Homotopy groups of generalized E(2)-local Moore spectra at the prime three

Ippei Ichigi and Katsumi Shimomura

To the memory of the late Professor Masahiro Sugawara

ABSTRACT. Let E(2) denote the Johnson-Wilson spectrum with homotopy groups π_{∗}(E(2)) = ℤ_{(3)}[v_{1}, v_{2}, v_{2}^{−1}]. Then the mod 3 Moore spectrum V (0) satisfies E(2)_{∗}(V (0)) = E(2)_{∗}/(3). We call a spectrum M generalized (E(2)-local) Moore spectrum if it satisfies E(2)_{∗}(M) = E(2)_{∗}/(3) = E(2)_{∗}(V (0)) as an E(2)_{∗}E(2)-comodule. We see that the Toda spectrum Σ^{−21}V (1 \frac{1}{2} ) is an example (cf. [10]) other than the Moore spectrum V (0). Here we introduce other generalized Moore spectra and determine the homotopy groups of them.

1. Introduction

Let S_{(p)} denote the stable homotopy category of p-local spectra for a prime p, and E(n) ∈ S_{(p)} denote the Johnson-Wilson spectrum characterized by the homotopy groups π_{∗}(E(n)) = E(n)_{∗} = v_{n}^{−1}ℤ_{(p)}[v_{1}, v_{2}, …, v_{n}] ⊂ v_{n}^{−1}BP_{∗} = v_{n}^{−1}ℤ_{(p)}[v_{1}, v_{2}, …]. Here, BP denotes the Brown-Peterson spectrum. We denote L_{n}: S_{(p)} → S_{(p)} as the Bousfield localization functor with respect to E(n). We write L_{n} as the image of L_{n}. We call a spectrum X ∈ L_{n} invertible if there exists a spectrum Y such that X ∧ Y = L_{n}S for the sphere spectrum S. Then Hovey and Sadofsky [6] showed that the collection of isomorphism classes of invertible spectra forms a group, which is called the Picard group of L_{n} and denoted by Pic(L_{n}). We call an invertible spectrum X strict if Hℚ_{0}(X) = ℚ, and proper if X is strict and X ≄ L_{n}S. The strict invertible spectra define the subgroup Pic(L_{n})^{0} ⊂ Pic(L_{n}). Hovey and Sadofsky also showed Pic(L_{n}) = Pic(L_{n})^{0}⊕ℤ, which means that an invertible spectrum is isomorphic to a suspension of a strict one.

In [6] and [10], it is shown that a spectrum X ∈ L_{n} is strict invertible if and only if E(n)_{∗}(X) = E(n)_{∗} = E(n)_{∗}(S) as an E(n)_{∗}E(n)-comodule. We generalize this. We call X ∈ L_{n} a generalized (E(n)-local) k-th Smith-Toda spectrum if E(n)_{∗}(X) = E(n)_{∗}/(p, v_{1}, …, v_{k}) as an E(n)_{∗}E(n)-comodule. In particular, the point spectrum ∗ is a generalized k-th Smith-Toda spectrum for k ≥ n, and the E(n)-localization of the k-th Smith-Toda spectrum V (k) is a generalized one if V (k) exists. Note that the 0-th Smith-Toda spectrum V (0) is the mod p Moore spectrum, which is a cofiber of p: S → S. So we call a generalized 0-th Smith-Toda spectrum a generalized (E(n)-local mod p) Moore spectrum, which exists for any n ≥ 0 and any prime p. A generalized (−1)-st Smith-Toda spectrum X is an invertible spectrum as above. Let V_{n}(k) denote the collection of the isomorphism classes of generalized k-th Smith-Toda spectra in L_{n}. In particular, V_{n}(−1) = Pic(L_{n})^{0} and V_{n}(k) = {∗} if k ≥ n. Strickland shows that if X has a finitely generated E(n)_{∗}-homology, then X is a small object in L_{n}. (See Theorem 2.1.) This shows that V_{n}(k) is also a set. In [20] and [19], Yosimura, Yokotani and the second author showed the existence and uniquness of a generalized Smith-Toda spectrum L_{n}V (k) if k < n and n^{2} + n < 2p.

Proposition 1.1. ([20], [19], [6]) If n^{2} + n < 2p, then V_{n}(k) = {L_{n}V (k)}. In particular, if n^{2} + n < 2p, then each generalized mod p Moore spectrum is nothing but the E(n)-localization of the mod p Moore spectrum.

In the same manner as the sphere spectrum acts on any spectrum, Pic(L_{n})^{0} acts on V_{n}(k) by xv = [X ∧ V] for x = [X] ∈ Pic(L_{n})^{0} and v = [V] ∈ V_{n}(k). Since a generalized Smith-Toda spectrum V is small, the Spanier-Whitehead dual D defines an action on V_{n}(k) by D_{∗}(V) = Σ^{v(n)}DV = Σ^{v(n)}F (V, L_{n}S) with D_{∗}D_{∗}(V) = V. Here, v(n) = ∑_{k=0}^{n}(2p^{k} − 1). Indeed, E(n)_{∗}(DV) = E(n)^{∗}(V) = E(n)_{∗}/(p, v_{1}, …, v_{k}). Note that D_{∗}(V (k)) = L_{n}V (k) for the Smith-Toda spectrum V (k) if it exists. Let Pic(L_{n})^{0}{V } denote the orbit of V: {[V ∧ X] : X ∈ Pic(L_{n})^{0}}. Hereafter, we write V ∈ V_{n}(k) for [V] ∈ V_{n}(k).

Conjecture A. For V ∈ V_{n}(k), Pic(L_{n})^{0}{D_{∗}(V)} = Pic(L_{n})^{0}{V }.

This is true if k = −1.

2000 Mathematics Subject Classification. 55Q99.

Key words and phrases. Moore spectrum, Bousfield localization, Johnson-Wilson spectrum, E(2)-based Adams spectral sequence.

Now we consider the generalized Moore spectra at the prime three. Then, Proposition 1.1 says that the generalized mod 3 Moore spectrum is L_{n}V (0) if n < 2. So we work in L_{2}. In [10], Kamiya and the second author showed that ℤ/3 ⊂ Pic(L_{2})^{0} ⊂ ⨁_{r>1} E_{r}^{r,r−1}(S), which is isomorphic to ℤ/3 ⊕ ℤ/3 by [18]. Let P be the proper invertible spectrum that generates ℤ/3 ⊂ Pic(L_{2})^{0}. It is constructed with a map f: P → Σ^{−21}L_{2}V (1\frac{1}{2}) that induces the projection f_{∗}: E(2)_{∗} → E(2)_{∗}/(3). There is a problem that asks if there is another proper invertible spectrum Q, which corresponds to the other summand of E_{5}^{5,4}(S). Since each E(2)_{∗}-homology sphere is an invertible spectrum by [10], X ∧ V (0) for each invertible spectrum X is an example of generalized Moore spectra other than L_{2}V (0). We have no idea if an invertible spectrum X is an E(2)-localization of a finite spectrum, though X is a retract of E(2)-localization of a finite spectrum [6]. There is another generalized Moore spectrum L_{2}V (1\frac{1}{2}) (cf. [10]) other than L_{2}V (0), which is an E(2)-localization of a finite spectrum. Here V (1\frac{1}{2}) is the Toda spectrum given in [21]. The construction is generalized as follows: It is shown the existence of a map B^{′(i)} : Σ^{16i}S → V (1) that induces v_{2}^{i}: BP_{∗} → BP_{∗}/(3, v_{1}) for i = 0, 1, 5 (cf. [12]). Since the order of B^{′(i)} is three, B^{′(i)} extends to B^{(i)} : Σ^{16i}V (0) → V (1). Let Σ^{16i+5}V_{i} denote the cofiber of B^{(i)}. Then, L_{2}V_{i} ∈ V_{2}(0) for i = 0, 1, 5. Note that V_{0} = V (0) and V_{1} = Σ^{−21}V (1\frac{1}{2}).

