On the homotopy groups of an invertible spectrum in the E(2)-local category at the prime

3

Ippei Ichigi and Katsumi Shimomura

Abstract. Let L_{2} denote the Bousfield localization functor with respect to the second Johnson-Wilson spectrum E(2). A spectrum L_{2}X is called invertible if there is a spectrum Y such that L_{2}X ∧ Y = L_{2}S^{0}. Then M. Hovey and H. Sadofsky showed that every invertible spectrum is a suspension of the sphere spectrum L_{2}S^{0} if the prime p is greater than three. At the prime three, Y. Kamiya and the second author constructed an invertible spectrum P other than a suspension of L_{2}S^{0}, and showed a possibility of existence of another invertible spectrum Q such that every invertible spectrum has a form Σ^{k}P^{∧p} ∧ Q^{∧q} for integers k ∈ Z and p, q ∈ Z/3, where X ∧ X ∧ X = L_{2}S^{0} for X = P, Q. In this paper, we consider the homotopy groups of the invertible spectrum Q under the assumption that Q exists, and determine the homotopy groups π_{∗}(Q^{∧q} ∧ V (1)) and π_{∗}(P^{∧p} ∧ Q^{∧q} ∧ V (1)) for the Smith-Toda spectrum V (1). The results make the authors conjecture that Q does not exist.

1. Introduction

Let S_{p} for a prime number p and L_{n} for an integer n ≥ 0 denote the category of p-local spectra and its full subcategory of E(n)-local spectra, respectively, and L_{n} : S_{p} → L_{n} be the Bousfield localization functor with respect to E(n), where E(n) denotes the Johnson-Wilson spectrum with the homotopy groups π_{∗}(E(n)) = E(n)_{∗} = Z_{(p)}[v_{1},…, v_{n−1}, v_{n}^{±1}]. We call a spectrum X ∈ L_{n} invertible if there exists a spectrum Y ∈ L_{n} such that X ∧Y = L_{n}S^{0}. M. Hopkins introduced the Picard group Pic_{(n)} = Pic(L_{n}) consisting of isomorphism classes of invertible spectra in L_{n} with multiplication defined by the smash product. Then in [5] (cf. [4]), it is shown by M. Hovey and H. Sadofsky that Pic_{(n)} is a well defined abelian group, that Pic_{(n)} ≅ Z if n^{2} + n < 2p − 2 and that Pic_{(1)} ≅ Z ⊕ Z/2 at the prime 2. Note that suspensions Σ^{k}L_{n}S^{0} of the sphere spectrum for k ∈ Z form a subgroup of Pic_{(n)} isomorphic to Z.

Let E_{r}^{s,t}(X) denote the E_{r}-term of the Adams-Novikov spectral sequence converging to π_{∗}(L_{n}X). Then Y. Kamiya and the second author [8] constructed a monomorphism

(1.1)  φ : Pic_{(n)}/Z ⊂ ⨁_{r≥2} E_{r}^{r,r−1}(S^{0}) = T.

In particular, it is well known that T = 0 if n^{2} +n < 2p−2 and T = E_{3}^{3,2}(S^{0}) = Z/2 if n = 1 and p = 2, which imply the above results of M. Hovey and H. Sadofsky’s. At the prime 3, T = E_{5}^{5,4}(S^{0}) = Z/3 ⊕ Z/3 by [13] (cf. [14]). In [6], we determined the structure of the homotopy groups π_{∗}(L_{1}QM ∧ V (0)), where V (0) denotes the mod 2 Moore spectrum, and the question mark complex QM = S^{0} ∪_{2} e^{1} ∪_{η} e^{3} represents the generator of Z/2 in Pic_{(1)} at the pirme two. In this paper, we study homotopy groups of an invertible spectrum in L_{2} at the prime three. Let α and β denote the generators of E_{5}^{5,4}(S^{0}) = Z/3 ⊕ Z/3. In [8], it is shown that α is pulled back to an invertible spectrum P under the monomorphism φ of (1.1). We determined the homotopy groups π_{∗}(P^{∧p} ∧V (1)) for an integer p ∈ Z/3 in [6]. Here V (1) denotes the first Smith-Toda spectrum, and X^{∧n} denotes the n-fold smash product of X. In [7], we also determined π_{∗}(P^{∧p}) for p ∈ Z/3. We further, in this paper, assume the existence of an invertible spectrum Q such that φ(Q) = β. Then Pic_{(2)} = Z ⊕ Z/3 ⊕ Z/3. Note that Q ∧ Q ∧ Q = L_{2}S^{0} by the definition corresponding 3β = 0. Since E_{2}^{∗,∗}(P^{∧p} ∧ Q^{∧q} ∧ V (1)) = E_{2}^{∗,∗}(V (1)) for integers p, q ∈ Z/3, we can compute the homotopy groups π_{∗}(P^{∧p} ∧Q^{∧q} ∧V (1)) by studying the differentials d_{5}(g_{p,q}) and d_{9}(g_{p,q}) on the generator g_{p,q} ∈ E_{2}^{0,0}(P^{∧p} ∧ Q^{∧q} ∧ V (1)) of the E_{2}-term.

