Shimomura, Katsumi

Tottori University Research Result Repository

J Fac Educ TottOri Univ.(Nat,Sci),40(1991)25-32On the F2‐terコn of the Novikov sPectral sequence jora Thom sPectrumKatsumi SHIMoMuRA*(Rcccived August 31,1991)§1・ IntrOducttonln his papcr[2],?rahowald shows thc way to construct a ring spcctrum fl・omThom cOmplcxes of a nbration.As an cxamplc,thcrc is a ring spectrumズ(1〉obtained froIIn the llbration ctassincd by thC canonica1 8Cncrator世2S2_→BO.  Consid rthe Brown‐Pctcrson homology B亀(―)at thC p?me 2, whose cocrfcicnt is thepolynomial algcbra BP.(S)=Z(動[υl,υ2,…・],and wc sec thatBP.(/(1〉)=BP./(2)[ォ1]as a subcOmodulc algcbra of B4BP/(2),whcrC BttBP=B呉[す1,ケ2,…・]・ We alsOconsidcr the ing spectrum E(2) whose COCttcient is a pOlynomial ringZ(2)[υl,υ2,υJl].In this notc wc study about thc E2~terrn of thc Adams―Novikovspcctral scqucnce for thc E(2)‐locahZation L2ズ(1〉Of thC Spectrumズ(1〉and Obtainthc nrst fcw lines.Hcrc the E2…tCrm is Ext≧P*】P(BPx,(υデlBP./(2))[ヶl]〉 In OrdCr tOstatc our thcorenl 、vc prcpare somc notation.  Lctδ:Ext≧P*BP(B亀,(υJl】み /(2,υ評))[オ1])→EXtとP*BP(BPx,(υJl BPx/(2))[ιl]dcllotc thc boundary homomorphism assOciatcd to thc cxact scqucnccO→(υ」l BPx/(2))[ιl]→ψ「 lυデ2B4/(2))[ヶl]→(υ豚lβP./(2,υF))[ιl]―→ 0.Wc also usc the integcrs Иた and cたdcnned byИO=1, И2た=4И21-2+2, and И2た+1=242た;andc。=-1, and cl+1=4cた+4,and an algcbra K(2)*=Z/2[υ2,υデ1]・THEOREM.T力θど2‐どθr陶9ガ″ル Иttα陶∫ぃⅣουttου・Fp?どr′α′∫ど?クθttθ?力r L2X〈1〉α′rルデカrαr'ο刀滋σκどル郎チカ防3ねσデυ9刀 ,∫ヵんν∫:0)EB'*=Z/2[υl,υ2]1)Eと'*ねα力?ど簡ク町げZ/2[υl,υ2](力20}αカプν.2)βち'*ねα瀬ル?じど∫ク陶げν{力31),α/rιθ K(2)*‐胸οカルσθ々θrαど″妙* Departinent oF Mathematics, Faculty Of Education, TOtttO? University, Tottori, 680, JapanKatsumi SHIMOMURAδυとん30/υl  and  δυをρ/υl)/′≧0,pηプリすた K(2)*[υl]―胸οとカル∫σ?乃?′α′ゼ′″δυ:S+1カ20/υl,δυ:ι力20/υ?(ケ≠0)αrT′ δυち力21/υl力rS≧Oα乃′ιlG=(22た+ls+Cた+11た≧0}.打θ′θちαη″ρα′"ルθ′9用ιηサ∫′Ψ′θ∫効じ′妙ヮじねど:すα刀プυεヶ4+すユげサルじοbαrCο型ルヽαカプ〃滋刀ο′g∫α″節 ∫"“げり訪じK(2)*[υl]―"οカル∫σθη9′α彦″妙δχ4/υrm力 ′Ю≧ 0,αtt ο"力′じσど前 >0,α刀′ ∫ο陶9θル“どガ χ刀∫"ε力 励αチχ打=υζtt mod(2,υl)(Sθ?§3).This provcd by using υl Bockstcin spcctral scquencc.  First wc givc the structureOf the E2~tCrm of thc Adams―No?kov spcctral sequcnce for thc E(2)-localiZationL2N(1,1)Of thC COnber N(1,1)Of thC map υl:ガ2ズ(1〉→/(1),WhiCh is a tcnsorproduct of an exterior and a polyno■lial algebras(scC PrOp.2.5) We notC that wcstudy in[3]about the homotopy group of L2N(1,1)and sCC that thc E∞―tcH■of thespcctral scqucncc is a tensor product of an extcrior and a stuntcd polynoH?al.  