We have another construction: Let X be an invertible spectrum and ι_{X} ∈ π_{0}(X) denote the element detected by 3g_{X} ∈ E_{2}^{0,0}(X) = ℤ_{(3)}{g_{X}}, and write W X as the cofiber of ι_{X}. In particular, W L_{2}S^{0} = L_{2}V (0). Note that W does not seem a good operation. Indeed, even though the Adams-Novikov differentials d_{5} on the generators of E_{2}^{0,0} = ℤ/3 for W X ∧ X′ and W (X ∧ X′) agree for any strict invertible spectra X and X′, these spectra are not always homotopy equivalent. (If the Adams-Novikov differentials d_{5} on the generators of E_{2}^{0,0} = ℤ_{(3)} for invertible spectra X and X′ agree, then they are homotopy equivalent by [10].) For the proper invertible spectrum P ∈ Pic(L_{2})^{0}, we have W P^{j} for j ≥ 0. Here U^{j} for a spectrum U denotes the j-fold smash product of U for j > 0 and U^{0} = L_{2}S. Note that W P^{2} = V_{1} ∧ P, since P^{2} → P →^{f} Σ^{−21}L_{2}V_{1} is a cofiber sequence [8]. It follows that Pic(L_{2})^{0}{W P^{2}} = Pic(L_{2})^{0}{L_{2}V_{1}}. If the other proper invertible spectrum Q exists, then we also have W Q, and V_{2}(0) contains the orbit Pic(L_{2})^{0}{W Q, W Q^{2}}. Put

V_{2}(0)^{0} = Pic(L_{2})^{0}{L_{2}V_{0}, L_{2}V_{1}, L_{2}V_{5}, W P, (W Q, W Q^{2} if they exist)}.

Proposition 1.2. V_{2}(0)^{0} ⊂ V_{2}(0).

The size of Pic(L_{n})^{0} = V_{n}(−1) has an upper bound, but we have no idea about the size of V_{n}(k) for k ≥ 0. Indeed, an generalized k-th Smith-Toda spectrum is not always a V (k)-module spectrum if k ≥ 0.

Conjecture B. V_{2}(0)^{0} = V_{2}(0).

For the Spanier-Whitehead dual D, D(X) = X^{2} for an invertible spectrum X (cf. [5], [10], Theorem 2.1), and we see that D_{∗}((W X) ∧ X) = W X ∧ X, since the dual of the cofiber sequence X ⟶^{ι∧X} X^{2} → (W X) ∧ X

is X ⟶^{D(ι∧X)} X^{2} → ΣD((W X) ∧ X). For V_{i}, we also see that D_{∗}(L_{2}V_{i}) = L_{2}V_{i}. It follows that D_{∗}(V) ∈ Pic(L_{2})^{0}{V } for V ∈ V_{2}(0)^{0}.

Proposition 1.3. If Conjecture B holds, then so does Conjecture A for k = 0.

In this paper, we determine the homotopy groups of all spectra in V_{2}(0)^{0} other than W Q^{i} for i = 1, 2. The structure of the homotopy groups of L_{2}U for a spectrum U is more complicated than that of π_{∗}(L_{K(2)}U). Here L_{K(2)} denotes the Bousfield localization functor with respect to the second Morava K-theory K(2). So we recall [13] the chromatic spectra M_{n}U and N_{n}U for a spectrum U, which are defined inductively by

N_{0}U = U, M_{n}U = L_{n}N_{n}U and the cofiber sequence N_{n}U → M_{n}U → N_{n+1}U.

Then the homotopy groups π_{∗}(L_{K(2)}U) are closely related with π_{∗}(M_{2}U), and the structure of π_{∗}(M_{2}U) is less complicated than that of π_{∗}(L_{2}U). Thus we consider π_{∗}(M_{2}U) instead of π_{∗}(L_{2}U). In [16], the homotopy groups π_{∗}(M_{2}V_{0}) for the mod 3 Moore spectrum V_{0} = V (0) are determined by use of the E(2)-based Adams spectral sequence. The E_{2}-term of it is a direct sum of modules A, B_{h} and B_{t}, and the homotopy groups are a direct sum of A, B̃_{h} and B̃_{t}. Here, B̃_{j} denotes the permanent cycles of B_{j}. By use of the decomposition of the E_{2}-term, we obtain the homotopy groups in [8]:

π_{∗}(M_{2}V_{0} ∧ P^{k}) = A ⊕ v_{2}^{9−3k}B̃_{h} ⊕ v_{2}^{9−3k}B̃_{t} and π_{∗}(M_{2}V_{1}) = A ⊕ v_{2}^{3}B̃_{h} ⊕ v_{2}^{6}B̃_{t}.

Note that v_{2}^{9}B̃_{j} ≅ B̃_{j}. These make us to predict the following theorem:

Theorem 1.4. π_{∗}(M_{2}V_{1} ∧ P^{k}) = A ⊕ v_{2}^{3−3k}B̃_{h} ⊕ v_{2}^{6−3k}B̃_{t}.

These results let us ask if there is a spectrum U_{k} such that π_{∗}(U_{k}) = A ⊕ v_{2}^{6−3k}B̃_{h} ⊕ v_{2}^{3−3k}B̃_{t}. The elements of the orbit Pic(L_{2})^{0}{W P } give the affirmative answer.

Theorem 1.5. π_{∗}(M_{2}W P ∧ P^{k+1}) = A ⊕ v_{2}^{6−3k}B̃_{h} ⊕ v_{2}^{3−3k}B̃_{t}.

For V_{5}, we have

Theorem 1.6. π_{∗}(M_{2}V_{5} ∧ P^{k}) = A ⊕ v_{2}^{3−3k}B̃_{h} ⊕ v_{2}^{9−3k}B̃_{t}.

Remark. π_{∗}(M_{2}V_{5}) = π_{∗}(M_{2}W P ∧ P^{2}), while L_{2}V_{5} ≄ W P ∧ P^{2} by Lemma 5.3.

Remark. If Q exists, then the homotopy groups π_{∗}(M_{2}W Q) agree with none of above (cf. [9]).

This paper is organized as follows: In the next section, we include the result of Strickland’s. In section 3, we consider a decomposition of the E(2)-based Adams E_{2}-term, which plays a crucial role to determine the homotopy groups. Sections 4 and 5 are devoted to define the generalized Moore spectrum, and to show some properties of them, which show Proposition 1.2. In section 6, we prove Theorems 1.4 and 1.5. Theorem 1.6 is proved in section 7.

The authors would like to thank the referee not only for suggesting them to add the basic facts on generalized Smith-Toda spectra but also introducing the result of Strickland’s. The authors would also like to thank Neil Strickland who kindly allow them to include his result in this paper.

