E(2)-INVERTIBLE SPECTRA SMASHING WITH THE SMITH-TODA SPECTRUM V (1) AT THE PRIME 3

IPPEI ICHIGI AND KATSUMI SHIMOMURA

Abstract. Let L_{2} denote the Bousfield localization functor with respect to the 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}. Hovey and Sadofsky [4] showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L_{2}S^{0} at a prime p > 3. At the prime 3, it is shown in [5] that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L_{2}S^{0}. In this paper, we show the homotopy equivalence v_{2}^{3} : Σ^{48}L_{2}V (1) ≃ V (1) ∧ X for the Smith-Toda spectrum V (1). In the same manner as this, we also show the existence of the self map β: Σ^{144}L_{2}V (1) → L_{2}V (1) that induces v_{2}^{9} on the E(2)_{∗}-homology.

Introduction

Let S_{p} denote the stable homotopy category of spectra localized away from the prime number p, and E(n), the Johnson-Wilson spectrum such that π_{∗}(E(n)) = E(n)_{∗} = v_{n}^{−1}Z_{(p)}[v_{1}, …, v_{n}]. We denote L_{n} the full subcategory of E(n)-local spectra, and have the Bousfield localization functor L_{n}: S_{p} → L_{n} ⊂ S_{p} with respect to E(n). We call a spectrum X ∈ L_{n} (E(n)-)invertible if there exists a spectrum Y ∈ L_{n} such that X ∧ Y = L_{n}S^{0}. In [4], Hovey and Sadofsky showed that every E(n)-invertible spectrum is homotopy equivalent to a suspension of L_{n}S^{0} if n^{2} + n < 2p − 2, and that every E(1)-invertible spectrum is homotopy equivalent to a suspension of L_{1}S^{0} or L_{1}QM if p = 2. Here QM denotes the so-called question mark complex S^{0} ∪_{2} e^{1} ∪_{η} e^{3}. In [5], Kamiya and the second author constructed an E(2)-invertible spectrum X such that X ≄ Σ^{k}L_{2}S^{0} for any k ∈ Z and X ∧ X ∧ X = L_{2}S^{0} at the prime 3. Unfortunately we do not know whether X is an E(2)-localization of a finite spectrum. This case is different from the case where p = 2 and n = 1. These spectra L_{1}QM and X are, so far, the only known examples of E(n)-invertible spectra other than the sphere spectrum. For QM at the prime 2, there is a homotopy equivalence v_{1}^{−2} : L_{1}V (0) → Σ^{4}L_{1}V (0) ∧ QM (see Proposition 3.3), where V (0) denotes the mod 2 Moore spectrum. This is deduced from the structure of the homotopy groups π_{∗}(L_{1}QM). We study here the invertible spectrum X in L_{2} at the prime 3. Consider the E(2)-based Adams spectral sequence E_{r}^{s,t}(W) for a spectrum W converging to π_{∗}(L_{2}W). Then it is shown [5] that d_{5}(g) = v_{2}^{−2}h_{11}b_{10}^{2}g ∈ E_{5}^{5,4}(X) for the generator g of E_{2}^{0,0}(X) = Z_{(p)}. We compute the E_{∞}-term E_{∞}^{∗,∗}(V (1)∧X), and then determine the homotopy groups π_{∗}(V (1)∧X) (see Corollary 2.13). This shows that v_{2}^{3}g detects a homotopy element v_{2}^{3} ∈ π_{∗}(V (1) ∧ X). Furthermore, this extends to the map Σ^{48}L_{2}V (1) → V (1) ∧ X.

2000 Mathematics Subject Classification. Primary 55Q99; Secondary 55Q45, 55Q51.

Key words and phrases. Invertible spectrum, Smith-Toda spectrum, Homotopy groups.

Theorem A. There is a homotopy equivalence v_{2}^{3} : Σ^{48}L_{2}V (1) ≃ V (1) ∧ X.

Here, the map v_{2}^{k} denotes a map f such that E(2)_{∗}(f) = v_{2}^{k}. In the same manner as we obtain the map v_{2}^{3}, we also obtain the self map v_{2}^{9}.