ThcE2‐tCrms E身*fOr any s are non‐trivial in our casc,though most of thcm arc trivial atan odd prilnc p.  In a samc way as thc proof of our thcorcm it sccms to bc computedthC E2‐tCrm for highcr nltration, but here wc only givc somc dimcrcntials for them(Lemma 3.7).?lost part of this is donc during thc author's visiting at NorthwestcrnUnivcrsity.  Thc author would likc to exprcss his gratitudc to Profcssor CヽarkWIahowald for his suggcstions and hearty hospitality.§2.  Change of ringsln this scction wc g?e somc thcorcms for the cbmodulc(υテ1】ニト/(2,υl))[ιl]similarly to thosc of[6]and[7].Lct BP dcnotc thc Brown―Peterson spcctrum at thc p?mc 2 and K(lCl thC ?IOravaK―spcctrum whosc cocmcient rings arc Zl劾[υl,υ2・中]and Z/(2)[υゎυ√1],rCSpect?cly,on Hazcwinkcl's gcncrators υ々's with dcgrcc 2た+1-2.Thcn Γ=B■*BP=BPx[ιl,ケ2,・…]fOr thc gcncratorsケデs with dCgree 2:+1_2,and wc dcaneガ(0=K(た)*K(た)to bC K(o*c⊃】P孝】PIBP①】PヰK(/cl*.ThCSc givc risc to Hopf algcbroids(BPx,Γ)and(K(た)*,Σ( ))and thc natural map r→ガ(rt1 0f HOpF algebroids.Fora HopF algcbroid(/4,L),thc Ext‐group ExtL(ス,Ar)of a cOmOdule M with structuremap ψ iS dCined to bc a cohomology of thc cobar complcx Ω:″with ΩLれイ=ν  ① ИL①x…① ИL(s copiCS Of L)providCd by the dircrcntial冴:ΩLM→Ωゴlνgivcn by冴(附① χ)=ψ附 ① χ+Σi=1(-1)1削① Иκχ―(-1)S町① χ ① l WhCrc Zκ=111①И ① ls_κ ttr thc idenity mれ1::LOι→LOtt Wc havc thc followttg changc of ingstheorem:E2~tCrm of ANSS for a Thom spcctrum(2.1)[4,Th.2.10]r/ν,s υl―んどα′αηガ2 =0,どル々Extl(】P.,M)些Ext数1)(K(1)*,K(1)*①BPネM)クη力r√ル 刀αどクrα′陶η,αヵプアMねυ2~んじα′α刀プ2ν=0=υl九r,どヵ9狩I              Extl(BP.,Ar)冬Extち(勢(K(2)*,K(2)*①BPキAr)ク乃瘍?r √力θ η,彦ク′α′ηtttp.Then for Hopf algcbroid(K(1),Σ(1)),WC havc(2.2)[8,Cor.6.5.6]ExtΣ(1)(K(1)*,K(1)*[ιl])=K(1)*[υ2]OE(力20)・Consider thc Hopf algcbroid (K(2)*,ガ)=(K(2)*,Σ(2)/(ォ1))Such that thepr?CCtiOn Σ(2)→Σ is a map of Hopf algebroids.Wc thcn havc thc following(2.3)[8,Th,6.5,5]Ext歩9)(K(2)*,K(2)*[ιl])=K(2)*[υ3]①氣2>ExtΣ(K121*,K(2)*)。As in[8,Th.6.3.7], thc manncr dcvclopcd in[6]and[7]to computcExtΣ。)(K(2)*,K(2)*)is applicd to obtain(2.4)7功ι′をね α呼,“rrty′∫θTクθ乃じ?E(力20,力21,力30,力31,力40,ん41)①P(b20うう21)=⇒ExtΣ(K(2)*,K(2)*)″′肋 輔 trι刀チヵ′冴2カ4J=力ち,J+う2,デ+1,ψ力″θ力,デαヵプう2ブカ ′デCZ/2α′?′攣 rθ∫例 ′″ 妙 ギ'α乃″ιちJ①ιちJ,′?ッ“チブυゼル,andヵちJ=b2J,カチカθじοbαr εοttpルχΩ各(2)K(2)*.rffr?E αη″P∫ど,η′ズ9膚力?