2. Some results on generalized Smith-Toda spectra

Let BP and E(n) denote the Brown-Peterson and the n-th Johnson-Wilson spectra, respectively, at a prime p. We write L_{n} as the stable homotopy category consisting of E(n)-local spectra. Then BP_{∗}BP = BP_{∗}[t_{1}, t_{2}, …] has a structure of a Hopf algebroid over BP_{∗} = π_{∗}(BP) = ℤ_{(p)}[v_{1}, v_{2}, …]. Besides, it induces a Hopf algebroid structure on E(n)_{∗}E(n) over E(n)_{∗}, since E(n)_{∗}E(n) = E(n)_{∗} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} E(n)_{∗} for E(n)_{∗} = v_{n}^{−1}ℤ_{(p)}[v_{1}, v_{2}, …, v_{n}] ⊂ v_{n}^{−1}BP_{∗}.

We consider the E(n)-based Adams spectral sequence E_{r}^{s,t}(U) for computing the homotopy groups π_{∗}(L_{n}U) of a spectrum U. The E_{2}-term is isomorphic to the Ext group

E_{2}^{s,t}(U) = Ext_{E(n)_{∗}E(n)}^{s,t}(E(n)_{∗}, E(n)_{∗}(U))

of E(n)_{∗}E(n)-comodules.

The k-th Smith-Toda spectrum V (k) is a spectrum with BP_{∗}(V (k)) = BP_{∗}/(p, v_{1}, …, v_{k}). It is known to exists if k < 4 and 2k < p (cf. [22]). We call a spectrum V a generalized (E(n)-local) k-th Smith-Toda spectrum if E(n)_{∗}(V) = E(n)_{∗}/(p, v_{1}, …, v_{k}) as an E(n)_{∗}E(n)-comodule. The E_{2}-term E_{2}^{∗,∗}(V) of the E(n)-based Adams spectral sequence agrees with the E_{2}-term E_{2}^{∗}(V (k)) if V (k) exists. We write V_{n}(k) as the collection of isomorphism classes of generalized k-th Smith-Toda spectra. Then

V_{n}(k) ≅ ∗ if n^{2} + n < 2p

by [20] and [19].

Since E(n)_{∗}(V) is finitely generated, we see that V_{n}(k) is a set by [5, Th. 2.1.3] and the theorem of Strickland’s:

Theorem 2.1 (N. Strickland) If E(n)_{∗}(X) is finitely generated, then X is small in the E(n)-local category.

Proof. Let E be an S-algebra such that E ≃ E(n), whose existence is certified in [2]. (A. Lazarev also has another argument.) Let C_{X} denote a full subcategory consisting of spectra Z such that the natural map

⨁_{i}[X, Z ∧ Y_{i}] w[X, Z ∧ ⋁_{i} Y_{i}]

is an isomorphism for all families {Y_{i}} of E-local spectra. Then, C_{X} is a thick subcategory. Note that X is small if L_{n}S ∈ C_{X}. Since E_{∗} is a ring of finite global dimension and E_{∗}(X) is finitely generated, E ∧ X has a finite resolution by finitely generated free modules in the derived category D_{E} as in [1, Chap. IV], and hence E ∧ X is small in D_{E}. Since D_{E}(E ∧ X, E ∧ W ∧ Y) = [X, E ∧ W ∧ Y], every spectrum of the form E ∧ W is in C_{X}. Furthermore, since L_{n}S is E-nilpotent (cf. [14]), L_{n}S is in the thick subcategory generated by the spectra of the form E ∧ W. It follows that L_{n}S ∈ C_{X} as desired. □

3. A decomposition of H^{∗}M_{1}^{1}

From this section on, we set the prime p = 3 and work in L_{2}, the stable homotopy category of spectra localized with respect to the second Johnson-Wilson spectrum E(2), whose coefficient ring is E(2)_{∗} = v_{2}^{−1}ℤ_{(3)}[v_{1}, v_{2}] ⊂ v_{2}^{−1}BP_{∗}. Consider the mod 3 Moore spectrum V (0) defined by the cofiber sequence

(3.1)  S ^{3} wS ^{i} wV (0) ^{j} wΣS.

and the first Smith-Toda spectrum V (1) defined by the cofiber sequence

Σ^{4}V (0) ^{α} wV (0) ^{i_{1}} wV (1) ^{j_{1}} wΣ^{5}V (0).

Here α denotes the Adams map.

Let U(n) denote an n-th generalized Smith-Toda spectrum such that E(2)_{∗}(U(n)) is isomorphic to E(2)_{∗}(V (n)) as an E(2)_{∗}E(2)-comodule, where V (n) denotes the n-th Smith-Toda spectrum. Note that U(n) = ∗ if n ≥ 2. For n = 0, we call a spectrum U(0) ∈ L_{2} a generalized Moore spectrum, and consider M_{2}U(0) defined by the cofiber sequence

U(0) wL_{1}U(0) wΣ^{−1}M_{2}U(0)

(cf. [13]). Consider the E(2)_{∗}E(2)-comodule M_{2}^{0} and M_{1}^{1} defined by M_{2}^{0} = E(2)_{∗}/(3, v_{1}), which is also denoted by K(2)_{∗}, and the short exact sequence

0 wE(2)_{∗}/(3) wv_{1}^{−1}E(2)_{∗}/(3) wM_{1}^{1} w0.

Then we have the short exact sequence

(3.2)  0  wM_{2}^{0} ^{1/v_{1}} wM_{1}^{1} ^{v_{1}} wM_{1}^{1}  w0

of E(2)_{∗}E(2)-comodules. The E_{2}-terms of the E(2)-based Adams spectral sequences converging to π_{∗}(U(1)) and π_{∗}(M_{2}U(0)) are H^{∗}M_{2}^{0} and H^{∗}M_{1}^{1}, respectively. Here H^{∗}M for an E(2)_{∗}E(2)-comodule M denotes Ext_{E(2)_{∗}E(2)}^{∗}(E(2)_{∗}, M). The E_{2}-term H^{∗}M_{2}^{0} (resp. H^{∗}M_{1}^{1}) is shown in [15] (resp. [16]) to be isomorphic to the tensor product of Λ(ζ_{2}) and the module M (resp. the direct sum of modules (K(1)_{∗}/k(1)_{∗}) ⊗ Λ(h_{10}), ⨁_{n≥0}F_{n} and (F ⊕ F^{∗}) ⊗ (ℤ/3)[b_{10}]). Here the modules M, F, F^{∗} and F_{n} are given by

M = K(2)_{∗}[b_{10}]{1, h_{10}, h_{11}, ξ, ψ_{0}, ψ_{1}, b_{11}ξ};
F = E(2, 1)_{∗}{v_{2}^{±1}/v_{1}, v_{2}h_{10}/v_{1}^{2}, v_{2}^{2}h_{11}/v_{1}^{2}, v_{2}^{±1}b_{11}/v_{1}},
F^{∗} = E(2, 1)_{∗}{ξ/v_{1}^{2}, v_{2}^{2±1}ψ_{0}/v_{1}, v_{2}^{±1}ψ_{1}/v_{1}, b_{11}ξ/v_{1}^{2}} and
F_{n} = E(2, n + 2)_{∗}{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−1}, v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}+1},
v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}}}

for

k(1)_{∗} = (ℤ/3)[v_{1}], E(2, n)_{∗} = k(1)_{∗}[v_{2}^{±3^{n}}], K(1)_{∗} = v_{1}^{−1}k(1)_{∗} and K(2)_{∗} = (ℤ/3)[v_{2}^{±1}].