We recall [11] (cf. [3], [10]) that the E_{2}-term E_{2}^{∗,∗}(V (1)) is isomorphic, as K(2)_{∗}-module, to the tensor product of K(2)_{∗}[b_{10}] and the direct sum of

\begin{matrix}F̃ = K(2)_{∗}\{1, h_{10}, h_{11}, b_{11}\}, & & F̃ ζ_{2} = K(2)_{∗}\{ζ_{2}, h_{10}ζ_{2}, h_{11}ζ_{2}, b_{11}ζ_{2}\}, \\ F̃^{∗} = K(2)_{∗}\{ξ, ψ_{0}, ψ_{1}, b_{11}ξ\} & and & F̃^{∗}ζ_{2} = K(2)_{∗}\{ξζ_{2}, ψ_{0}ζ_{2}, ψ_{1}ζ_{2}, b_{11}ξζ_{2}\}. \end{matrix}

Here K(2)_{∗} = Z/3[v_{2}^{±1}] and the bidegrees of generators are given as

\begin{matrix}||v_{2}|| = (0, 16), & ||h_{10}|| = (1, 4), & ||h_{11}|| = (1, 12), & & ||b_{10}|| = (2, 12), \\ ||b_{11}|| = (2, 36), & ||ξ|| = (2, 8), & ||ψ_{0}|| = (3, 16) & and & ||ψ_{1}|| = (3, 24). \end{matrix}

In order to describe the homotopy groups of the invertible spectrum Q^{∧q} for q ∈ Z/3, we introduce some modules: (1.2)

H_{q} = ∑_{j≢0,1,5 (9)} G_{2}{v_{2}^{j−2}h_{11}} ⊕ G_{3}{v_{2}^{j+3}h_{10}},
H_{q}ζ_{2} = ∑_{j≢0,1,5 (9)} G_{2}{v_{2}^{j−2}h_{11}ζ_{2}} ⊕ G_{3}{v_{2}^{j+3}h_{10}ζ_{2}},
H_{q}^{∗} = ∑_{j≢0,1,5 (9)} G_{2}{v_{2}^{j−1}b_{11}ξ} ⊕ G_{3}{v_{2}^{j+5}ξ} and
H_{q}^{∗}ζ_{2} = \{ \begin{matrix}∑_{j≢0,1,5 (9)} G_{2}\{v_{2}^{j−1}b_{11}ξζ_{2}\} ⊕ G_{3}\{v_{2}^{j+5}ξζ_{2}\} for q = 0 \\ ∑_{j≡3,4,8 (9)} Z/3\{v_{2}^{j−1}b_{11}ξζ_{2}\} ⊕ Z/3\{v_{2}^{j+5}ξζ_{2}\} \\ ⊕ ∑_{j≡2,6,7 (9)} G_{2}\{v_{2}^{j−1}b_{11}ξζ_{2}\} ⊕ G_{3}\{v_{2}^{j+5}ξζ_{2}\} for q = 1, 2 \end{matrix} and

(1.3)

\overline{I_{q}} = \{ \begin{matrix}∑_{j≡0,1,5 (9)} G_{5}\{v_{2}^{j}\} ⊕ G_{4}\{v_{2}^{j+3}b_{11}\} for q = 0 \\ ∑_{j≡0,1,5 (9)} G_{3}\{v_{2}^{j}b_{10}^{2}, v_{2}^{j+3}b_{11}b_{10}\} for q = 1, 2, \end{matrix}
\overline{I_{q}ζ_{2}} = ∑_{j≡0,1,5 (9)} G_{5}\{v_{2}^{j}ζ_{2}\} ⊕ G_{4}\{v_{2}^{j+3}b_{11}ζ_{2}\},
\overline{I_{q}^{∗}} = ∑_{j≡0,1,5 (9)} G_{4}\{v_{2}^{j+2}ψ_{0}\} ⊕ G_{5}\{v_{2}^{j+6}ψ_{1}\} and
\overline{I_{q}^{∗}ζ_{2}} = ∑_{j≡0,1,5 (9)} G_{4}\{v_{2}^{j+2}ψ_{0}ζ_{2}\} ⊕ G_{5}\{v_{2}^{j+6}ψ_{1}ζ_{2}\},