Theorem B. There is a homotopy equivalence v_{2}^{9} : Σ^{144}L_{2}V (1) → L_{2}V (1).

Recently, M. Behrens and S. Pemmaraju show the existence of the self map v_{2}^{9} : Σ^{144}V (1) → V (1) [1].

The β-element β_{s} ∈ ^{A}E_{2}^{2}(S^{0}) for an integer s > 0 is defined to be the image of v_{2}^{s} ∈ ^{A}E_{2}^{0}(V (1)) under the composite of the connecting homomorphisms AE_{2}^{0}(V (1)) → ^{A}E_{2}^{1}(V (0)) and ^{A}E_{2}^{1}(V (0)) → ^{A}E_{2}^{2}(S^{0}). Here ^{A}E_{r}^{∗}(W) denotes the E_{r}-term of the Adams-Novikov spectral sequence converging to π_{∗}(W). In [9], Oka showed how to prove the ‘if’ part of the conjecture of Ravenel’s: The element β_{s} ∈ ^{A}E_{2}^{2}(S^{0}) for an integer s survives to π_{∗}(S^{0}) if and only if s ≡ 0, 1, 2, 3, 5, 6 mod 9. The ‘only if’ part is shown in [11, Th. F]. In Oka’s arguments, the self map v_{2}^{9} : Σ^{144}V (1) → V (1) plays the principal role. Here we play the same game in π_{∗}(L_{2}S^{0}).

Corollary C([13]). The element β_{s} ∈ E_{2}^{2}(S^{0}) for an integer s survives to π_{∗}(L_{2}S^{0}) if and only if s ≡ 0, 1, 2, 3, 5, 6 mod 9.

Note that it is, so far, not known whether β_{s} ∈ ^{A}E_{2}(S^{0}) for s ≡ 3 mod 9 survives to π_{∗}(S^{0}), even if there is the self map v_{2}^{9} : Σ^{144}V (1) → V (1) in [1].

By the definition of β_{s}, Theorem A seems to indicate that if the element β_{s} survives to π_{∗}(L_{2}S^{0}), then β_{s+3} survives to π_{∗}(X).

Corollary D. The element β_{s+3} ∈ E_{2}^{2}(X) for an integer s survives to π_{∗}(X) if s ≡ 0, 1, 2, 5, 6 mod 9.

Note that the non-existence of β_{s+3} for s ≡ 4, 7, 8 mod 9 follows from [11, Th. F] together with the equivalence v_{2}^{3} in Theorem A. Here we do not conclude the case where s ≡ 3 mod 9 (see Remark 3.6).

In the next section, we consider the self maps on L_{2}V (1) using ring spectra V (1)_{k} with k > 1 and show Theorem B and Corollary C. In section 2, we recall some facts on invertible spectra and show that v_{2}^{3}g: E_{r}^{∗}(V (1)) → E_{r}^{∗}(V (1) ∧ X) is an isomorphism of spectral sequences, which induces an isomorphism of homotopy groups π_{∗}(L_{2}V (1)) ≅ π_{∗}(V (1) ∧ X). In the last section, we verify the equivalences Σ^{−4}L_{1}V (0)∧QM = L_{1}V (0) for the invertible spectra QM at the prime 2. Then we construct a map v_{2}^{3}g: L_{2}V (1) → V (1) ∧ X by the use of the result of the previous sections, which shows Theorem A. In the last, we show Corollary D.

E(2)-INVERTIBLE SPECTRA SMASHING WITH SMITH-TODA SPECTRUM V (1) 3

1. The self maps on the spectrum L_{2}V (1)_{k}

Let V (0) denote the mod 3 Moore spectrum and V (1)_{k} be a cofiber of α^{k}: Σ^{4k}V (0) → V (0) for the Adams map α: Σ^{4}V (0) → V (0). Here V (1)_{1} = V (1) is the Smith-Toda spectrum. Then we have cofiber sequences

(1.1)  \begin{matrix}S^{0} ⟶^{3} S^{0} ⟶^{i} V (0) ⟶^{j} V (1) and \\ Σ^{4k}V (0) ⟶^{α^{k}} V (0) ⟶^{i_{k}} V (1)_{k} ⟶^{j_{k}} Σ^{4k+1}V (0). \end{matrix}