θχサιガοr α刀プ′ルpor/ヵ。麗,α′αヵθbrα∫ουθr K(2)*[υ3]・This shows thc followingPROPOSITION 2.5.ExtBみ】P(B亀,(υデlB亀/(2,υl))[ケ1])=K(2)*[υ3,力20]①И(力21ン30シ31,ρ)・】θ′ゼカ2',力3'αη′ρね′ηrθ♂θttrθ′″ヶちt,オζt,乃′υ】ι4+ケヱ,rglsP“″′υ9炒・|I                          §3. The Bockstein spectral sequencel            ln this scction wc computc thc Bockstein spcctral sequence to g?c thc hromaticEl―tcrms for computing our targct Ext】PネBP(BP.,(υJlBPx/(2))[ιl] ,and COnsidcr thc十        following Γ=BttBP―COmodules:I                И =υ豚lBP./(2),】=υ「lИ,C=B/И, and D=υデlB=*/(2,υl),and thc notation28 Katsumi SH二、rOMuttAHS'M=欧げlB亀,ν).Thcse nOtatiOns give rise to Short exact scquenccs:o→Иど恥】_→c_→0ぅ andO→D→C4c→0,which inducos thc long cxact scquenccs:0-→打V→ fr°】→ 打OC 4 π14-→・―(3.1)→ ,たИ→汀切→ 汀たC4打4・lИ→ …andO→汀OD―→Fr°C―→打°C二4,lD_,1…(3.2)→ 打たИ→ 汀 C々―→πIC 4汀+々lD―),…We call the scqucnces 13.1)and 13.2)the clarOmatic and thc Bockstcln cxactsequences,respcct?ely,Now dcflnc i4tegers:,。=1, αz十二=4銑+2;bO=0, b打+1=4うぇ+3;c。=-1, cた+1=4oκ+4;and element of υJl=亀:死1=υ:一υ?υ」lυ4χ2=χ?~υ♀巧υ3~υtυ2 υξXl~υfυ万395+υ:υ2S υЭ嵯and労2■+1=χをた一υ子'4υデ1+2br υ:+2clcυ4χ2t+2=瑶1+1-υtrたttκυな争0「6υをO x2+υ「lυ:υ:+υ4υ万3υ:υ5+Diυデ5,:υ身+υ「1。υみヽ3+ク子υ二電υ4+υJ6υ:―χl)forた≧1,whcre貫2=死i~クニυttυ3~υこクεχⅢ The O―血 diIFcrential′Of thc cObarE2~term or ANSS for a Thom spcctrumcomplcx is ηR~ηL Where ηL iS the indusion B亀→ BttBP,Sincc comodulcs which wctrcat hcrc havc thc struCture maps induccd froAn the right unit 14ap ηR・  ThCn、vc haveLEMMA 3.3.肋(υ豚1】Px/(2))[ヶ2,ι3,…],冴x。=υlιち―υtケ2冴χl≡υ〔υttι2+υユυJlケ: mod(υt)冴χ2た=υttυttυき十cたιち十υttυち十bたυ子ヶ3 mOd(υ:+αり冴χ2たキ1=υfaたυB+2うたυζ+2●たヶ2+υ44たυち+2bたυ:oたιζ+υl+2,たυJ二十2bたυζ+2cたιζ mOd(υそ+2αた)PROOF. ThiS fomows from a dircct calculation using冴υl=0'υ2=0(3.4)       冴υ3=υlιち―υtι2冴υ4≡υ2ケち~υを才2+υlケζ―υこυ3ιζ mOd(υ二)冴υ5≡υ3ケB~υζι2+υ2姑~υBケ3 mOd(υl),which arc vcrincd easily by Hazewinkcと's and Quillen's formulac(丁 [8]):υ″=フ脇“一 Σ :ゴ!翻〆'と,and ηR脇“=Σ j十ザ=“粗'fi・ThC arst One in thc lcmma is thc immcdiateconscqucncc of thc third cquality in (3.4) Noting that ttχ三ノ mod (のimplies冴χ2=ノ2 mOd(IT2)cnablCS us to go forwara by inductivc stcps, Thc rlrst rcsult shows冴υζ tt υ?ιをmOd(υ二)and a dircct calculation brings ttυttυJlυ4三υttυデ1(υ2ι参~υちι2+υlιζ―υttυ3ιち)mOd(υこ),the sum of which is thc sccond one of thc lemma.This shows ttx4≡υtυ;ι身+υttυ万2姑+υ二υデ2υ:ιtt mOd(υl°).ThCn we gct痴τ=υttυ万2ι會+υ二υをιζ+υttυ万2υ:ιtt mOd(υ!)by the rcsults ttυユυ】υ3=υユυ'(υlι身―υtι2)and ttυ二υ3χl三υ二υ〕(υ4υヨι2+υユυ豚lιζ)mOd(υ二)Obtaincd by(3,4).Thc third one forた=l now f llows from morc calculation冴υtυデ6υttχl=υ4υ豚6υ會(υ?