in which G_{k} = Z/3[b_{10}]/(b_{10}^{k}). Put

\begin{matrix}\overline{F_{q}} = H_{q} ⊕ \overline{I_{q}}, & & \overline{F_{q}ζ_{2}} = H_{q}ζ_{2} ⊕ \overline{I_{q}ζ_{2}}, \\ \overline{F_{q}^{∗}} = H_{q}^{∗} ⊕ \overline{I_{q}^{∗}} & and & \overline{F_{q}^{∗}ζ_{2}} = H_{q}^{∗}ζ_{2} ⊕ \overline{I_{q}^{∗}ζ_{2}}. \end{matrix}

Then π_{∗}(Q^{∧q} ∧ V (1)) for q = 0, which is π_{∗}(L_{2}V (1)), is shown in [11] (cf. [3], [1]) to be the direct sum of the four subgroups \overline{F_{0}}, \overline{F_{0}ζ_{2}}, \overline{F_{0}^{∗}}, and \overline{F_{0}^{∗}ζ_{2}}.

Theorem I. At the prime three, the homotopy groups π_{∗}(Q^{∧q} ∧ V (1)) for q ∈ Z/3 are isomorphic to the direct sum of the subgroups \overline{F_{q}}, \overline{F_{q}ζ_{2}}, \overline{F_{q}^{∗}} and \overline{F_{q}^{∗}ζ_{2}}.

Note that this theorem shows an isomorphism π_{∗}(Q^{∧1} ∧ V (1)) ≅ π_{∗}(Q^{∧2} ∧ V (1)) while π_{∗}(Q^{∧0} ∧ V (1)) ≅ π̸_{∗}(Q^{∧1} ∧ V (1)). This is the reason why the authors are skeptical about the existence of Q, though the authors of [2] seem to believe the existence.

Recall [6] the equivalence v_{2}^{3} shown by determining the homotopy groups of P^{∧p} ∧ V (1).

Theorem II. ([6]) There exists a homotopy equivalence v_{2}^{3} : Σ^{48}L_{2}V (1) ≃ P^{∧1} ∧ V (1).

Theorems I and II give rise to the homotopy groups of an invertible spectrum in the E(2)-local category L_{2} smashing with the Smith-Toda spectrum V (1) at the prime 3. Indeed, each invertible spectrum has the form Σ^{k}P^{∧p} ∧ Q^{∧q} for integers k ∈ Z and p, q ∈ Z/3.

Corollary III. The homotopy groups π_{∗}(P^{∧p} ∧ Q^{∧q} ∧ V (1)) for p, q ∈ Z/3 are isomorphic to v_{2}^{3p}π_{∗}(Q^{∧q} ∧ V (1)).

In sections 2 and 3, we compute the differentials d_{5} and d_{9}, which give us the E_{9}- and E_{13}-terms of the Admas-Novikov spectral sequence converging to the homotopy groups of Q^{∧q} ∧ V (1), respectively, and show the Theorem I.

2. The Adams-Novikov E_{9}-term for the invertible spectrum Q^{∧q}

Let E_{r}^{∗,∗}(X) denote the E_{r}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{2}X). Then E_{2}^{∗,∗}(X) is an E_{2}^{∗,∗}(S^{0})-module with the action induced from the pairing X ∧ S^{0} →^{≈} X. Let Q be an invertible spectrum such that φ(Q) = β = v_{2}^{−1}ξb_{10}ζ_{2} ∈ E_{5}^{5,4}(S^{0}) for φ in (1.1). Then E_{2}^{∗,∗}(Q^{∧q}) is isomorphic to E_{2}^{∗,∗}(S^{0}) as an E_{2}^{∗,∗}(S^{0})-module on the generator g_{q} = g_{0,q} ∈ E_{2}^{0,0}(Q^{∧q}) and d_{5}(g_{q}) = qβg_{q} ∈ E_{5}^{5,4}(Q^{∧q}) by the definition of φ [8]. We determine the E_{9}-term converging to the homotopy groups π_{∗}(Q^{∧q} ∧ V (1)) for q = 1, 2 by computing d_{5}(g_{q}) with help of the relations given in [11, Prop. 5.9]:

(2.1)  \begin{matrix}h_{10}h_{11} = 0, h_{10}ξ = 0, h_{11}ξ = 0, \\ v_{2}^{2}h_{10}b_{10} = h_{11}b_{11}, v_{2}h_{11}b_{10} = −h_{10}b_{11}, \\ b_{11}ξ = v_{2}h_{10}ψ_{1} = v_{2}h_{11}ψ_{0}, b_{10}ξ = −h_{10}ψ_{0} = v_{2}^{−1}h_{11}ψ_{1}, \\ v_{2}^{3}b_{10}^{2} = −b_{11}^{2}, b_{10}ψ_{1} = −v_{2}^{−1}b_{11}ψ_{0} and b_{10}ψ_{0} = v_{2}^{−2}b_{11}ψ_{1}. \end{matrix}

For conveniences, we write down the chart of the E_{2}-term.

\overline{}^{1} ^{2} ^{3} ^{4} ^{5} ^{6} ^{7} ^{8} ^{9} ^{10} ^{11} ^{12} ^{13} ^{14} ^{15}^{1}^{2}^{3}^{4}^{5}^{6}^{7}^{8}^{9}^{10}^{11}^{12}^{13}^{14}^{15}^{16}^{17}^{18}^{19}^{20}  b_{10} b_{10}^{2}  b_{10}^{3} b_{10}^{4} b_{10}^{5}  b_{10}^{6} b_{10}^{7} b_{10}^{8}  b_{10}^{9} b_{10}^{10} h_{10}  ^{b_{10}h_{10}}^{b_{10}^{2}h_{10}}^{b_{10}^{3}h_{10}}  ^{b_{10}^{4}h_{10}}^{b_{10}^{5}h_{10}}  ^{b_{10}^{6}h_{10}}^{b_{10}^{7}h_{10}}^{b_{10}^{8}h_{10}}  ^{b_{10}^{9}h_{10}} h_{11}^{b_{10}h_{11}}  ^{b_{10}^{2}h_{11}}^{b_{10}^{3}h_{11}}^{b_{10}^{4}h_{11}}  ^{b_{10}^{5}h_{11}}^{b_{10}^{6}h_{11}}^{b_{10}^{7}h_{11}}  ^{b_{10}^{8}h_{11}}^{b_{10}^{9}h_{11}} b_{11}  ^{b_{10}b_{11}}^{b_{10}^{2}b_{11}}^{b_{10}^{3}b_{11}}  ^{b_{10}^{4}b_{11}}^{b_{10}^{5}b_{11}}  ^{b_{10}^{6}b_{11}}^{b_{10}^{7}b_{11}}^{b_{10}^{8}b_{11}}  ^{b_{10}^{9}b_{11}} ξ^{b_{10}ξ}  ^{b_{10}^{2}ξ}^{b_{10}^{3}ξ}  ^{b_{10}^{4}ξ}^{b_{10}^{5}ξ}^{b_{10}^{6}ξ}  ^{b_{10}^{7}ξ}^{b_{10}^{8}ξ}^{b_{10}^{9}ξ}  ψ_{1}  ^{ψ_{1}b_{10}}^{ψ_{1}b_{10}^{2}}^{ψ_{1}b_{10}^{3}}  ^{ψ_{1}b_{10}^{4}}^{ψ_{1}b_{10}^{5}}^{ψ_{1}b_{10}^{6}}  ^{ψ_{1}b_{10}^{7}}^{ψ_{1}b_{10}^{8}}  ψ_{0}^{ψ_{0}b_{10}}^{ψ_{0}b_{10}^{2}}  ^{ψ_{0}b_{10}^{3}}^{ψ_{0}b_{10}^{4}}  ^{ψ_{0}b_{10}^{5}}^{ψ_{0}b_{10}^{6}}^{ψ_{0}b_{10}^{7}}  ^{ψ_{0}b_{10}^{8}}^{b_{11}ξ}^{b_{10}b_{11}ξ}  ^{b_{10}^{2}b_{11}ξ}^{b_{10}^{3}b_{11}ξ}^{b_{10}^{4}b_{11}ξ}  ^{b_{10}^{5}b_{11}ξ}^{b_{10}^{6}b_{11}ξ}  ^{b_{10}^{7}b_{11}ξ}^{b_{10}^{8}b_{11}ξ}  ζ_{2} [figure]

Here each little circle denotes Z/3 generated by the indicated element, and the dot on the upper left of a circle is Z/3 whose generator is the multiplication of the element and ζ_{2}.