The element b_{10} acts on (F ⊕F^{∗})⊗(ℤ/3)[b_{10}] freely. The action of b_{10} on F_{n} is seen as follows: Consider the exact sequence H^{s}M_{2}^{0} → H^{s}M_{1}^{1} → H^{s}M_{1}^{1} →^{δ} H^{s+1}M_{2}^{0} associated to (3.2) and suppose that δ(x) = y and δ(w) = yb_{10} for x ∈ F_{n}, w ∈ F ⊕ F^{∗} and y ≠ 0 ∈ H^{∗}M_{2}^{0}. Then there exists an element u ∈ H^{∗}M_{1}^{1} such that xb_{10} = w + v_{1}u. Thus replacing w by w +v_{1}u, we have an isomorphism (ℤ/3){x}⊕(ℤ/3)[b_{10}]{w} = (ℤ/3)[b_{10}]{x}. We also write x = \overline{w}. In this way, we compute the b_{10}-action in [8] and rewrite here the direct sum of the modules ⨁_{n≥0}F_{n} and (F ⊕ F^{∗}) ⊗ (ℤ/3)[b_{10}] as the direct sum of ⨁_{n≥0}F′_{n} and F′ ⊗ (ℤ/3)[b_{10}]. Here

F′ = (ℤ/3)_{∗}\{v_{2}^{3(3t±1)}/v_{1}^{3} = \overline{v_{2}^{9t±3−2}b_{11}/v_{1}}, v_{2}^{3^{n}(3t±1)}/v_{1}^{4×3^{n−1}−1} = \overline{v_{2}^{3^{n−1}(9t±3−1)−1}b_{11}/v_{1}} (n > 1),
v_{2}^{9t−2}b_{11}/v_{1}, v_{2}^{3^{n−1}(3t+1)−1}b_{11}/v_{1}, v_{2}^{3^{n−1}(9t−1)−1}b_{11}/v_{1},
v_{2}^{3^{n}(3t+1)}h_{10}/v_{1}^{2×3^{n}+1} = \overline{v_{2}^{3^{n+1}t+(3^{n}−1)/2}ψ_{1}/v_{1}} (n > 0),
v_{2}^{3^{n}(9t+8)}h_{10}/v_{1}^{10×3^{n}+1} = \overline{v_{2}^{3^{n+2}t+5×3+(3^{n}−1)/2}ψ_{1}/v_{1}} (n ≥ 0), v_{2}^{3t−1}ψ_{1}/v_{1},
v_{2}^{3^{n}(9t+5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}} = \overline{v_{2}^{3^{n+1}(3t+1±1)+3(3^{n}−1)/2}b_{11}ξ/v_{1}^{2}} (n ≥ 0), v_{2}^{3t}b_{11}ξ/v_{1}\}
⊕E(2, 1)_{∗}{v_{2}^{±1}/v_{1}, v_{2}h_{10}/v_{1}^{2}, v_{2}^{2}h_{11}/v_{1}^{2}, ξ/v_{1}^{2}, v_{2}^{2±1}ψ_{0}/v_{1}} and
F′_{n} = E(2, n + 2)_{∗}{v_{2}^{±3^{n+1}}/v_{1}^{4×3^{n}−2}, v_{2}^{3^{n+1}}h_{10}/v_{1}^{6×3^{n}},
v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}}, v_{2}^{3^{n}(5±3)+(3^{n}−1)/2}ξ/v_{1}^{4×3^{n}−1}}.

Put

(3.3)  \begin{aligned}A & = & ((K(1)_{∗}/k(1)_{∗}) ⊗ Λ(h_{10}) ⊕ ⨁_{n≥0}F′_{n}) ⊗ Λ(ζ_{2}) and \\ B & = & F′ ⊗ (ℤ/3)[b_{10}] ⊗ Λ(ζ_{2}). \end{aligned}

Then we have a decomposition of H^{∗}M_{1}^{1}:

H^{∗}M_{1}^{1} = A ⊕ B.

By observing the construction of modules, these modules satisfy the following:

(3.4)  1) A ⊂ ⨁_{s=0}^{3}H^{s}M_{1}^{1}. 2) If x ∈ B ∩ H^{s}M_{1}^{1} for s > 5, then x = yb_{10} for some y ∈ B. 3) the generator b_{10} acts trivially on A and freely on B.

4. The generalized Moore spectrum W X for an invertible spectrum X at the prime three

We call a spectrum X invertible if there is a spectrum X′ such that X ∧X′ = L_{2}S. An invertible spectrum X is called strict if Hℚ_{0}(X) = ℚ, and proper if Hℚ_{0}(X) = ℚ and X ≄ L_{2}S. We denote by Pic(L_{2})^{0} the collection of isomorphism classes of strict invertible spectra, which is shown to be a group with multiplication defined by the smash product and with the unit L_{2}S. Note that every invertible spectrum is a suspension of a strict one (cf. [6]). It is shown in [6] and [10] that X is a strict invertible spectrum if and only if E(2)_{∗}(X) = E(2)_{∗} as an E(2)_{∗}E(2)-comodule. For a strict invertible spectrum X, the E(2)-based Adams E_{2}-term E_{2}^{∗}(X) is isomorphic to E_{2}^{∗}(S). In particular, E_{2}^{0,0}(X) = E_{2}^{0,0}(S) = ℤ_{(3)}. Take the generator g_{X} ∈ E_{2}^{0,0}(X). Then by [10], X is characterized by d_{5}(g_{X}) ∈ E_{2}^{5,4}(X). For example, d_{5}(g_{X}) = 0 if and only if X = L_{2}S. If d_{5}(g_{X}) ≠ 0, then g_{X} does not detect a homotopy element but 3g_{X} does. We denote ι_{X} as the homotopy element detected by 3g_{X}, and write W X as the cofiber of ι_{X}.

Let V (1\frac{1}{2}) denote the Toda spectrum defined in [21] by the cofiber sequence

Σ^{16}V (0) ^{B} wV (1) wV (1\frac{1}{2}) wΣ^{17}V (0),

in which the map B induces the homomorphism v_{2} : E(2)_{∗}/(3) → E(2)_{∗}/(3, v_{1}), the multiplication by v_{2}. The element B is denoted by [βi_{1}] in [21]. This is an example of generalized Moore spectra studied in the next section. In [10], a proper invertible spectrum P is constructed as well as the existence of a map f: P → Σ^{−21}L_{2}V (1\frac{1}{2}) that realizes the projection E(2)_{∗} → E(2)_{∗}/(3). The spectrum P is characterized by the differential of the E(2)-based Adams spectral sequence as follows:

d_{5}(g_{P}) = v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{2}^{5,4}(P) = E_{2}^{5,4}(S),

where the E_{2}-term E_{2}^{5,4}(S) for π_{∗}(L_{2}S) is isomorphic to (ℤ/3){v_{2}^{−2}h_{11}b_{10}^{2}, v_{2}^{−1}ξb_{10}ζ_{2}}. Note then that P^{2} = P ∧ P is an invertible spectrum characterized by d_{5}(g_{P^{2}}) = −v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{2}^{5,4}(P^{2}) = E_{2}^{5,4}(S). Now the generalized Moore spectrum W P^{k} for k = 1, 2 fits in the cofiber sequence

(4.1)  S ^{ι_{k}} wP^{k} ^{i_{k}} wW P^{k} ^{j_{k}} wΣS,

where ι_{k} is the abbreviation of ι_{P^{k}}.

Proposition 4.2. W P and W P^{2} are generalized Moore spectra, which are not V (0)-module spectra.

