Shimomura, Katsumi

Tottori University Research Result Repository

Note on invariant regular ideals in BP*Katsumi SHIMoMuRA・      I(R¢t19ブッ?'A,Ibβ′,7,79∂∂)§1.  IntrOductionLet BP be thc Brown…Petcrson spcctrun at a pri=nc p.  Thcn its homotopy group犠(B⊃=BP*iS the polynomial ring over Z。)witt the Haze?nk l's generators υl,υ2,・・・,It also g?es thc Hopf algebroid(BP*,BP*】P)with the right and the left units ηR andηL(Cf・[1],[4]).Letデdeno e an ideal of lengthヵgenerated byれhOmogeneous elcmcnts,。,,1,・",',_1 0f BPキand put弟=(,0,,1,・・・,αた_1)fOr k≦■.Thc idcalデis said to bertg,肋r if,。is a power ofthe prime p and α々 is not a zero d?isor in】P*/Jz for eachた,andto bc加ッ,ri,ヵ才if deg,。<deg,1<…・<deg,._1,7R'0=7L'o and 7R'た=7L'たmOd tt forcachた。Pi S.Landweber[21 studied sOme propertics of invariant rCgular idcals and deteHninedan the invariant regular idcais of lcngth l and 2.  Wc can rcad off an thc invariant regularideals of length 3仕om e results of H,Millcr,D.Ravcncl,and S.Wilson[31 for an Oddp?me P,and ttom[5]for thC prime 2(scc PrOpOsition 2.7).E.Tsukada[7]found all thcinvariant rcgular idcals of lcngth η≧≧l in the casc that each generator,たofデis a somepower ofthe Hazcwinkel'sυたfor O≦た≦胞(υO=2).In thiS note we give a result similar toTsukada's using thc clcmcntsぇた,giVCn in[3〕instCad Of υた(seC PrOposition 3.8).Wc nOtCthat invariant rcgular sequences give a periodic faHlily in the E2‐term of Adans‐N vikovspcctral sequcnce converging to the stable homotopy gЮups of sphcrcs([1],[31).Wealso notc that there exists the BP―local pcctrum yJ such that BP*yJ=υ「18P*〃for eachinvariant rcgular idealデat a large pttme p comparing with the length tt ofデby[61, oughwc do not know the e?stence of a spectrum yJ such that】P.7J=BP./J forデwhiOh wchave constructed here.§2.  Invariant regular sequencesThe coefacient ring BP* of thc Brown―POterson spectrum BP at a primc P is thepoly■omial ttng Z。)[υl,υ2,・…]and the BP*‐homology】PキBP is the polynomial BP*[■,ら,…・〕,Where deg υそ=deg残=効メ_2.Then(BP*,BP.B⊃iS the Hopf algebroid(cf.[11,[41),whOSe right and left units 7R,7L:】P■~→】P*BP are g?cn by the fo1lowing cqualities:(2.1.1) 7L υた=υた,7R上=Σ′+デ=れ牙D(ち=1)Dcpartmcnt of Mathcmatics,Facully Of Education,Totto? Univcrsity,Tot o?,680SmMOMURA,K,for後:】P*①2-】P*BP⑬2,where BP*⑬2=2[rt,J2,中●]and(2.1,21              υたとフ晩―Σ!Ξ:υ!