Lemma 2.2. The differential d_{5} of the Adams-Novikov spectral sequence converging to π_{∗}(Q^{∧q} ∧ V (1)) acts trivially expect for

d_{5}(v_{2}^{j}Xg_{q}) = v_{2}^{j−1}Y_{k}g_{q} if j ≡ 3k, 3k + 1, 3k + 5 (9),

where the elements X and Y_{k} for k ∈ Z/3 are given in the following table:

\begin{matrix}X & Y_{0} & Y_{1} & Y_{2} \\ 1 & qξb_{10}ζ_{2} & −v_{2}^{−1}h_{11}b_{10}^{2} + qξb_{10}ζ_{2} & v_{2}^{−1}h_{11}b_{10}^{2} + qξb_{10}ζ_{2} \\ ζ_{2} & 0 & −v_{2}^{−1}h_{11}b_{10}^{2}ζ_{2} & v_{2}^{−1}h_{11}b_{10}^{2}ζ_{2} \\ b_{11} & v_{2}h_{10}b_{10}^{3} + qb_{11}ξb_{10}ζ_{2} & qb_{11}ξb_{10}ζ_{2} & −v_{2}h_{10}b_{10}^{3} + qb_{11}ξb_{10}ζ_{2} \\ b_{11}ζ_{2} & v_{2}h_{10}b_{10}^{3}ζ_{2} & 0 & −v_{2}h_{10}b_{10}^{3}ζ_{2} \\ v_{2}^{2}ψ_{0} & 0 & −b_{11}ξb_{10}^{2} & b_{11}ξb_{10}^{2} \\ v_{2}^{2}ψ_{0}ζ_{2} & 0 & −b_{11}ξb_{10}^{2}ζ_{2} & b_{11}ξb_{10}^{2}ζ_{2} \\ ψ_{1} & −ξb_{10}^{3} & ξb_{10}^{3} & 0 \\ ψ_{1}ζ_{2} & −ξb_{10}^{3}ζ_{2} & ξb_{10}^{3}ζ_{2} & 0 \end{matrix}

Proof. Since the identity map Q^{∧q} ∧ V (1) →^{=} (Q^{∧q} ∧ V (1)) gives a natural pairing, we have a derivation formula

(2.3)  d_{5}(xy) = d_{5}(x)y + (−1)^{t−s}xd_{5}(y)

for x ∈ E_{2}^{s,t}(Q^{∧q}) and y ∈ E_{2}^{s′,t′}(V (1)) [10, Th. 2.3.3]. Then the formula (2.3) and the equations on d_{5} given in [11, Prop.s 8.4, 9.9, 9.10] show us the desired differentials. Note that we set λ in [11] to be 1 in this paper. q.e.d.

By Lemma 2.2, we have

Lemma 2.4. In the E_{2}-term of the Adams-Novikov spectral sequence converging to π_{∗}(Q^{∧q} ∧V (1)), the elements ṽ_{2}^{j}b_{10}^{2}g_{q} = v_{2}^{j}b_{10}^{2}g_{q} +qv_{2}^{j}ψ_{1}ζ_{2}g_{q} for j ≡ 0, 1, 5 and ṽ_{2}^{j}b_{11}b_{10}g_{q} = v_{2}^{j}b_{11}b_{10}g_{q} + qv_{2}^{j+2}ψ_{0}ζ_{2}g_{q} for j ≡ 3, 4, 8 are d_{5}-cycles.

Proof. The lemma follows from the equations

\begin{aligned}d_{5}(v_{2}^{j}b_{10}^{2}g_{q}) & = & qv_{2}^{j−1}ξb_{10}^{3}ζ_{2}g_{q}, \\ d_{5}(qv_{2}^{j}ψ_{1}ζ_{2}g_{q}) & = & −qv_{2}^{j−1}ξb_{10}^{3}ζ_{2}g_{q}, \\ d_{5}(v_{2}^{j}b_{11}b_{10}g_{q}) & = & qv_{2}^{j−1}b_{11}ξb_{10}^{2}ζ_{2}g_{q} and \\ d_{5}(qv_{2}^{j+2}ψ_{0}ζ_{2}g_{q}) & = & −qv_{2}^{j−1}b_{11}ξb_{10}^{2}ζ_{2}g_{q} \end{aligned}

shown in Lemma 2.2. q.e.d.

Lemmas 2.2 and 2.4 show the structure of the E_{9}-term of the Adams-Novikov spectral sequence.