Proof. By definition, the homotopy element ι_{k} ∈ π_{0}(P^{k}) for k = 1, 2 induces (ι_{k})_{∗} = 3: E(2)_{∗} = E(2)_{∗}(S) → E(2)_{∗}(P^{k}) = E(2)_{∗}, and so E(2)_{∗}(W P^{k}) is isomorphic to E(2)_{∗}/(3) as an E(2)_{∗}E(2)-comodule.

We show that 3id ≠ 0 ∈ [W P, W P]_{0} for the identity map id ∈ [W P, W P]_{0}. Consider the diagram

[W P, W P]_{0} [P, S]_{0}  [P, P]_{0}  [P, W P]_{0}  [P, S]_{−1}_{u_{i_{1}^{∗}}}_{^{ι_{1}_{∗}} w}  _{^{i_{1}_{∗}} w}  _{^{j_{1}_{∗}} w}

The behavior of the map ι_{1}_{∗}: [P, S]_{0} → [P, P]_{0} is observed by the one of (ι_{1} ∧ P^{2})_{∗}: [S, P^{2}]_{0} → [S, S]_{0}, since P is an invertible spectrum and P^{2} is its inverse. The generator ι_{2} ∈ ℤ ⊂ [S, P^{2}]_{0} is detected by 3g_{P^{2}} ∈ E_{2}^{0,0}(P^{2}) and ι_{1} induces also a multiplication by 3 on the E_{2}-terms. Therefore, the induced map (ι_{1} ∧ P^{2})_{∗} assigns the element 3g_{P^{2}} to 9g_{S} on the E_{2}-terms, and so we have (ι_{1} ∧ P^{2})_{∗}(ι_{2}) = 9 ∈ π_{0}(S) on the homotopy. It follows that the map i_{1} generates ℤ/9 ⊂ [P, W P]_{0}. Therefore, i_{1}^{∗}(3id) = 3i_{1} ≠ 0 ∈ [P, W P]_{0}, and 3id ≠ 0. □

The results of [10] and [17] imply the following Lemma 4.3. If U = W X for an invertible spectrum X, then d_{5}(g_{U}) ∈ (ℤ/3){v_{2}^{−2}h_{11}b_{10}^{2}, v_{2}^{−1}ξb_{10}ζ_{2}} ⊂ E_{2}^{5,4}(U).

Proof. By the definition of W X, there is an exact sequence

E_{2}^{5,4}(S) ^{(ι_{X})_{∗}=3}wE_{2}^{5,4}(X) ^{(i_{X})_{∗}}wE_{2}^{5,4}(U) ^{δ} wE_{2}^{6,4}(S)

associated to the cofiber sequence S ⟶^{ι_{X}} X ⟶^{i_{X}} U. Since E_{2}^{5,4}(X) = E_{2}^{5,4}(S) = (ℤ/3){v_{2}^{−2}h_{11}b_{10}^{2}, v_{2}^{−1}ξb_{10}ζ_{2}} by

[17], (i_{X})_{∗} is a monomorphism. Now the lemma follows from the naturality of the Adams differentials. □

5. The generalized Moore spectrum V_{5} related to v_{2}^{5}

It is well known that there is a generator v_{2}^{i} ∈ π_{16i}(V (1)) for each i = 0, 1, 5 ([12]), which are of order three. It follows that there exists a map B^{(i)} : Σ^{16i}V (0) → V (1) that induces v_{2}^{i}: E(2)_{∗}/(3) → E(2)_{∗}/(3, v_{1}). Note that B^{(0)} = i_{1} and B^{(1)} = β, the Smith element. First we show that

Lemma 5.1. The element v_{2}^{5} ∈ π_{80}(V (1)) does not extend to a self-map v_{2}^{5} : Σ^{80}V (1) → V (1).

Proof. Suppose that such a self-map exists. Then we have a map v_{2}^{10} : Σ^{160}V (1) → V (1). Since there is an equivalence v_{2}^{9} : Σ^{144}L_{2}V (1) ≃ L_{2}V (1) by [7], we obtain a self map v_{2} : Σ^{16}L_{2}V (1) → L_{2}V (1). This contradicts to the non-existence of the self map v_{2} : Σ^{16}V (1) → V (1), whose obstruction survives after localizing it with respect to E(2). □

Now we write Σ^{16i+5}V_{i} as the cofiber of B^{(i)}.

Proposition 5.2. L_{2}V_{5} is a generalized Moore spectrum.

Proof. Consider the diagram

Σ^{84}V (0)  Σ^{80}V (0)  Σ^{80}V (1)  Σ^{85}V (0) Σ^{−1}V_{5}  Σ^{80}V (0)  V (1)  Σ^{85}V_{5}. u  ^{α} w  ^{i_{1}} w  ^{j_{1}} w_{u_{v_{2}^{5}}}  u_{^{j_{B^{(5)}}} w}  _{^{B^{(5)}} w}  _{^{i_{B^{(5)}}} w}

Here the broken arrows exist after smashing with E(2) so that the diagram commutes. Since v_{2}^{5} is an isomorphism on the E(2)_{∗}-homology of V (1), we have an isomorphism E(2)_{∗}/(3) = E(2)_{∗}(V (0)) = E(2)_{∗}(V_{5}) by the Five Lemma. □

Lemma 5.3. L_{2}V_{5} ≄ W X for any invertible spectrum X.

Proof. Lemma 5.1 shows that B^{(5)}α ≠ 0 ∈ π_{84}(L_{2}V (1)). Write x ≠ 0 ∈ E_{2}^{∗,∗}(V (1)) for an element that detects B^{(5)}α. Then x ∈ E_{2}^{4,88}(V (1)) or x ∈ E_{2}^{8,92}(V (1)). By [15], we see that E_{6}^{4,88}(V (1)) = (ℤ/3){v_{2}^{4}b_{10}h_{11}ζ_{2}} and E_{6}^{8,92}(V (1)) = 0. It follows that x = ±v_{2}^{4}b_{10}h_{11}ζ_{2} ∈ E_{2}^{4,88}(V (1)). Observe the cofiber sequence that defines V_{5}, and we see that d_{5}(g_{V_{5}}) = δ(x) = ±v_{1}v_{2}^{−3}b_{11}b_{10}ζ_{2} ∈ E_{5}^{5,4}(V_{5}). In fact, δ(v_{2}^{4}b_{10}h_{11}ζ_{2}) = [v_{1}^{−1}d(v_{2}^{−1}b_{10}h_{11}ζ_{2})] = v_{1}v_{2}^{−3}b_{11}b_{10}ζ_{2} read off from [16, Lemma 3.3]. Therefore, Lemma 4.3 shows that there is no invertible spectrum X such that L_{2}V_{5} = W X. □

Proof of Proposition 1.2. Since E(2)_{∗}(X) for a strict invertible spectrum X is isomorphic to E(2)_{∗} as an E(2)_{∗}E(2)-comodule, we see that E(2)_{∗}(U ∧ X) = E(2)_{∗}(U) ⊗_{E(2)_{∗}} E(2)_{∗}(X) = E(2)_{∗}(U) as an E(2)_{∗}E(2)- comodule. So it suffices to show that W X and L_{2}V_{i} are generalized Moore spectra. For L_{2}V_{0}, it is trivial, and for L_{2}V_{1}, it is shown in [10]. L_{2}V_{5} and W X are generalized Moore spectra by Propositions 5.2 and 4.2, respectively. □