とi!,Cn thiS papcr(o in the exponcnt dcttOtCS Pた)・ F r example,we deducc thc follo?ngcongruences:(2.1.3)    ηR銑+1=υた+ょキυが筆)―中l mOd fκ for た≧1,andηRυた+2=銑+lι筆十り_υぞ+1ケ1+υ々ιtt mod lrt,り forた≧2,in whichみdcnOtCS the ideal ψ ,01,P"ぅυた_1)Of BP*(Cf.[4;p.145]).Consider the following BP.BP‐comodules de?ved from thc comodule】P*deaned by:o.2)¶=BP,れand the exact sequenceO―N4-→υ評たN与一―>?;+1_→O for た≧0.he coactions of thesc comodulcs are the oncs induced from thc right unit 7R of】PキansalsO dcnOtcd by 7R・WC Sh証1五bbreviatc Nok to r.Each homogeneous element χ of珂存is writtOn by a linear conibinadon of fractions:12.3)       χ=w/9 fOr w∈BP* and υ=Πμ噺~lαぅ andχ=O if wcr“ o  α.lw fOrsOmc ,,where α,(′≧■)are Clements of BP*such that deg,ど<dcg,テキl and he ridical of the idealO吼,αヵ,・・,α'+た~1)iS島十々・Let Ar denote a cOmodule dinned abOヤe.  We denne汀°″=Ker冴for′=7R-7L・田随 mbdule打°Vi`dosely related to he E2‐Crm f the Adams―Novikov spcctral scquencc converging to thc stable homotopy groups of spheres(cf.[31),Let y=I,たlκ≧o be a sequencc ofinanite elements of BP*with deg αた<dcg αた+l fOrた≧0,and九=枠I10≦た<,denote thO swぃOqucncc ofデ.九 will alsO be writtcn as,0,・中 ,,ヵ_ⅢThcn thc scquence=η is caned/9g,肋′if lL)iS a proper ideal,,。ョトO an ,たis O non‐zcrodivisbr in BP*κ残)fOr eachた<打,and called加ッαri2肪if 7R,。=9LαO and 7R,た=7L'たmOdげ∂ fOr eachた<η.For an invariant regular sequcncc.九千1,COn ider the clemcnt   ヽテ           ■σ,力)=,/αo…2Pa_1∈^μ.The fo1lowing is an easy consequence of o.3)|LEMMA 2.4. Lすち+l bι,力′れソ,′力ヵサrgg,力r♂ιT"ι刀じじ,,刀tr,αtt ι力脇ι刀′9デBP*.劉佐刀,餌σ,刀)=0げ'“′οカウゲ'∈σ″)・           'Invariant regular ideals in BP*PR00F.If,∈α″) then(2.3)implies ttσ,ヵ)=0,On the other hand,研σ,用)=0implies the equahty,,″=Σデ<″均勺 by o.3),and SO α銑=O mod αう。 HenCe α=o modα″)by the regularity ofち+1.                           q.c,d.LEMLIA 2.5.Spppο♂ι ttα′ち ね α力滋ソα′力乃′r9g"協′∫θ?Йιれじιr9/刀≧′.勁 3η α r9g"肋rdι9'翻α 島 ■1ね加ツαri2肪丁伽 冴οカウ 丁χ傷 つ C打°/3.PROOF.  First we provc the fonowing by the induction onた:(2.5.1) ′(1カ。…αた)=O On♂離+l if,。,中0,αたis invariant.Since,。=P?fOr somc ι by[2,Prop.2.5],we haVC 7Rα。=化αo,and so′(1胞。)=OOn ? .Both 7R and ηL are algebra Fnaps,which imphcs(2.5。2) ′(αた+1胞。・・,そ)=7R'そキ生′(1/2。…α∂十冴(αた+1)/1Jo・・αた.It turns into′(αた+ゴ多。。・。αD=′(,そ+1)力0・・,そ by thc inductive hypothesis.Bcsides,ガ(αた+1)=O mOd残+l implies冴(,た+1)=αoE10+…十,そ晦 fOr somc,ど∈BP*.lhereforc'(αそ+1)ル。