Proposition 2.5. For q = 1, 2, the subgroup F_{q} ⊕ F_{q}ζ_{2} (resp. F_{q}^{∗} ⊕ F_{q}^{∗}ζ_{2}) of the E_{9}-term E_{9}^{∗,∗}(Q^{∧q} ∧ V (1)) originating from F̃ ⊕ F̃ ζ_{2} (resp. F̃^{∗} ⊕ F̃^{∗}ζ_{2}) is H_{q} ⊕ I_{q} ⊕ H_{q}ζ_{2} ⊕ I_{q}ζ_{2} (resp. H_{q}^{∗} ⊕ I_{q}^{∗} ⊕ H_{q}^{∗}ζ_{2} ⊕ I_{q}^{∗}ζ_{2}). Here H_{q}, H_{q}ζ_{2}, H_{q}^{∗} and H_{q}^{∗}ζ_{2} are the modules of (1.2), and I_{q}, I_{q}ζ_{2}, I_{q}^{∗} and I_{q}^{∗}ζ_{2} are modules defined by

\begin{aligned}I_{q} & = & ∑_{j≡0,1,5 (9)} G\{ṽ_{2}^{j}b_{10}^{2}, v_{2}^{j−2}h_{11}, v_{2}^{j+3}h_{10}, ṽ_{2}^{j+3}b_{11}b_{10}\}, \\ I_{q}ζ_{2} & = & ∑_{j≡0,1,5 (9)} G\{v_{2}^{j}ζ_{2}, v_{2}^{j−2}h_{11}ζ_{2}, v_{2}^{j+3}h_{10}ζ_{2}, v_{2}^{j+3}b_{11}ζ_{2}\}, \\ I_{q}^{∗} & = & ∑_{j≡0,1,5 (9)} G\{v_{2}^{j+2}ψ_{0}, v_{2}^{j−1}b_{11}ξ, v_{2}^{j+5}ξ, v_{2}^{j+6}ψ_{1}\} and \\ I_{q}^{∗}ζ_{2} & = & ∑_{j≡0,1,5 (9)} G\{v_{2}^{j+2}ψ_{0}ζ_{2}, v_{2}^{j−1}b_{11}ξζ_{2}, v_{2}^{j+5}ξζ_{2}, v_{2}^{j+6}ψ_{1}ζ_{2}\}, \end{aligned}

in which G denotes Z/3[b_{10}].

3. The Adams-Novikov E_{13}-term for the invertible spectrum Q^{∧q}

In this section, we compute the Adams-Novikov differential d_{9} and determine the E_{13}-term for the homotopy groups π_{∗}(Q^{∧q} ∧ V (1)), and prove Theorem I by showing that the E_{∞}-term is isomorphic to the E_{13}-term.

Consider the spectra N^{1} and M^{2} defined by the cofiber sequences

L_{2}S^{0} → L_{0}S^{0} → N^{1} →^{j_{0}} ΣL_{2}S^{0} and N^{1} → L_{1}N^{1} → M^{2} →^{j_{1}} ΣN^{1}.

In [13, Lemma 6.2], it is shown that

d_{9}(v_{2}^{4}h_{10}/3v_{1}) = v_{2}b_{10}^{5}/3v_{1} ∈ E_{2}^{∗}(M^{2}).

Sending this to E_{2}^{∗}(S^{0}) under the map (j_{0}j_{1})_{∗}, we see that

(3.1)  d_{9}(α_{1}β_{4}) = β_{1}^{6} ∈ E_{2}^{∗}(S^{0}),

since α_{1} = h_{10}, β_{1} = (j_{0}j_{1})_{∗}(v_{2}/3v_{1}) = b_{10} and β_{4} = (j_{0}j_{1})_{∗}(v_{2}^{4}/3v_{1}) = v_{2}^{3}b_{10} by definition (cf. [9]).

Lemma 3.2. The action of the differential d_{9} of the Adams-Novikov spectral sequence converging to π_{∗}(Q^{∧q} ∧ V (1)) for q = 1, 2 is trivial except for

d_{9}(X_{j}g_{q}) = Y_{j}g_{q} for j ≡ 0, 1, 5 (9),

where the elements X_{j} and Y_{j} are those in the table:

\begin{matrix}X_{j} & Y_{j} & X_{j} & Y_{j} \\ v_{2}^{j+3}h_{10} & ṽ_{2}^{j}b_{10}^{5} & v_{2}^{j+3}h_{10}ζ_{2} & v_{2}^{j}b_{10}^{5}ζ_{2} \\ v_{2}^{j+7}h_{11} & ṽ_{2}^{j+3}b_{11}b_{10}^{4} & v_{2}^{j+7}h_{11}ζ_{2} & v_{2}^{j+3}b_{11}b_{10}^{4}ζ_{2} \\ v_{2}^{j+5}ξ & v_{2}^{j+2}ψ_{0}b_{10}^{4} & v_{2}^{j+5}ξζ_{2} & v_{2}^{j+2}ψ_{0}b_{10}^{4}ζ_{2} \\ v_{2}^{j+8}b_{11}ξ & v_{2}^{j+6}ψ_{1}b_{10}^{5} & v_{2}^{j+8}b_{11}ξζ_{2} & v_{2}^{j+6}ψ_{1}b_{10}^{5}ζ_{2} \end{matrix}