6. The homotopy groups of generalized Moore spectra

We also work in L_{2} at the prime three. For a spectrum U, the spectra N_{n}U and M_{n}U are defined in [13] inductively by the cofiber sequence

N_{n}U wM_{n}U wN_{n+1}U,

setting N_{0}U = U and M_{n}U = L_{n}N_{n}U. Then the E(2)-based Adams E_{2}-term for the homotopy groups π_{∗}(M_{2}U) of a generalized Moore spectrum U is isomorphic to

E_{2}^{∗}(M_{2}U) = E_{2}^{∗}(M_{2}V (0)) = H^{∗}M_{1}^{1},

where H^{∗}M for an E(2)_{∗}E(2)-comodule M denotes Ext_{E(2)_{∗}E(2)}^{∗}(E(2)_{∗}, M), and the comodule M_{1}^{1} is the cokernel of the localization map E(2)_{∗}/(3) → v_{1}^{−1}E(2)_{∗}/(3). By observing the decomposition (3.3) of E_{2}^{∗}(M_{2}U) = H^{∗}M_{1}^{1}, we obtain

Lemma 6.1. The E(2)-based Adams differentials d_{5} and d_{9} are trivial on A ⊂ E_{2}^{∗}(M_{2}U) for any generalized Moore spectrum U.

Proof. Suppose that d_{r}(x) = y for an element x ∈ A. Then y ∈ B by (3.4) 1), since filt y ≥ 5. Now apply b_{10}, and we have

0 = d_{r}(xb_{10}) = yb_{10} ∈ E_{r}^{∗}(M_{2}U)

by (3.4) 3) and the naturality of the differential, since b_{10} detects the homotopy element β_{1} ∈ π_{10}(S). If r = 5, then b_{10} acts freely on B, and so y = 0.

If r = 9, then yb_{10} ∈ E_{2}^{∗}(M_{2}U) is zero or killed by an element u under the differential d_{5}. If yb_{10} = 0 in the E_{2}-term, then y = 0 by (3.4) 3). So we assume that d_{5}(u) = yb_{10} for some u ∈ E_{2}^{∗}(M_{2}U). Since filt y ≥ 9, filt u ≥ 6, and so u = wb_{10} for some w ∈ B by (3.4) 2). It follows that d_{5}(w) = y mod Ker b_{10}. Note that Ker b_{10} ⊂ B. Since b_{10} acts freely on B in the E_{5}(= E_{2})-term, we see that Ker b_{10} = 0, which shows that d_{9}(x) = 0.

□

Let P be the proper invertible spectrum constructed in [10], and consider the cofiber sequence S ^{ι_{k}} wP^{k} ^{i_{k}} w W P^{k} ^{j_{k}} wΣS for k = 0, 1, 2, where we abbreviate ι_{P^{k}} by ι_{k}. Then it induces the long exact sequence

π_{∗}(M_{2}S)^{ι_{k}_{∗}}wπ_{∗}(M_{2}P^{k})^{i_{k}_{∗}}wπ_{∗}(M_{2}W P^{k})^{j_{k}_{∗}}wπ_{∗−1}(M_{2}S)^{ι_{k}_{∗}}wπ_{∗−1}(M_{2}P^{k}).

The homotopy groups π_{∗}(M_{2}S) are computed by the E(2)-based Adams spectral sequence with E_{2}-term H^{∗}M_{0}^{2}. Here M_{0}^{2} is the comodule defined by the short exact sequences

0 wE(2)_{∗} w3^{−1}E(2)_{∗} wN_{0}^{1} w0 and 0 wN_{0}^{1} wv_{1}^{−1}N_{0}^{1} wM_{0}^{2} w0

of E(2)_{∗}E(2)-comodules, in which both of the inclusions are the localization maps. Since we have H^{∗}M_{1}^{1} = A ⊕ B, the structure of the E_{2}-term H^{∗}M_{0}^{2} is described by using A and B, which is seen by observing the long exact sequence

0 wH^{0}M_{1}^{1} ^{φ_{∗}}wH^{0}M_{0}^{2} ^{3} wH^{0}M_{0}^{2} ^{δ} wH^{1}M_{1}^{1} w⋯ wH^{s}M_{1}^{1} ^{φ_{∗}}wH^{s}M_{0}^{2} ^{3} wH^{s}M_{0}^{2} ^{δ} wH^{s+1}M_{1}^{1} w⋯

associated to the short exact sequence

(6.2)  0 → M_{1}^{1} →^{φ} M_{0}^{2} →^{3} M_{0}^{2} → 0.

In fact, let \overline{M} for M = A, B denote the module fitting in the exact sequence M →^{φ} \overline{M} →^{3} \overline{M} →^{δ} M. It is shown in [17] that B is decomposed into B_{h} and B_{t} so that φ(B_{h}) = \overline{B} and δ(\overline{B}) = B_{t}. Note that these are isomorphisms, that is, B_{h} ≅ \overline{B} ≅ B_{t}. Then, we obtain

H^{∗}M_{0}^{2} = \overline{A} ⊕ \overline{B}

by [11, Remark 3.11]. The behaviors of the differentials d_{5} and d_{9} on the spectral sequence for π_{∗}(M_{2}V (0)) are studied in [16] as follows:

(6.3)  1) the differentials d_{5} and d_{9} act trivially on A. 2) the survivors of B in the E_{10}-term have the filtration degree less than 13.

Let B̃_{h} and B̃_{t} denote the submodules consisting of the survivors of B_{h} and B_{t} in E_{10}-term, respectively. The properties (6.3) 2) and (3.4) 1) show that the E_{10}-term has the horizontal vanishing line s = 13, which means that the E_{10}-term is the E_{∞}-term. Therefore, we obtain

π_{∗}(M_{2}V (0)) = A ⊕ B̃_{h} ⊕ B̃_{t}.

A similar properties hold for the spectral sequence for π_{∗}(M_{2}S) [17]:

(6.4)  1) the differentials d_{5} and d_{9} act trivially on \overline{A}. 2) the survivors of \overline{B} in the E_{10}-term have the filtration degree less than 13.

Therefore, the same argument as above shows

π_{∗}(M_{2}S) = \overline{A} ⊕ ˜ \overline{B},

where ˜ \overline{B} denotes the submodule consisting of the survivors of \overline{B} in E_{10}-term.

Remark. In [17], the structure of \overline{A} was left undetermined. It is determined in [18].

Under this notation, we determined in [8] the structure of the homotopy groups π_{∗}(P^{k}) for k = 0, 1, 2 as

(6.5)  π_{∗}(M_{2}P^{k}) = \overline{A} ⊕ v_{2}^{9−3k} ˜\overline{B}.

Note that v_{2}^{9} ˜\overline{B} = ˜ \overline{B}. Consider now the cofiber sequence

(6.6)  M_{2}P^{l} ^{ι_{k}∧P^{l}} wM_{2}P^{k+l} ^{i_{k}∧P^{l}} wM_{2}W P^{k} ∧ P^{l} ^{j_{k}∧P^{l}} wM_{2}P^{l}

obtained by smashing P^{l} with the cofiber sequence (4.1). Then we compute the homotopy groups π_{∗}(M_{2}W P^{k} ∧ P^{l}) by the structure (6.5) of π_{∗}(M_{2}P^{k}).

Theorem 6.7. π_{∗}(M_{2}W P^{k} ∧ P^{l}) = A ⊕ v_{2}^{9−3l}B̃_{h} ⊕ v_{2}^{9−3(k+l)}B̃_{t}.