…βた=O by(2.3)and hCnce,(α評1/,。―。,た)=O in噺ギ♂F,which win thc exactsequence in(2,2)showS(2.5,1)forた+1.Now turn to the proof of thc lcmma.If島+l iS inva?ant,hen′(αИ)=O mOd tt and(2.5。1-2)implyどはσ,力))=0.COnversely if′はσ,η))と0,(2.5,1-2)again imply′(,,)力。中 ●α″_1=0,Which shows,(,,)=O mOdちan 九十 1 iS invanant.           q.C.d.LEMMA 2,6.Lすち+1=修たb≦た≦″α力冴 【確+1=|うたお≦た≦″bι肋ッαri2打′′q"肋′dιT,(%cS・r/σ確+1)=g囁+1),加ιtt σ“)=僻軌)(1≦“≦れ+1)α乃冴dCg,ど=dcg b,(0≦テ≦“).PROOF.Supposc ttst that α胸+1)=(「銑 +1)。lhen,(2.6.1) If αJ Of王秘 (′<“)SatiSnes,テ∈ (】銑 ),then deg α,≧dcg b″In fact,,,≡"う独mod(為)by thC assumption g協+1)=(〔胸+1)fOr a non_仕i?al elcmcnt"of】P*.Furthcrmore suppose α胸)+(〔“).If σw)⊃(ヽ“),there e?sts αJ of rtt So thatαテ∈(【胸)(′<脇).ThercfOre we scc hat dcg αれ>d g α,≧dcgうれby(2.6.1).On theother hand,う痺=″α/PI mod σか for SOme〃∈BPキby tte assumption.These imply ν=0and b″cc″).Then σ胸)⊃(玉碑+1)=α胸+1)WhiCh contradicts to tte rcgularity of王胸+1.ThuS α脚)φ(】銑).Similarly(石脚)φα閣)・ h this case thcre exist at of=w and b7 of【秘SOthat a,∈(〔胸)andちqEσw)・ ¶hen(2.6.1)is alSO applicd to showdcg α塑>dcg,,≧degう脇>deg by≧deg αれ,which is again a contradiction,Thercfore we have pЮved that o協+1)=(K確+1)impliesα胸)=(【胸).huS We Obtain the nrst statement,If dcg,サ<deg b,,then wc have々c(為),Sincc tt c(馬,X=儡+1))and先羊,あmod(馬)for any,∈】P*.Thereforc c+1)⊂(馬)=|り,ThiS also contradicts to thc regularity of105Jw+Ⅲ Thusdcgα甚≧degうJo Sinila』y deg rJJ:≦dcgう,. q.e.d.SⅢMOMURA,K.An ideal o④ generated by clemcnts in a scqucncc K=修お。≦κ≦,iS Said to be滋ッαrip肪r9g剪協r if F is invariant rcgular(cf・12,Cor.2.41), Let rR,denOtc thc sct of an inva?antregular ideals of lcngthヵ. For a Z。)―module♂И,IA4 denOtes the set of thc subscts卜l forall addit?e generators χ∈A ,wherc伊|=|′悦 lλ∈Z。)一PZ。♯.Then wc havePROPOSⅢN 2.7.例修κ 窃 ねね α寛 巧 ?前ツι ttψ∴:rR2+1-ド? I(乃>0)鯛炒 独1鳴+1)r9μσ,力)|.PR00F,First we shall show that ttσ,力)|=任(て,η)lif O,+1)=0転fOr invariantrcgular sequenccs九+1=|,たb≦≦ヵand KИ+1=lbたお≦≦″,Lemma 2.6 and the regularityiinply thc fonttringi(2.7,1)  うた=Åvた十 Σデ<た均々 fOrsOmc tt c Z.)一PZψ)and均∈ 】P*・Thcn by the deanition of^P,we havc χσ,ヵ)=λXK,刀)fOr sOmc λ∈Z。)一pZ。),There―fore the mapんis wcll denned.Now suppose ttat ttσ,海)|=卜(て,力)|・ Then we see that α∂=(〔確)by Lemma2.4,andwO can apply(2.7.1)to ShOW l胞。・。・,4_1=AlbO。・・う2_l for λ∈Z。)