On the chromatic Ext 0 (Mn 1 1 ) on Γ(m + 1) for an odd prime

Transcription

1 On the chromatic Ext 0 Mn 1 1 on Γm + 1 for an odd prime Rié Kitahama and Katsumi Shimomura Received Xxx 00, 0000 Abstract. Let Mn 1 1 denote the coernel of the localization map BP /I v 1 n 1 BP /I, here I denotes the ideal of BP generated by v i s for 0 i n 2. The chromatic Ext 0 Mn 1 1 on Γm + 1, hich e denote Ex0 m, n, is isomorphic to the 0-th line of the E 2 -term of the Adams-Noviov spectral sequence for computing the homotopy groups of a spectrum, hose BP -homology is Mn 1 1 BP BP [t 1, t 2,..., t m ] for the generators t i of BP BP. In [9], the homotopy groups of such a spectrum are determined for m + 1 nn 1 by computing Ex m, n. The 0-th line Ex 0 m, 3 is determined by Ichigi, Naai and Ravenel [1]. Here, e determine the 0-th line Ex 0 m, n under the condition: n 1 2 m + 1 < nn 1 and n Introduction Let BP be the Bron-Peterson spectrum at an odd prime p, and the pair BP, BP BP = Z p [v 1, v 2,... ], BP [t 1, t 2,... ] the associated Hopf algebroid. Ravenel [5] constructed the spectra T m for m 0 as ell as a map T m BP that induces the inclusion BP T m = BP [t 1,..., t m ] BP BP of BP BP -comodules. The Smith-Toda spectrum V is characterized by the BP -homology: BP V = BP /p, v 1,..., v. We consider a spectrum V m such that BP V m = BP /p, v 1,..., v [t 1,..., t m ]. Furthermore, e consider the Bousfield localization functor L n : S S ith respect to vn 1 BP on the stable homotopy category S of p-local spectra see [4]. If L n V exists, then L n V m = T m L n V. We notice that L n V n 1 exists if n 2 + n < 2p see [10]. We are interested in the homotopy groups of L n T m. The homotopy groups are determined from those of L n V m by virtue of the Bocstein spectral sequences. We study the homotopy groups by the Adams-Noviov spectral sequence converging to the homotopy groups π X of a spectrum X ith E 2 -term E2X = Ext BP BP BP, BP X. Our input is a result of Ravenel s: 1.1cf. [5, Cor ] If n < m + 2 and n < 2p 1m + 1/p, then here E 2L n V m n 1 = E m n Eh,j : m + 1 m + n, j Z/n, E m n = v 1 n Z/p[v n, v n+1,..., v n+m ] Mathematics Subject Classification. Primary 55T15; Secondary 55Q51. Key ords and phrases. Adams-Noviov spectral sequence, chromatic spectral sequence, Ravenel spectrum.

2 2 Rié Kitahama and Katsumi Shimomura In order to study the Adams-Noviov E 2 -term E2L n V m n 2, e consider a η spectrum V m n 2 defined as a cofiber of the localization map V m n 2 L n 1 V m n 2. Put Ex m, n = Ext BP BP BP, BP V m n 2 = Ext BP BP BP, Mn 1 1 BP BP [t 1, t 2,..., t m ], for Mn 1 1 = vn 1 BP /p, v 1,..., v n 2, v p n 1. Then the Adams-Noviov E 2 -term E2L n V m n 2 is obtained from Ex m, n and E2L n 1 V m n 2. Note that for n = 1, Ex 0, 1 is the Adams-Noviov E 2 -term for π L 1 S 0, hose structure is found in [4, Th.8.10] see also [3]. Ex 0, 2 is the E 2 -term for π L 2 V 0, hich is determined by the second author [6],[7],[8], and Ex m, 2 is determined by Kamiya and the second author in [2]. For m +1 nn 1, the second author also determined Ex m, n in [9]. In [1], Ichigi, Naai and Ravenel determined Ex 0 m, 3 for m > 1. In this paper, e determine the Ext group Ex 0 m, n for m, n ith n 1 2 m+1 < nn 1 and n 4. One of our tool is the change of rings theorem cf. [5]: Ex m, n = Ext Γm+1 BP, vn 1 BP /p, v 1,..., v n 2, v p n 1 for the associated Hopf algebroid BP, Γm+1 = BP, BP BP/t 1,..., t m = BP, BP [t m+1, t m+2,... ]. Let D m n denote the algebra 1.2 D m n = E m 1 n [v n 1 ] = v 1 n Z/p[v n 1, v n,..., v m+n 1 ]. Then, our main theorem is obtained from the folloing proposition hich is proved in the next section. Proposition 1.3. Suppose that 1 For each integer 0, there is an element x vn 1 BP such that x v p m+n mod I1 and dx v a n 1 va n v a m+ng mod Ia + 1 for nonnegative integers a, a and a and g {t pj m+i : 0 < i n, j Z/n}. Here dx = η R x x vn 1 Γm + 1, and I denotes the ideal of vn 1 BP generated by p, v 1,..., v n 2 and vn 1. 2 The elements v s 1p +a m+n g for nonnegative integers s and represent linearly independent generators over E m 1 n in E2L 1 n V m n 1. Then, Ex 0 m, n is the direct sum of vn 1 1 D mn /D m n and D m n -modules generated by x s /va n 1 isomorphic to D mn /v a n 1 for each integers, s ith 0 and p s > 0.