Proof. The cofiber sequence (6.6) induces the long exact sequence

H^{s}M_{1}^{1}  w^{(j_{k}∧P^{l})_{∗}} H^{s}M_{0}^{2}  ^{3} wH^{s}M_{0}^{2}  w^{(i_{k}∧P^{l})_{∗}} H^{s+1}M_{1}^{1}

of the E(2)-based Adams E_{2}-terms. Since there are isomorphisms (i_{k} ∧ P^{l})_{∗}(\overline{B}) = δ(\overline{B}) = B_{t} and (j_{k} ∧ P^{l})_{∗}(B_{h}) = φ(B_{h}) = \overline{B} in the E_{2}-terms, we see that v_{2}^{9−3(k+l)}B̃_{t} ⊕ v_{2}^{9−3l}B̃_{h} is a summand of E_{∞}-term for π_{∗}(M_{2}W P^{k} ∧ P^{l}) by the naturality of the Adams differential. It also shows the same result as (6.3) 2). Therefore, the differentials d_{r} on A are trivial by Lemma 6.1. □

Proof of Theorem 1.4. In [8], we show that P^{2} wP ^{f} wL_{2}V_{1} is a cofiber sequence for the proper invertible spectrum P. It follows that V_{1} ∧ P = W P^{2}, and V_{1} = W P^{2} ∧ P^{2}. Thus the theorem follows from Theorem 6.7.

□

Proof of Theorem 1.5. This is a corollary of Theorem 6.7. □

7. The homotopy groups π_{∗}(L_{2}V_{5})

Let Σ^{85}V_{5} denote the cofiber of B^{(5)} : Σ^{80}V (0) → V (1) as above. By definition, we have the cofiber sequence

Σ^{84}L_{2}V_{5} ^{j_{B^{(5)}}} wΣ^{80}L_{2}V (0) ^{B^{(5)}} wL_{2}V (1) ^{i_{B^{(5)}}} wΣ^{85}L_{2}V_{5}.

Then we obtain the cofiber sequence

L_{2}V (1) ^{φ_{5}} wΣ^{84}M_{2}V_{5} ^{A_{5}}wΣ^{80}M_{2}V (0) ^{∂_{5}} wΣL_{2}V (1)

by Verdier’s axiom. This induces the exact sequence

(7.1)  ⋯ wH^{∗}M_{2}^{0} ^{v_{2}^{−5}/v_{1}} w H^{∗}M_{1}^{1} ^{v_{1}} wH^{∗}M_{1}^{1} ^{δ_{5}} wH^{∗}M_{2}^{0} w⋯,

where δ_{5} = v_{2}^{5}δ for the connecting homomorphism δ associated to the short exact sequence (3.2).

We consider the Adams differentials on B ⊂ E_{2}^{∗}(M_{2}V_{5}) ≅ H^{∗}M_{1}^{1}. For this sake, it suffices to know the behavior on the elements

v_{2}^{3t±1}/v_{1}, v_{2}^{3t+1}h_{10}/v_{1}^{ε}, v_{2}^{3t}ξ/v_{1}^{ε}, and v_{2}^{3t±1}ψ_{1}/v_{1}

for t ∈ ℤ and ε = 1, 2. Indeed, using the multiplicative relations given in [15, Prop. 5.9], the behavior on the other elements is deduced by the actions of the homotopy elements β_{6/3} ∈ π_{82}(S) and β_{1} ∈ π_{10}(S), which are

detected by { \atop v_{2}^{3}b_{11}} ∈ E_{2}^{2,84}(S) and b_{10} ∈ E_{2}^{2,12}(S), respectively. Here, { \atop v_{2}^{3}b_{11}} ≡ v_{2}^{3}b_{11} mod (3, v_{1}).

Lemma 7.2. The behavior of the Adams differentials is given by:

d_{5}(v_{2}^{3t+1}/v_{1}) = (1 − t)v_{2}^{3t−1}h_{11}b_{10}^{2}/v_{1},
d_{5}(v_{2}^{3t−1}/v_{1}) = 0,
d_{9}(v_{2}^{3t+1}h_{10}/v_{1}) = ±t(t − 1)v_{2}^{3t−2}b_{10}^{5}/v_{1},
d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) = tv_{2}^{3t−1}b_{10}^{3}/v_{1} up to sign,
d_{9}(v_{2}^{3t}ξ/v_{1}) = ±(t^{2} − 1)v_{2}^{3t−3}ψ_{0}b_{10}^{4}/v_{1},
d_{5}(v_{2}^{3t}ξ/v_{1}^{2}) = (1 − t)v_{2}^{3t−2}ψ_{0}b_{10}^{2}/v_{1} up to sign,
d_{5}(v_{2}^{3t+1}ψ_{1}/v_{1}) = −tv_{2}^{3t}ξb_{10}^{3}/v_{1} and
d_{5}(v_{2}^{3t−1}ψ_{1}/v_{1}) = 0.

Proof. First consider the elements in the image of v_{2}^{−5}/v_{1}. In the E(2)_{∗}-based Adams spectral sequence for π_{∗}(L_{2}V (1)),

(7.3)

d_{5}(v_{2}^{j}) = \begin{cases}0 & j ≡ 0, 1, 5 (9) \\ −v_{2}^{j−2}h_{11}b_{10}^{2} & j ≡ 3, 4, 8 (9) \\ v_{2}^{j−2}h_{11}b_{10}^{2} & j ≡ 2, 6, 7 (9) \end{cases} d_{5}(v_{2}^{j}ψ_{1}) = \begin{cases}0 & j ≡ 2, 6, 7 (9) \\ −v_{2}^{j−1}ξb_{10}^{3} & j ≡ 0, 1, 5 (9) \\ v_{2}^{j−1}ξb_{10}^{3} & j ≡ 3, 4, 8 (9) \end{cases}

d_{9}(v_{2}^{j}h_{10}) = \begin{cases}v_{2}^{j−3}b_{10}^{5} & j ≡ 3, 4, 8 (9) \\ 0 & otherwise \end{cases} d_{9}(v_{2}^{j}ξ) = \begin{cases}v_{2}^{j−3}ψ_{0}b_{10}^{4} & j ≡ 1, 5, 6 (9) \\ 0 & otherwise \end{cases}

by [15, Prop. 8.4, Prop.9.10]. Note that the undetermined integer k in [15] is shown to be 1 in [4] (see also [3]). By the first two equations, we compute

d_{5}(v_{2}^{3t+1}/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{5}(v_{2}^{3t+6}) = (1 − t)(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+4}h_{11}b_{10}^{2}),
d_{5}(v_{2}^{3t−1}/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{5}(v_{2}^{3t+4}) = −(t + 1)(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+2}h_{11}b_{10}^{2}),
d_{5}(v_{2}^{3t+1}ψ_{1}/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{5}(v_{2}^{3t+6}ψ_{1}) = −t(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+5}ξb_{10}^{3}) and
d_{5}(v_{2}^{3t−1}ψ_{1}/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{5}(v_{2}^{3t+4}ψ_{1}) = (1 − t)(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+3}ξb_{10}^{3}).

Since δ_{5}(v_{2}^{3t±1}b_{10}^{2}/v_{1}) = ±v_{2}^{3t±1+4}h_{11}b_{10}^{2} and δ_{5}(v_{2}^{3t±1}ψ_{1}b_{10}^{2}/v_{1}) = ±v_{2}^{3t±1+5}ξb_{10}^{3} are the only relations related to our case by [16, Prop.3.4], we see the first, the second, the seventh and the eighth equalities.