一PZ。).Thus脂打=♭″mOd α″)and wC have he equality α滑1)=(【確+1)・ q.e,d,Hη>1,he mapんis not SuttCCt?C.In fact,we can and an elcmcnt修,ん。・・,,_110fFプ拗V4With,,a zero divisor of BPキ/(αo,"・,α″_1).For cxample,take(pυτ)+υΥ)+(劾υザ)(2)/p2υ?)十o)十(1)} if 猾=2.§3. The elements χヵ,,for an odd primeFrom hcre on wc assume thatthe pttmc p is odd.Thcn thc clcmcnts χ(れ,′)∈劣lBP*●≧1,'≧0)(=ガ,デin[3])arc dCancd as follows(cf.[3,p.4941):(3.1)    χ(れ,0)=υ,,χ●,′)=χ(れ,J-lχ―(υ,_1)う(″'りe,′)for'≧1・Hcrc υ。=P,thC CICmentsズヵ,,)are g?n by(3.1.1)           ノ(■,1)=υ√lυ″+l  if 秘≧2;ズ2:動=υ夢1'2)(υ2+υ:υJフυ3);ズ2,つ=2υ夢1'.1)+l  if ,≧3;ノ(■,テ)=υ身(1''~1)+l   if η≧3, ,=1(附-1)andテ>1,andノ(η,')=0   0thcrwisc,for the intcgcrs(3.1,2) ι(た,デ)=亀P'一P' 1and he integers♭(れ,つdenOte p'forヵ=生or′<刀,and(3.1.3) b●,つ=P/⑫κ(力~1)-1)0カ_1)/砂力~1-1)Invariant rcgular ideals in BPキfor η>l and J=た(れ-1)+デ1≧刀With O≦デ<カー 1・Calculations with the equalities(2,1.3)and ttυl=υl+ptt g?en by(2.1.1_2)show uSthat these elemcnts satisfy the followingPROPOSmoN 3.2([3,pp.492_495〕).とす乃,力′′うιPθ∫肪ソι加″g′3『.Яο′肋ι ttteだれ腕 ′'=7R-7L:巧」BP*一げ 1】P*BP,加(ヵ,つiS COmputed to be:′χ(乃,0)=υ打_lι?~1)血od(r“_1,υ子_1)(υ。=p);冴χ(1,')=p'+lυ二ヶl mOdば+2)1/ 9=P'-1;冴χ(2,1)三υ:υち~1'l mOd lp,υ:+1)i冴χ(2,,)=2υ4( ,Dυ夢1'ど~1をl mOd lp,υi+玉2,う),αη'冴χ(η,つ三υ身笠'Pυ,(1''1)'?mOd lf._1,υ'=f(・'D)ヵ′η≧3,,=た傲-1)十デ+lν′加0≦デ<刀-1,,η′加ι加宅y℃s(3.2,1)           ,(2,う=う(2,′)十P  ヵ= ′>1),(猾,′)=う(刃,つ     (刀>2,テくれ),●,つ=う(寛,つ+ノ+1(れ>2,ゴ≧刀)CONVEN覆O  3.3. Since υ「lBP.=Z。)[υ「1,91,…・I cOntains BP*=Z。)Iυl,・ ]Canonical…ly,ea(h element χ of υ「1】P*iS uniquely witten as:χ=ガ～十コ fOr χ～∈BP*and χl c υ「 lBPキsu(五hat χ=χl in υァ1】P*/BP*. Thcn a sequcnce J:α。,αl,・中With,。∈BP*and α,cυ丁lBP*(′≧1)iS COnSidered to be the sequcnce of BP*by repladng,,with,ど～,and so wchave the idcal αη)Of BP*.Considcr the sequcnce of positive intcgcrsざ:ιが1,・ ,∫わ "・with∫た=ιρη and P/θf々Orた>0.  We call the sequcnce S P′♂―うどRV if it satisnes(3.4)0<ι≦′二十1,"=々売―売_1-ι+1≧090<ιた_1≦,(た,,た),and ιた_1≦Prtt if ιた=1.A subsequencc弓″:9,dl,・・・ ∫″_1 0f a prc―MRW sequencc d is also called pre‐MRW.Fora prc‐MRW sequcncc S,wc have the sequcnce xo=枠た院≧。Of】P*givcn by勾=P?,'た=χ(た,,た/forた>0,/_ι″フ>P'(,=第_1+ι-1),and,た=υtt fOrた>O if θた=1.A subsequcnce of xo iS Said to bc a】T‐Jι?,ρπι if the every cntry,たis a powcr of υた。Nouce thatデ(S),iS regular for any洸.The follo?g is a result of[7].108                           SHIMOMuRA,K.PROPOSII10N 3.5.Lすれ>Oα乃′ぎうゼαPκ‐″RV∫ι?,ιヵ∝.r/ιた=1〕/α〃た″′協 0<た<れ,力ιηデ(S)ヵた,■加ソαri,ヵrr皓,力′BT―dθ?