3 On the chromatic Ext 0 M 1 n 1 on Γm + 1 for an odd prime 3 Consider the integers p n 1+j p 1 p 1 p a n+j = p j n+2 n 1 p 2n A + a n2 2n 1 p j p n pn n 1 p 2n A + a n 1 2 n < n n is even 0 n is odd 1 for A = p n a n 2 n + p m+1 p n 12. Then, there exist elements x satisfying the assumptions of Proposition 1.3 for the integers a, hich e sho in section three for n 1 2 m + 1 < nn 1 and n 3. Thus, e obtain our main theorem: Theorem 1.4. Let n 1 2 m + 1 nn 1 and n 3. Then, E2m, 0 n = vn 1 1 mn /D m n 0 p - s > 0 Dm n /v a n 1 x s /v a n 1. Note that if n = 3, then the result is the same as that in [1]. Remar. The computation on section three shos that the structure of Ex 0 m, n depends on such that n 1 m + 1 < + 1n 1. In [9], it is shon that Ex m, n has the same structure for n. In this paper, e consider the case = n Preliminaries Throughout the paper, e fix the positive integers m and n 4 satisfying the condition n 1 2 m + 1 < nn The structure map η R. In Γm + 1 = BP [t m+1, t m+2,... ], e compute the action of η R on v i s. We have the formulas of Hazeinel s and Quillen s: v n = pl n n 1 i=1 l iv pi n i BP Q = Q[l 1, l 2,... ], η R l n = l n + n m i=1 l n m it pn m i m+i Γm + 1 Q. In particular, mod l 1, l 2,..., l n 2, v i pl i i < 2n 2 and v = pl n+1 i=n 1 l iv pi i > 2n 2. Lemma 2.1. as follos : The right unit η R : : BP Γm + 1 acts on generators v i η R v i = v i Γm + 1 for i m, η R v m+ = v m+ + pt m+ 0 < < n