Here ṽ_{2}^{j}b_{10}^{5} = v_{2}^{j}b_{10}^{5} + qv_{2}^{j}ψ_{1}b_{10}^{3}ζ_{2} and ṽ_{2}^{j+3}b_{11}b_{10}^{4} = v_{2}^{j+3}b_{11}b_{10}^{4} + qv_{2}^{j+5}ψ_{0}b_{10}^{3}ζ_{2}.

Proof. Let κ be one of the generators of I_{q}, I_{q}ζ_{2}, I_{q}^{∗} and I_{q}^{∗}ζ_{2} such that

h_{10}κ ≠ 0. Then κ is one of the elements ṽ_{2}^{j}b_{10}^{2}, ṽ_{2}^{j+3}b_{11}b_{10}, v_{2}^{j}ζ_{2}, v_{2}^{j+3}b_{11}ζ_{2}, v_{2}^{j+2}ψ_{0}, v_{2}^{j+6}ψ_{1}, v_{2}^{j+2}ψ_{0}ζ_{2} and v_{2}^{j+6}ψ_{1}ζ_{2} for j ≡ 0, 1, 5 (9).

Since β_{4} ≡ v_{2}^{3}b_{10} mod (3, v_{1}), β_{1} = b_{10} and α_{1} = h_{10}, the derivation formula shows d_{9}(v_{2}^{3}h_{10}b_{10}κ) = b_{10}^{6}κ + v_{2}^{3}h_{10}b_{10}d_{9}(κ) by (3.1). Since κ and v_{2}^{3}h_{10}κ are G = Z/3[b_{10}]-free generators of the E_{9}-term, we have

(3.3)  d_{9}(v_{2}^{3}h_{10}κ) = b_{10}^{5}κ + v_{2}^{3}h_{10}d_{9}(κ).

From the picture of the E_{2}(= E_{5})-term in the previous section, we obtain the following table:

\begin{matrix}κ & C & κ & C \\ ṽ_{2}^{j}b_{10}^{2} & b_{10}^{7}, ψ_{1}b_{10}^{5}ζ_{2}, h_{11}b_{10}^{6}ζ_{2} & ṽ_{2}^{j+3}b_{11}b_{10} & b_{11}b_{10}^{6}, ψ_{0}b_{10}^{5}ζ_{2}, h_{10}b_{10}^{6}ζ_{2} \\ v_{2}^{j+2}ψ_{0} & ψ_{0}b_{10}^{5}, h_{10}b_{10}^{6}, b_{11}ξb_{10}^{4}ζ_{2} & v_{2}^{j+6}ψ_{1} & ψ_{1}b_{10}^{5}, h_{11}b_{10}^{6}, ξb_{10}^{5}ζ_{2} \\ v_{2}^{j+2}ψ_{0}ζ_{2} & b_{11}b_{10}^{6}, ψ_{0}b_{10}^{5}ζ_{2}, h_{10}b_{10}^{6}ζ_{2} & v_{2}^{j+6}ψ_{1}ζ_{2} & b_{10}^{7}, ψ_{1}b_{10}^{5}ζ_{2}, h_{11}b_{10}^{6}ζ_{2} \\ v_{2}^{j}ζ_{2} & b_{10}^{5}ζ_{2} & v_{2}^{j+3}b_{11}ζ_{2} & b_{10}^{5}b_{11}ζ_{2} \end{matrix}