υBι2), 冴υttυJ3υ5=υttυJ3(υ3ケB~υなど2+υ2ιな~υBケ3) and冴υttυJ5υ3υヱ=υttυ豚5υ3(υをιε―υBケζ)all mOdu10(υ子).Supposc the congrucnccc for 2た, whi h brings ttχちた=υ〔' υちうたυζ+2じた姑+υ:●たυ参+2bκ υ:ごたι: mod (υ:+2,■)。 we alsO compute ttυ?aたυちbた lυ:+2じたυ4=υ♀'たυをうたlυ:+2cx(υ2ιケ~υなす2+υlケ:)mod(υt+2,k)to Obtain thc congruencc for 2た+1.Thc inductivc hypothesis implies ttxζた+1=υ4'たυちbたυ:Cた(υBυをすち十υBケな+υ?υデ2υ:ιな)mod(υ4+4,た),Notc hcre that cたis  multiplc of 4 and so 4ol is of 24,which indicatesthat ttυttCた三O mOd(υ16)。 wc nOW CCrtiけthC last onc by computing mod(υユ)30                             Katsumi SHIMoMuRA冴υ」6υゥ0万三υ「6υ】0(υttυ32姑+υ二υζιζ+υ二υ豚2υζヶを)冴υ「lυBυ:=υ「lυBυ會(υl場)冴υ:υ」3υ:υ5三υttυデ39會(υ3ケB~υζι2+υ2姑~υ'ι3)冴υ4υJ5υ:υぇ=υ?υJ5υ:(υちι】―υB′を)冴υ「1°υ:χ3三υ「1°υa(υ12υ号(υBυζι2+υをケ:))'電υ二幡υ4= 賢υttυ:02を~υ参ケ2)冴υデ6υ:χl=υJ6υ:(υ〔υBι2)q.e.d.In order to study about thc δ in the sequencc(3.2),wc giVe anothcr rcpresentativcOf力31:力31=ケζ+υl υ3ι:+υtυ3ケ2+υ:才〕+υttι3,which cquals to υ「19υ4~υ2ケち~υをι2)・ SinCC υl acts monomorphically on thccomplcx and υ2 and ι2 arC priinitivcs, wc sceLEMMA 3.5.   冴力31=0カ(BPx/(2))[ι,ケ3,・~]・PROPOSITION 3.6。 T/tι?ル陶?乃葎 υ2,力2,】クr'CZ/2,αヵプ カ31 αrg β〃 じノεル∫ カ チカ?じοうαrどοη ルχ ΩIC.カサカゼ∫?¢クθηθ?(3.2)δ―デ胞ασ9∫`アr/2?θル胸?肪∫υ2,力2'α狩′力31α′?α〃 ″カカ′.In the ncxt lcmma,we do not writc the multiplcs of υ2'S,bCcause wc can tcn thcmby virtuc of dcgrees of clcmcnts and furthcrmorc thcy madc thc formula morccomplicatcd though thcy havc no inducnecc on thc modulc structurc which we、vant todctcrininc.LEMMA 3.7. Lιr δ bθチカ? 陶η カ サル ∫??クθηじ?(3.2), テル乃ぅんr α刀ノ ?力麗θηねα∈K(2)*[力20]OИ(力21,力31)αη′肋ど999r∫ι=2s+1>0.υζS力21αυttS力20αυ:И+lS+2●■+2ヵ20α(η=0)(η=1)1)    δ(X'α/υr4)=十υ:・+IS+2cゥ31α  (η=2た+1,た≧ 1)υ:泳+lS+Cた十二ヵ21α+υ:況+ls+cゥ30α   (η=2た,た≧ 1)動 地 期 り =拶雛評+兆印移≡8E2‐tCrm of ANSS For a Thom spectrum勤  δは,ρα/υl)={み:二統比∴4咎枠】与2i'とτ力20カ21α ttΞ8rr9rι И“ねαη力r?σ9rαたアη=2たαη′2,たアη=2た+1.PROOF.  By virturc of Proposition 3.6,wc sce thatδ(χα)=δ(χ)α,and so it is cnough to show that thc lemma for α=1.Then l)follOWS immcdiatclyfrom the den?tion of δ togethcr with Lcmma 3.3.Using QuillCn's formulaΣ,+J=“附,Zιtti=Σ,十Jキた=Иttiι:こ①ケ?i十デ  and  Hazcwinkcl's  formula  υ“=p“.