4 4 Rié Kitahama and Katsumi Shimomura η R v m+n+ v m+n+ + j=0 v n 1+j t pn 1+j m+1+ j vpm+1+ j n 1+j t m+1+ j ω mod I n 1 0 n here I denotes the ideal generated by p, v 1,..., v 1, and 0 < n 1, 2.2 ω = v n 1 m+n,n 2 = n 1, v n 1 m+n+1,n 2 + v n m+n,n 1 = n for the elements m+n,i and m+n+1,i defined by dvm+n pi+1 v pi+1 tpn+i n 1 m+1 tpi+1 vpm+i+2 m+1 n 1 + p m+n,i mod p 2, v 1,..., v n 2, and dv pi+1 m+n+1 vpi+1 n 1 tpn+i Proof. m+2 + vpi+1 n t pn+i+1 m+1 t pi+1 m+1 vpm+i+2 n +p m+n+1,i mod p 2, v 1,..., v n 2. This follos from routine computation: t pi+1 vpm+i+3 m+2 n 1 η R v m+ = pl m+ + t m+ m+ n+1 i=n 1 l i v pi m+ i = v m+ + pt m+ 0 < < n η R v m+n+ pl m+n+ + j=0 l n 1+jt pn 1+j m+1+ j + t m+n+ m i=n 1 l iv pi m+n+ i i=1 l m+i + t m+i v pm+i n+ i v m+n+ + j=0 v n 1+jt pn 1+j m+1+ j i=1 t m+iv pm+i n+ i mod I n 1 0 < n 1 η R v m+2n 1 pl m+2n 1 + n 1 j=0 l n 1+jt pn 1+j m+n j + t m+2n 1 l n 1 v pn 1 m+n + v pn 1 n 1 tp2n 2 m+1 vpm+n n 1 tpn 1 m+1 + p m+n,n 2 m i=n l iv pi m+2n 1 i n 1 i=1 l m+i + t m+i v pm+i 2n 1 i l m+n + l n 1 t pn 1 m+1 + t m+nv pm+n n 1 v m+2n 1 + n 1 j=0 v n 1+jt pn 1+j m+n j n i=1 t m+iv pm+i 2n 1 i v n 1 m+n,n 2 mod I n 1 η R v m+2n pm m+2n + n j=0 l n 1+jt pn 1+j m+1+n j + t m+2n 1 l n 1 v pn 1 m+n+1 + vpn 1 n 1 m+2 + vpn 1 n t p2n 1 v pm+n n t pn 1 m+1 + p m+n+1,n 2 l n v pn m+n + v pn n 1 tp2n 1 m+1 vpm+n+1 n 1 m+1 vpm+n+1 n 1 t pn 1 m+2 t pn m+1 + p m+n,n 1 m i=n l iv pi m+2n i n 1 i=1 l m+i + t m+i v pm+i 2n i l m+n + l n 1 t pn 1 m+1 + t m+nvn pm+n l m+n+1 + l n 1 t pn 1 m+2 + l nt pn m+1 + t m+n+1v pm+n+1 n 1 v m+2n + n j=0 v n 1+jt pn 1+j m+1+n j n+1 i=1 t m+iv pm+i 2n i v n 1 m+n+1,n 2 v n m+n,n 1 mod I n 1

5 u u On the chromatic Ext 0 M 1 n 1 on Γm + 1 for an odd prime Proof of Proposition 1.3. The proof is based on [3, Remar 3.11], hich states, in our case, that if the commutative diagram of to exact sequences 1/u 0 E 0 D 0 D 0 E 1 1/u 0 E 0 Ex 0 m, n Ex 0 m, n E 1 commutes, then f is an isomorphism. Hereafter, e set u = v n 1. Let D 0 be the module of the proposition, that is, the D m n -modules generated by 1/u j for j > 0 and x s /ua for each integers, s ith 0 and p s > 0. Then, D 0 Mn 1 1 = vn 1 BP /p, v 1,..., v n 2, vn 1. Note that Ex 0 m, n is the ernel of the map d: Mn 1 1 Mn 1 1 BP Γm + 1 given by dx/u j = η R x x/u j. Thus, the first supposition shos that the every element x s /ua belongs to Ex 0 m, n. We also see that 1/u j Ex 0 m, n by Lemma 2.1. No e define the map f to be the inclusion. Then, the map 1/u: Ex 0 D 0 is ell defined, since the elements vm+n/u s belong to D 0. Since D 0 is a D m n -module, D 0 admits the self map u. The map δ : D 0 E 0 is defined by the composite δf. These sho the existence of the commutative diagram. We ill sho that the upper sequence is exact. Since the diagram commutes and the loer sequence is exact, the sequence 0 E 0 1/u D 0 u D 0 is exact and the composite δu is trivial. Suppose that δx = 0 for x = 0,p s>0 a,sx s /ua + j a j/u j for a,s, a j D m n. Then, by the first supposition, 0 = δx = 0,p s>0 a,sδx s /ua = 0,p s>0 sϕa,sv a f u u f δ δ n v s 1p +a m+n g. u ϕ for the map ϕ in the exact sequence D m n Dm n Em 1 n. Then the second assumption shos that sϕa,s = 0 E m 1 n for every, s, and hence a,s = ub,s for an element b,s D m n. It follos that x = uy for y = 0,p s>0 b,sx s /ua + j a j/u j+1, as desired. 3. The elements x n In the sequel, e put v n = 1, and use the notation u = v n 1, = v m+n+, s = t m+, I = I n 1 = p, v 1,..., v n 2 and I = p, v 1,..., v n 2, v n 1 for the sae of simplicity. We also introduce integers: P = p n 1, Q = p m+1 e P = P 1 P 1, e = p 1 p 1, A = P e + 1 and