We also compute with (7.3) as

d_{9}(v_{2}^{3t+1}h_{10}/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{9}(v_{2}^{3t+6}h_{10}) = ±t(t − 1)(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+3}b_{10}^{5}) and
d_{9}(v_{2}^{3t}ξ/v_{1}) = (v_{2}^{−5}/v_{1})_{∗}d_{9}(v_{2}^{3t+5}ξ) = ±(t^{2} − 1)(v_{2}^{−5}/v_{1})_{∗}(v_{2}^{3t+2}ψ_{0}b_{10}^{4}).

Furthermore, δ_{5}(v_{2}^{3t+1}h_{10}b_{10}^{4}/v_{1}^{2}) = v_{2}^{3t+5}b_{10}^{5} and δ_{5}(v_{2}^{3t}ξb_{10}^{4}/v_{1}^{2}) = −v_{2}^{3t+4}ψ_{0}b_{10}^{4} are the only relations, and we see the third and fifth equalities.

These arguments are not applied to the fourth and the sixth equalities. Since d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) ∈ E_{2}^{6,16(3t+1)}(L_{2}V_{5}) = (ℤ/3){v_{2}^{3t−1}b_{10}^{3}/v_{1}}, we put d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) = kv_{2}^{3t−1}b_{10}^{3}/v_{1} for some k ∈ ℤ/3. (v_{2}^{−5}/v_{1})_{∗}(kv_{2}^{3t+4}b_{10}^{3}) = kv_{2}^{3t−1}b_{10}^{3}/v_{1} and d_{5}(v_{2}^{3t+4}b_{10}^{3}) = −(1 + t)v_{2}^{3t+2}h_{11}b_{10}^{5}. On the other hand, δ_{5}(d_{9}(v_{2}^{3t+1}h_{10}/v_{1})) = t(t + 1)δ_{5}(v_{2}^{3t−2}b_{10}^{5}/v_{1}) = t(t + 1)v_{2}^{3t+2}h_{11}b_{10}^{5}. Therefore, k(t + 1) = t(t + 1) ∈ ℤ/3 up to sign. It follows that k = t ∈ ℤ/3 if t ≠ 2 ∈ ℤ/3. If t = 2 ∈ ℤ/3, then ∂_{5}_{∗}(v_{2}^{9s+7}h_{10}/v_{1}) = kv_{2}^{9s+10}b_{10}^{3}. By applying b_{10}^{3},

kv_{2}^{9s+10}b_{10}^{6} = ∂_{5}_{∗}(v_{2}^{9s+7}h_{10}b_{10}^{3}/v_{1}) = ∂_{5}_{∗}(d_{5}(v_{2}^{9s+7}b_{11}/v_{1}))
= d_{9}(δ_{5}(v_{2}^{9s+7}b_{11}/v_{1})) = d_{9}(v_{2}^{9s+13}h_{10}b_{10})
= v_{2}^{9s+10}b_{10}^{6}

up to sign by [16, Prop. 8.5], and k ≠ 0. Thus the fourth equality follows. In the same manner as this, we obtain the sixth. In fact, δ_{5}(d_{9}(v_{2}^{3t}ξ/v_{1})) = t(t − 1)v_{2}^{3t}ξb_{11}b_{10}^{4} = ktv_{2}^{3t}ξb_{11}b_{10}^{4} = d_{5}(kv_{2}^{3t+3}ψ_{0}b_{10}^{2}) if we put d_{5}(v_{2}^{3t}ξ/v_{1}^{2}) = kv_{2}^{3t−2}ψ_{0}b_{10}^{2}/v_{1} for some k ∈ ℤ/3. Thus, k = t − 1 if t ≠ 0. If t = 0, we see that k ≠ 0 by the equation kv_{2}^{9s+3}ψ_{0}b_{10}^{5} = ∂_{5}_{∗}(d_{5}(v_{2}^{9s+1}ψ_{1}/v_{1})) = d_{9}(δ_{5}_{∗}(v_{2}^{9s+1}ψ_{1}/v_{1})) = v_{2}^{9s+3}ψ_{0}b_{10}^{5} up to sign. □

Proof of Theorem 1.6. For the homotopy groups π_{∗}(L_{2}V (0)), the Adams differentials act as follows:

\begin{aligned}d_{5}(v_{2}^{3t+1}/v_{1}) & = & −tv_{2}^{3t−1}h_{11}b_{10}^{2}/v_{1}, \\ d_{5}(v_{2}^{3t−1}/v_{1}) & = & 0, \\ d_{9}(v_{2}^{3t+1}h_{10}/v_{1}) & = & ±t(t + 1)v_{2}^{3t−2}b_{10}^{5}/v_{1}, \\ d_{5}(v_{2}^{3t+1}h_{10}/v_{1}^{2}) & = & tv_{2}^{3t−1}b_{10}^{3}/v_{1}, \end{aligned}  \begin{aligned}d_{9}(v_{2}^{3t}ξ/v_{1}) & = & ±t(t − 1)v_{2}^{3t−3}ψ_{0}b_{10}^{4}/v_{1}, \\ d_{5}(v_{2}^{3t}ξ/v_{1}^{2}) & = & (1 − t)v_{2}^{3t−2}ψ_{0}b_{10}^{2}/v_{1}, \\ d_{5}(v_{2}^{3t+1}ψ_{1}/v_{1}) & = & −(t + 1)v_{2}^{3t}ξb_{10}^{3}/v_{1}, \\ d_{5}(v_{2}^{3t−1}ψ_{1}/v_{1}) & = & 0. \end{aligned}

We decompose B into eight summands, each of which is generated by the one of the above eight elements as E(2)_{∗}[β_{1}, β_{6/3}]/(β_{6/3}^{2} − v_{2}^{9}β_{1}^{2})-modules. We name them F_{h}^{0}, F_{t}^{0}, F_{h}^{1}, F_{t}^{1}, F_{h}^{∗}^{0}, F_{t}^{∗}^{0}, F_{h}^{∗}^{1} and F_{t}^{∗}^{1}, respectively. Corresponding ones of L_{2}V_{5}, we name them G instead of F. Then Lemma 7.2 shows the isomorphisms

\begin{matrix}G_{h}^{0} = v_{2}^{3}F_{h}^{0}, G_{t}^{0} = F_{t}^{0}, G_{h}^{1} = v_{2}^{3}F_{h}^{1}, G_{t}^{1} = F_{t}^{1}, \\ G_{h}^{∗}^{0} = v_{2}^{3}F_{h}^{∗}^{0}, G_{t}^{∗}^{0} = F_{t}^{∗}^{0}, G_{h}^{∗}^{1} = v_{2}^{3}F_{h}^{∗}^{1} and G_{t}^{∗}^{1} = F_{t}^{∗}^{1} \end{matrix}

as spectral sequences. Since B = ⨁_{x=h,t,ε=0,1}G_{x}^{ε} ⊕ G_{x}^{∗}^{ε}, the E_{10}-term has the same horizontal vanishing line as the one for π_{∗}(L_{2}V (0)). It follows that every element of A survives to the E_{∞}-term by Lemma 6.1. □

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Ippei Ichigi, Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan E-mail address: 95sm004@math.kochi-u.ac.jp

Katsumi Shimomura, Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan E-mail address: katsumi@math.kochi-u.ac.jp