―Σ:fl“:ケ'と,,We computcИ ι3=ι3①l+l ① ι3+υlι2 ① ι2∠ι4=ι4①l+ι2①姑+l①ケ4+υlケ30ケ3+υ2ιち①ιちZ ι5=ケ5①l+ケ3①'3+オ2①姑+l ① ι4+υ2ιζ ①電十υ3ιち①おThe nrst forl■ula givcs the oncδ(χ力30/υl)=δ(χ/υl)力30+渤30'and by thc others togcthcr with(3.4)we computcδ(ρ/υl)=υン21カ31+υち:力20カ21・Thus wc obtain 2)and 3)from Lcmma 3.3.                  q.c.d.It sccms that this cnablcs us to computc all dilncnsion of thc groups frた(テ,but thccomplcxity of intcgcrs appcars in the cxponcnt of υ3 prCVen s us t  writc down theexplicit gcnerators of ffたC.Oncc we witc down them,we cotteCture that we ncedonty a fe、v morc diIFcrentials to computc al1 8roups rrたC.  Hcrc wc give the nrst twOgroups as follows,though wc can writc down ffえC fof l ore small valueた,inductivcly,by routinc computation.PROPOSITION 3.8. 1)π°Cね α Jぞη∫οr prο励 何 げ K(2)*α刀プ α ttrιc・r∫ク麗 げ(Z72[υl,「1])/(Z72[υl])αη冴ヮ訪じZ/2[υl]嘲ο励ル∫99η?rαr″″χ,/υr澱力′Ю≧Oα刀′ι=2s+1崩r力 s≧0。2)耳lCねα ′?力∫Or prθ力じチげ K(2)*αヵプ α 加′“チ∫"陶 げff°COZ/2{力31),η 2EV3]{力30/υl,ρ/υl),αη′ υ じ′力 Z/2[υl]初ο肋 ル∫ σι乃?′α彦ぞ′ 妙υζs+lヵ20/υlク カ20/υl,υ:rヵ20/υ争(ι≠0)α刀′υち力21/υl力rS≧0,カプす挙G=(22え+ls+cぇ+11た≧0).Rccall thc exact scqucncc(3.1)Katsumi SH「MoMuRA(3.9)0→打切 → 打°】→ frOc 4汀弦 → frl B→rfl c4打物 → 打2】_→..・.In thc scqucncc,thc structure of r*B and汀*C arc givcn in(2.2)and PrOposition 3.8.PROOF OF THEOREMo Sincc lf*B is isomorphic to K(1)*[υ2]①E(力20)by(2,1)and(2.2),汀°B=K(1)*[υ2],河lB=K(1)*[υ2]{力20}αη冴打κB=O forた>1.Noticethat υ参力20/υ?=力31/υl SCCn by thc sccond congruence in Lcmma 3.3.Then Thcoremfollows immediately from thc cxact scqucncc(3.9).               q.e.d.References[1] J F.Adams, Sサαうル カοttο′οPメ伽′σθ力¢r,776θ′力Ottοわ♂ノ, U?Versity oF Chicago Prcss,1974[2] M MahOWald, Ring spectra which are Thom complexes, Dr/t¢MarFD ェ 46(1979)[3] M MahOWald and K Shimomura, The Adams‐No?kov spectral sequence For the L2 1°Calization ofa υ2 SpeCtrunl, to appear.[4]H,Miller and D.Ravencl, Morava Stabilizer Algcbras and thc localization of No?kov's E2‐tCrm,Dクたθ Maカエ 44(1977),433447[5]D.C Ravcncl, The structure of】∴βP mOdulo an invariant prime ideal.T9Pοんσノ 15(1976),149-153.[6]D C Ravenel, The structure oF Morava Stabinzcr Algebras, 力υθ″′力″θ∫〕ど,ど力.37(1976),109120,E7]D,C Ravenel, The cohomology or the Morava Stabilizer Algebras, Math Z 152(1977),287-297[8] D.C Ravenel,(b/zpルχ θοbοr'ism翻′∫と,う降 力οttοrDPッ′rOf/pτデΨ力θ′θs,Academic Prcss,1986

或るトムスペクトラムに対するノビコフのスペクトル系列のE2項について

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或るトムスペクトラムに対するノビコフのスペクトル系列のE2項について

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