6 6 Rié Kitahama and Katsumi Shimomura No e introduce elements X and X for n, hich correspond to the elements x n and x n = 0 X = X 1 P u pp 1A 1 X 1 u pp 1A 1+Q P 1 X 1 0 < n X = X p 1 W 1 W = u pa 1 P i u pi+1 P i A 1 i v P 1 n+i i=1 u P i 1 A 1 i v p i 1 P 1 Q n+i X 1 i P i 1. The element ω of 2.2 is a multiple of u, and e put: X pi+1 P i 1 i 3.2 σ u P 1 ω P 1 and σ = σ n 1 = P n 2 m+n,n 2 u pp n 2 X p 1 n 2 spp n 2 P 1 mod I2pP n 2. Note that σ = 0 if < n 1, and = σ if = n 1. X Lemma 3.3. for 0 as follos: The differential d: v 1 n BP vn 1 Γm + 1 acts on X and dx 1 u A s P σp u Q P s P σ, mod I2Q + A e P + 1 for n 1, dx 1 u pa s P up s P +2 σ mod IQ + pa P 1 for n 2, dx n 1 1 n+1 u pa n 1 s P n 1 n mod IpA n 1 + P n 1, and dx n 1 n u pp A n 1+Q P n 1 σ mod IQ + pp A n 1. Here, e set P 1 = 0. Remar. In the case m + 1 nn 1, dx n 1 n+1 u pp A n 1 σ P n 1 mod IpP A n 1 + Q P n 1 in our notation. Proof. Since dx 0 = d 0 us pn 1 1 u pm+1 s 1 = us P 1 u Q s 1 mod I by Lemma 2.1, e have the first step of the induction. Suppose that e have the congruences on dx i for i < n 1. Since n 3, e see that < m and dv n+i 0 mod I n 1 for i. Then e compute

7 On the chromatic Ext 0 M 1 n 1 on Γm + 1 for an odd prime 7 by Lemma 2.1 as follos: d X p 1 u pa pp s u pq pp s pp mod I2pQ + pa pe P + 1 d W u pa d P + u pa + u pa 1 +i+1 d u pi+1 P i A i v P u P i 1 A i v p i P Q n+i X P i 1 i i=1 pp u P s P +2 + s s P i=1 i=1 v P n+is pi+1 P i v P n+is pi+1 P i u P s P +2 n+ix pi+1 P i i v p i P Q n+i s P i ω P v p i P Q n+i s P i + spp s P up σ mod IpA + Q P 1, hich yields the congruence on dx. dx P 1 u pp A s P up s P σp mod IP Q + pp A P d u pp 1A X 1 u pp A s P uq P s P mod I2Q + pp A e P + 1 d u pp 1A +Q P X 1 u pp A +Q P s P up s P +2 σ mod I2Q + pp A P P 1, and e have d X 1 u pp A u P s P σp u Q s P +2 σ mod I2Q + pp A e P + 1. Since A = pp A + P and pp A e P + 1 = A P e P + 1 = A e P + 2, e obtain dx. This completes the induction to obtain the congruences on dx n 2 and dx n 1. A similar computation as above shos the congruence on dx n 1, here e tae the ideal in the congruence so that ω P n 1 n is annihilated. For dx n, using the congruences on dx n 1 and dx n 1, e compute

8 8 Rié Kitahama and Katsumi Shimomura d X n 1 P 1 n+1 u pp A n 1 s P n n mod IpP A n 1 + P n d u pp 1An 1 X n 1 s 1 n u pp A n 1 P n n σ P u Q P n 1 s P n 1 n σ mod I2Q + pp A n 1 e P n d u pp 1A n 1+Q P n 1 X n 1 1 n u pp A n 1+Q P n 1 s P n 1 n mod IpP A n 1 + Q to obtain dx n. pp n 2 Consider an element Y = X 0 X n 2, and e see that it is congruent to zero modulo IpP 1pP n 3, and 3.4 dx p 1 n 2 Y Xp 1 n 2 u pp n 2 s pp n 2 P 1 dx n 2 σ X p 1 n 2 dx n 2 mod I2pP n 2 pp n 3. We introduce integers e 2, A, A, c and C defined by e 2 2 = pp 2 1 pp 2 1, A = pp A n 1 + Q P n 1, P e + 1 < n A = e na + A n 2 n is even 0 { pp e na + A n 1 n is odd 1 pp e 2 1A + pp 1A n 2 is odd 1, c = pp 2 e 2 2A + pp 1A n 1 is even 2, We replace X n by X n 1 n u A X p 1 n 2 Y, and define inductively the elements X n+ for > 0 by 3.5 X n+ = X pp n+ 1 1n u pp A X p 1pP n+ 3 Y. for Here e notice that pp = p n. Y = X pp n+ 3 X n+ 2 Lemma 3.6. The element Y is a multiple of u c for > 0. Proof. By the definition 3.1, e see that Y 1 and Y 2 are divisible by u pp 1An 2 and u pp 1An 1, respectively. For > 2, e see that c = pp 2 A+ c 2 by 3.5, and verify the lemma by induction.