肱%Cι・LEMMA 3.6. Lιr S bι,Pκ‐7RV∫留"ι刀じια刀′刀>0. 例佐乃″9カαソθPυ身=O mOd σ(0,+1)丁∫≧∫2,α乃′υ完=O mOd σ( )2+1)丁∫≧転,力 ′肋ι ttr9gι澪 ∫″=9/1Ptt α刀冴転=d,(+P嘉~?+1+P嘉一?げιz>1).PROOF.For η=1,デ(S)2 iS a Sequence of he form p9,υi Or s=狗啄~1(た>0),and SOυt=O mOdly(02)・In CaSe ιヵ=1,the lcmma is clear.Now suppose that thc lemmaholds for刀,ιヵ>1.Then thc ideal〔確=ゆι,pυン,υン)iS COntained in σ(S)ヵ+1)・We alSOconsider thc ideal Lz+1=0鈴,ら+1)I Put′=二十1 and′=ら.By thC dCnnition(3.1),weobtain thc congruence(3.6.1) χ(Ю+1,た)=(υ″+1/)一υ肝)~←~打 1ち洋猛1ゥmOd(p)inら汗l BP*fOr someノ∈】P*,whcreノiS a multiple of υ″+1~O ifた<ヵ+1.Thecongruence(3.6.1)implies(υ打+1)Jた+1=り¢~Iυ群)(た~21)υ″隼rpたPた1ノmod L刀+1in BP*with Convention 3.3.Since L″+1⊂σ(ざ)刀+2)and(υ群)~(た-2-1))2=O mOdLヵ+1,we have the lemma.                                                       q.e.d.LEMMA 3,7.とす 猾>l αttlr d bι,pκ_vRV∫ιT,ι用じι. P,ど′=崩_1,ヵど Cο淋 虎ル′力ι∫ιTttιttεじ【確_1:P?,υ7),…,巧と3,υ身-2,WhCrc c=Pι tt C>1,α刀′ =2Pど丁 ι=1.例修刀g吼_1)⊂σ(d)ヵ)・PROOF.Put′=売for O<たくれ-1.If ι々=1,then αた=?″)■r′=残(0<た<刀-1)・Therefore we have υ″)∈σ(S),)Since,≧′by the inequality′―′―ι+1≧0,NouCe that,(用,0)=l fOr盟>1・ Thus if ιた>1,thenテー′-9+1≧"た+1>0,and SO′>′.Therefore,P'>,(た+1, ,そ+1光pr十ノ十Pr_1≧∫々if ι>l orた<乃-2 by thc assumption, and henceυ″)∈lJKざ)】)by Lcmma3.6.Simila■y wc scc that υ,壁ちc(プ(S)И)in thc case ι=1.q.e,d.followingc)た≦2o■1く,たキ1(たと1).         ・A subseqwence S″: θ,sl, 中●,∫И_1 0f a■江｀RW‐seqttence S iS Said to be an″RV―∫ιTttι力Cゼげ″町脆虎・PROPOSITION 3.8.と,P bιαれ ο冴,Prim9,ヵ′軌 ,刀 νRV‐∫ι9,9乃露 げ 力れg加れ>1,Tんι刀腕ι s99,9ηCι′(ざ)ヵね′れソαri,ヵ′rtg"肋/.  rA pre‐MRW sequcnce S is said to be anれrRv_∫ιT,9刀cι if S S tisnes thcconditions a)or b)fOr eaChた,and c)if ι=1.a)θそ>l and ιた_1<α(た,,た)(-l if ι=1)・b)ιた=l and銑_1<P'(-l if θ=1)(,=冴た事第一嘉_1-ι+1).Inva?nt regular ideals in BP*PROOF.r力≦3,then the results of[3〕with Lcmma 2.5 1ead us to the proposition,Suppose that η≧3 andデ(S)コiS an invariant regular sequence.  It is cnough to show that7RαИ=7Lα7 mOd σ(ぎ)″)・ We put′=ちand′=,,_Ⅲ We nlst shOw itin the case 9PI>1.Now we noticc the following:(3.8.1)If加=O mOd砂,αl,中ちα″),then加(た)=0即od ψ ,α争),…,,V)),and(3.8.2) If加≡O mod ψ,つ,thCn trx(た)=o modゅた+1,つ.Here冴=7R-7L・COnsidcr thc invariant regular sequcnce二:P,υl,中●,υ″_3,υえ-2,υ身_land thc element χ=χ(れ,ど一′―ι+1),Wherc c=3-min 12,ιl and α='(れ,′―ケー ι+1).The