Here the column name C stands for candidates of v_{2}^{3}h_{10}d_{9}(κ) up to a multiple of v_{2}. Using the relations in (2.1), the possible non-trivial differentials are as follows:

\begin{aligned}d_{9}(ṽ_{2}^{j}b_{10}^{2}) & = & k_{1}v_{2}^{j−4}b_{11}b_{10}^{5}ζ_{2}, \\ d_{9}(ṽ_{2}^{j+3}b_{11}b_{10}) & = & k_{2}v_{2}^{j+2}b_{10}^{6}ζ_{2}, \\ d_{9}(v_{2}^{j+2}ψ_{0}) & = & k_{3}v_{2}^{j−1}b_{10}^{6} + k_{4}v_{2}^{j−1}ψ_{1}b_{10}^{4}ζ_{2}, \\ d_{9}(v_{2}^{j+6}ψ_{1}) & = & k_{5}v_{2}^{j+2}b_{11}b_{10}^{5} + k_{6}v_{2}^{j+4}ψ_{0}b_{10}^{4}ζ_{2}, \\ d_{9}(v_{2}^{j+2}ψ_{0}ζ_{2}) & = & k_{7}v_{2}^{j−1}b_{10}^{6}ζ_{2} and \\ d_{9}(v_{2}^{j+6}ψ_{1}ζ_{2}) & = & k_{8}v_{2}^{j+2}b_{11}b_{10}^{5}ζ_{2} \end{aligned}

for skalars k_{i} ∈ Z/3. All generators of the E_{5}-term on the right hand sides of the equations die in E_{9}-term by Lemma 2.2 except for the possibility

d_{9}(v_{2}^{3}ψ_{0}ζ_{2}) = k_{7}b_{10}^{6}ζ_{2} and d_{9}(v_{2}^{7}ψ_{1}ζ_{2}) = k_{8}v_{2}^{3}b_{11}b_{10}^{5}ζ_{2}.

The relation (3.3) for κ = ζ_{2} (resp. κ = v_{2}^{3}b_{11}ζ_{2}) is d_{9}(v_{2}^{3}h_{10}ζ_{2}) = b_{10}^{5}ζ_{2} (resp. d_{9}(v_{2}^{6}h_{10}b_{11}ζ_{2}) = v_{2}^{3}b_{11}b_{10}^{5}ζ_{2}), and so we replace v_{2}^{3}ψ_{0}ζ_{2} (resp. v_{2}^{7}ψ_{1}ζ_{2}) with v_{2}^{3}ψ_{0}ζ_{2} −k_{7}v_{2}^{3}h_{10}b_{10}ζ_{2} (resp. v_{2}^{7}ψ_{1}ζ_{2} −k_{8}v_{2}^{6}h_{10}b_{11}ζ_{2}). After the replacement, we see that d_{9}(κ) = 0, and so d_{9}(v_{2}^{3}h_{10}κ) = b_{10}^{5}κ. It gives rise to the table of the theorem by the relations (2.1). q.e.d.

Therefore, we obtain the E_{13}-term of the Adams-Novikov spectral sequence.

Proposition 3.4. In the homotopy group π_{∗}(Q^{∧q} ∧ V (1)) for each q = 1, 2, the subgroup \overline{F_{q}} ⊕ \overline{F_{q}ζ_{2}} (resp. \overline{F_{q}^{∗}} ⊕ \overline{F_{q}^{∗}ζ_{2}}) of the E_{13}-term originating from F_{q}⊕F_{q}ζ_{2} (resp. F_{q}^{∗}⊕F_{q}^{∗}ζ_{2}) is H_{q}⊕\overline{I_{q}}⊕H_{q}ζ_{2}⊕\overline{I_{q}ζ_{2}} (resp. H_{q}^{∗}⊕\overline{I_{q}^{∗}}⊕H_{q}^{∗}ζ_{2}⊕\overline{I_{q}^{∗}ζ_{2}}). Here the modules H_{q}, H_{q}ζ_{2}, H_{q}^{∗} and H_{q}^{∗}ζ_{2} (resp. \overline{I_{q}}, \overline{I_{q}ζ_{2}}, \overline{I_{q}^{∗}} and \overline{I_{q}^{∗}ζ_{2}}) are given in (1.2)(resp. (1.3)).

Proof of Theorem I. For q = 0, the theorem is shown in [11] (cf. [3], [1]). For q = 1, 2, Proposition 3.4 says that E_{13}^{s,t}(Q^{∧q} ∧ V (1)) = 0 for s > 12, and so E_{13}^{s,t}(Q^{∧q} ∧ V (1)) = E_{∞}^{s,t}(Q^{∧q} ∧ V (1)) since the differential d_{r} = 0 for r ≥ 13. The homotopy group π_{k}(Q^{∧q} ∧ V (1)) for each k ∈ Z is a Z/3-vector space since Q^{∧q} ∧ V (1) is a V (0)-module spectrum. Therefore, there is no extention problem and we obtain the theorem from Proposition 3.4. q.e.d.

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Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan

E-mail address,

95sm004@math.kochi-u.ac.jp katsumi@math.kochi-u.ac.jp