9 On the chromatic Ext 0 M 1 n 1 on Γm + 1 for an odd prime 9 We further introduce integers A and elements G defined by { A 0 < n = pp 2 p 1 + A n Then, e have G = Lemma 3.7. For i 0, {s p < n G 2 n dx n+i 1 n u pp ia X p 1 n+i 2 dx n+i 2 mod IA n+i + P n εi 1 n u A n+i A n+i 0 G n+i mod IA n+i + 1. Here εi = max{0, 1 i}. Proof. We sho the first congruence by induction. The congruence for i = 0 follos from Lemma 3.3 and 3.4. Inductively suppose that the congruence holds for i. Then, e compute dx pp n+i 1n u pp i+1a X p 1pP n+i 2 d 1 n+1 u pp i+1a X p 1pP n+i 2 Y i+1 1 n+1 u pp i+1a X p 1pP n+i 2 2 dx pp n+i 2 mod IpP A n+i + pp n εi dx pp n+i 2 dx n+i 1 mod IpP i+1 A + c i+1 + pp A n+i 2. Since pp i+1 A + pp A n+i 2 = pp A n+i, c i+1 > pp n εi and pp A n+i + pp n εi > A n+i+1 + P n 1, e obtain the congruence for i + 1. Thus, the induction completes. No e define integers a and a, and elements x, g and g as follos: a in+j = p j A i a in+j = p j A i x in+j = X pj i g in+j = G pj i for i 0 and 0 j < n. We notice that the integer a is the same as the one in the introduction. These elements and integers satisfy the assumption of Propo- Lemma 3.8. sition 1.3. Proof. The first part of the assumption follos from the definition of elements and Lemmas 3.3 and 3.7. We prove the second part by shoing the folloing assertion on l The elements v s 1p +a l m+n g for non-negative integers s, ith p s and < l are linearly independent over E m 1 n.

10 10 Rié Kitahama and Katsumi Shimomura For l < n 2, it is trivial, since g g if. Suppose that it holds for l, and v s 1pl +a l m+n g l = l 1 =0 λ v s 1p +a m+n g for λ E m 1 n, here s is a nonnegative integer prime to p such that s 1p l + a l = s 1p + a. Then, since g i s are generators, l mod n, and s 1p l + a l = s 1p + a. Then, a l s 1p + a mod pl. If = in + j, then A l s 1p in + A n mod p l j for l such that l = l n + j. The definition of integers A i implies that p divides s, and there is no integers s satisfying the congruence. Hence, 3.7 l+1 holds, and the induction completes. References [1] I. Ichigi, H. Naai and D. C. Ravenel, The chromatic Ext groups Ext 0 Γm+1 BP, M 1 2, Trans. Amer. Math. Soc., 354, [2] Y. Kamiya and K. Shimomura, The homotopy groups π L 2 V 0 T, Hiroshima Math. J , [3] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-Noviov spectral sequence, Ann. of Math , [4] D. C. Ravenel, Localization ith respect to certain periodic homology theories. Amer. J. Math , [5] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, [6] K. Shimomura, On the Adams-Noviov spectral sequence and products of β-elements, Hiroshima Math. J , [7] K. Shimomura, The Adams-Noviov E 2 -term for computing π L 2 V 0 at the prime 2, Topology and its Applications , [8] K. Shimomura, The homotopy groups of the L 2 -localized mod 3 Moore spectrum, J. Math. Soc. of Japan , [9] K. Shimomura, The homotopy groups π L nt m V n 2, Contemp. Math , [10] K. Shimomura and Z. Yosimura, BP -Hopf module spectrum and BP -Adams spectral sequence, Publ. Res. Inst. Math. Sci , Rié Kitahama Department of Mathematics, Faculty of Science, Kochi University, Kochi, , Japan Katsumi Shimomura Department of Mathematics, Faculty of Science, Kochi University, Kochi, , Japan atsumi@math.ochi-u.ac.jp