucncc.For a ring spcctrum E,we have an】‐Adams rcsolution/←ズ.,1←/E,2←… OF a spcctrumズ,in whichχE,■+l is a nbcr of thc induccd mapズE,И~→E∧ズE,.fronl the unit map テ:S―→E ofE.Then thc E―Adams spcctral sequencc{E,'ι(χ)}fOr a specttum χ is the oncassociated with the exact couplc π*(ズE,.+1)→π*(/E,.)→E*(/E,.)induCed from the EぃAdams rcsolution. Bousncld E2]givcS a dilfcrcnt resolution fron this E‐A amsonc.Lct χとbC thC Conbcr Of thc mapズど,.すズ and COnsidcr the associatedresolution pι←/と←χ乙←….ThCn we construct thc homotopy invcrse limitSmMOMURA,K.Im←ズとOf this rcsolution,We call this E―刀J″οr?刀r cο/mp″ガοtt of/and dcnotc it by】A/.Thcn wc havcTHEOREM 3.1([2]). Lθι E α刀プ/bθじοη寛9じケカ?→ριじ″α. k97ppο∫θ チカ,チヵ r?αじ力 Sαη′ヶ,ど力θr9 9χたな αデ崩セ rdクじカ チカαr E,'サ(/)=0,7カθη 彦力?EぃИttα陶∫呼夕ιす/α′∫?T夕θ刀じθ{E,'1(ズ)}εοηυ?rσど∫ √Oπ,s(EA χ 〉Lct BP be the Brown‐Pctcrs n spcctrul■at a primc p.Then it is known[2]notonly that BP satisnes the hypOthcsis of Thcorcm 3.l but also that BPA/=LBP/=勇f(″)lor a conncctivc spcctrum /, and so wc havcTHEOREM 3.2([1]and[2]).Л9r θαε力prガ胸θpりθ力αυ"力ιИ】2盟ざ―Ⅳουttου sPどすrαチ∫じ?ク9ηθ?じοηυθrσ力♂ゲOα力οttο√?pノσ′Ott π*(χ)げαp司リカ ′じθ刀ヵじo力ど颯9“サ′ク麗 χ ψカカE2~′θrtt Extと貧(DP)(BP*,BP*(χ)).Lct E(rt)10r η>O and a primc p dcnotc the ring spcctrum introduccd by Johnsonand Wilson whosc cocmcicnt ring is ZID[υl,…・,υ",υ√1](8虎[4,Cor.2.16]),We alSOusc the notation L.for thc B6usncld iocahzation LE(“)following Ravcncl.  For small Ю,E(D SatiSncs thc condition of Bousncld's Convergcncc Thcorcrl and we havcTHEOREM 3.3([1],[2],and[句〉 L'pう?αP/デ陶θ乃ク陶うθr αη′η αPοすガυ9加r?σθrνゴ励 乃<p-1.η`θヵ ψ?力αυθ E(η)Ax=L“ズ α猾″ ど力θE(つ‐Иttα陶∫・9p?じrrα′∫??クιηじ9じOttυ9rσ力σ どOπ*(Lヵメ)″カカ E2~サθ′陶 ExtとぉЦE(け)(E(η)*,E(η)*(ズ)),§4. Relation between the mapsLct E and F be ?ng spcctra andデ:E→F a map of ?ng spectra.Thcn this mapデinduccs thc map of resolutionsデ*:E*ズ→F*Й「and so we have an induccd mapデΛ:EAχ→FAズfor a spectrum χ.Hcrc』*χ and F*〆dcnotc the rcsolutionsgivcn in thc prcvious scction.That is,E“ズd notcsズち.Lct ηx:ズ→ L“ズ be the E(Ю)-10CaliZation map of a spcctrum χ and Φ:BP→E(η)thc Thom map.  Supposc that X is p-local and conncctivc.  Then as notcd in thcprcvious scction, wc havcBPAズ=ズ and E骨)Aズ=L〃χ.Thercfore wc scc that thc induccd map φ Λ:】PAズ→ E(η)A/is Oハ:ズ→L./.Nowthe theorcnl in the introduction mcans thc follo、vingTHEOREM 4.1.  Lθ′ηう?Pο∫カデυじ′η√θσtPr α乃′ズ ap―′οじα′じοηttθεどカゼΨ?εrrクηl.  7駒θE(れ)―んじαttαttη ttη χ:χ→ LИズ ね √ル ∫α陶じ斜 どル 肋力ε9プ陶η ΦA:/→L“χ .PR00F,Supposc thatthe map φΛ:ズ→ Lた/induccs an isomorphism E(η)*(ΦA):E(η)*(χ)→E(η)*(L″ズ).Then by deanition,wc have a map σ:LИ/→L打/SuCh th tσOA=ηx,since E(η)*(x)and E(■)*(0人)arC bOth iSomorphism, wc havc anisomorphism E(η)*(σ〉 Now apply Corollary 2.2 to obtain thc thc homotopycquivalcncc σ.Bouslleld localization w r.t J(η)Thercforc it is suttcicnt to show that thc map φA induCcs an isomorphismE(η)*(φ∧〉 ThC map φ induces thc map of E2~termSφ*:Ext苫質(BP)(BP*,BP*(E1/P)∧χ))一→Ext言ぉ*(E("))(E(η*,E(η)*(E(η)∧ズ )).By the changc of rings thcore■1, wc hav  isomorphismsExt苫糞(BP)(BP*,】P*(E(η)∧ズ))=E(η)*(ズ), andExtぉぉ項E仰)(E(猾)*,E(η)*(E(乃)∧χ))=E(η)*(ズ),since wc havc isomorphisms】P*げ(乃)∧ズ)=BP*(BP)①】Ptt E(つ* χ)and E傲)*(E(乃)∧ ズ)=E骸)*(E(η))OE(.)*ElrT)*(χ).The Thom map Φ induccs the cano?cal ri gmap:BP*(BP)一→ E(猾)*(E(η))=E(η)*(⊃】P*】P*(】P*)(DBP*E(η)*,whcrc BP*acts on E傲)*by SCnding υi to υ,forテ≦η and O for the othcr,. Thercforeobscrving thc changc of rings thcOrem shows that thc map Φ* is an isomorphism.Sincc these spcctral scqucncc for E(η)∧ズ C01lapse,this isomorphism induccs thedeslrcd onc.                                                             q.e.dReferences1] J F Adams, Sどαb乃力ο/TPοι07/'カプσθ〃?′αrisθ′ヵοttοんσノ, University oF Chicago Prcss, 19742] A K Bousncld, Thc localization oF spcctra with respect to homology, 冨9pοんヮノ18(1979),257-2813] D,C.Johnson and W S Wlson, βP operation  a d Morava's extraordinary K―theories, Лrαど力.zj144(1975), 55-754]D C Ravcncl, Localization with respect to ccrtain periodic homology theOrics, 2脇θr Jル「αι力106(19841,351-414

Note on the Bousfield Localization with Respect to E(n)

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Note on the Bousfield Localization with Respect to E(n)

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