condition c)guarantees that tt iS invattant even if ι=l since plα.ThCn wc haveが=O mod σ身)fOr d>l by PЮposition 3.2,and by the condition c)if ι=1.Thcrefore加′=O mod C)fOr′=spJ・¢l and the sequence二:P9,υP,…,υ解3,υ理2,υたl With α′=ψどby(3.8.1-2).SinCeゴ(ざ)ヵ atiSnes a),α′≧ι″_lp′十P′(十prif ι=1)>♂れ 1.Thus wehaVC σЭ⊂σ(ざ)ヵ)by Lemmas 3.6-7,and ttr=O mod σ(S),)・Take nOw s=ιヵ,and WCsec that J(ざ),+l iS invattant.Next suppose ιヵ=1.In this casc wc havc tr17伊=o mOd傷)for the sequence巧:P?,υ争),●中,υ誉21(た=テー ι+1)by(3.8.1-2),sinCe冴ち=O modち,The assumption b)andLemmas 3.6-7 show that σ二)⊂σ(0.).ThuS We provc thc casc cPI=1.    q.c.d.PROPOSITION 3.9.Lιチρ bι αη ο′′ Privι'力冴 ぎ'dι?′ι“じιて√ 肋r9gιrd.T力ι∫ЙttιT,9,cιざz JiF ttο′P′ι‐MR":ヵι々刀♂(d),ねηοr′れソαr力れど/9gヵ肋′.PROOF,  Ifざ,is not prc‐MRW, thcn we havc a positive intcgerた≦ζtt such that Sたisprc‐MRW and Sた+l iS nOt.  If た=l or 2, then the proposition is the coronary OfProposition 2.7 by ?rtuc of the results on fダNt(た=1,2)of[3].Now supposcた>2.Consider the scquence of integers S′:ι,∫4,…。wi h=ヽP力~′for′>0.Then Цざ')たiSinvariant regular by[7],and the idcal o(ぎ′)た_1,υえ_1)●=dた_1)COntains the ideal σ(Sン)sincc,,=ら一方_1-ι+1≧O fOr′<たand,た_1≡υ完_lmOd σ(S′)々_1).If駐dOeS nOt satisfythe condition(3.4),thcn PrOposition 3.2 implies冴♭た半O mod σ(ぎ′)た)and sO♂(S)たiS nOtinvattant. q.e.d.ReferencesJ F Adams, ざ勉うカ カοttοЮ碑 '4′γ″¢“Jiz?,力?脇ο′ο=ノ,Un?ersity of Chicago Press, Chicago,1974.P.S,Landweber, Inva?ant rcgular ideals in Brown‐Petcrson homology, Duke Math J.42(1975),499-505,H.R Miner,1)。c Ravcncl,and W,S,Wilson,  Periodic Phenomena in the Adams‐Novikov spectralsequencc, An■,of Math.,106(1977),469-516D,C Ravenel, Cο″T,力χじοうοィOis閉,ヵ′d勉う″力οttοttJ grO″p∫9テψ力¢r9d, Academic Press,1986.K,Shimomura, No?kov's Ext2 at the primc 2, Hiroshima Math.J ll(1981),499-513,K.Shirnomura and Z.Yosimura,  Br)_Hopf module spectrum and BP*―Adams spcctral s quence,Publ.RIMS,Kyoto Univ.21(1986),925-947.SmMOMURA,K.[71 E.Tsukada, Invariant sequencc in Brown‐Petcrson homology and some applications, HiroshimaMatI J 10(1980),385-389

Note on Invariant regular ideals in BP*

http://repository.lib.tottori-u.ac.jp/files/public/0/2474/20190712112502972161/jfens37(2)_103(1).pdf

Note on Invariant regular ideals in BP*

Abstract

Similar works

Full text

Available Versions

Tottori University Research Result Repository