In [8, Th. 5.6], Oka showed that V (1)_{k} is a ring spectrum for k > 1, in which a ring spectrum means a spectrum V equipped with a unit map ι: S^{0} → V and a multiplication μ: V ∧V → V such that μ(ι∧1_{V}) = 1_{V} = μ(1_{V} ∧ι). In other words, a ring spectrum here is not assumed to satisfy the associative law. By a module spectrum, we also do not assume the associative law. Lemma 1.2. V (1) is a V (1)_{k}-module spectrum with ν_{k}: V (1) ∧ V (1)_{k} → V (1) for k = 2, 4.

Proof. Consider the exact sequence

[V (1) ∧ V (1)_{k}, V (1)]_{0} ⟶^{i_{k}^{∗}} [V (1) ∧ V (0), V (1)]_{0} ⟶^{(α^{k})^{∗}} [V (1) ∧ V (0), V (1)]_{4k}

associated to the cofiber sequence Σ^{4k}V (0) →^{α^{k}} V (0) →^{i_{k}} V (1)_{k}. It is shown in [16, Th.6.11] that [V (1), V (1)]_{l} = 0 if l = 8, 9, 16, 17. Therefore, [V (1)∧V (0), V (1)]_{4k} = 0 if k = 2, 4, and so the V (0)-module structure ν ∈ [V (1) ∧ V (0), V (1)]_{0} is pulled back to a V (1)_{k}-module structure ν_{k} ∈ [V (1) ∧ V (1)_{k}, V (1)]_{0}. □

Let ĩ_{k} and j̃_{k} be the maps in the cofiber sequence

(1.3)  Σ^{4k}V (1) \underrightarrow{α̃^{k}} V (1)_{k+1} \underrightarrow{ĩ_{k}} V (1)_{k} \underrightarrow{j̃_{k}} Σ^{4k+1}V (1),

obtained from 3 × 3 Lemma (Verdier’s axiom), and i_{0} = i_{1}i: S^{0} → V (1) denote the inclusion to the bottom cell. Then Lemma 1.4. ν_{2}(i_{0} ∧ 1) = ĩ_{1} + kβ′i_{0}jj_{2} for some k ∈ Z/3.

Proof. ν_{2}(i_{0} ∧ 1)i_{2} = ν_{2}(1 ∧ i_{2})(i_{0} ∧ 1) = ν(i_{0} ∧ 1) = i_{1} and ĩ_{1}i_{2} = i_{1}. It follows that ν_{2}(i_{0} ∧ 1) − ĩ_{1} ∈ [V (1)_{2}, V (1)]_{0} is in the image of (j_{2})^{∗}: [V (0), V (1)]_{9} → [V (1)_{2}, V (1)]_{0}. Since [V (0), V (1)]_{9} = Z/3{β′i_{0}j} by [16, Prop.6.9], we have the desired equation. □

Lemma 1.5. Let U be an invertible spectrum with E(2)_{∗}(U) = E(2)_{∗}, and ξ, a homotopy element of π_{∗}(V (1)_{2} ∧ U) that induces v_{2}^{k} for k ∈ Z on E(2)_{∗}-homology. Then it induces the map ξ̂: V (1) → V (1) ∧ U such that E(2)_{∗}(ξ̂) = v_{2}^{k}.

Proof. The map ξ̂ is defined as the composite Σ^{|ξ|}V (1) = V (1) ∧ S^{|ξ|} ⟶^{1∧ξ} V (1) ∧ V (1)_{2} ∧U ⟶^{ν_{2}∧1} V (1)∧U. Since E(2)_{∗}(j) = 0, we have E(2)_{∗}(ν_{2}(i_{0} ∧1)) = E(2)_{∗}(ĩ_{1}) by Lemma 1.4. Then we compute i_{0}^{∗}(E(2)_{∗}(ξ̂)) = E(2)_{∗}(ξ̂i_{0}) = E(2)_{∗}((ν_{2} ∧ 1)(1 ∧ ξ)i_{0}) = E(2)_{∗}((ν_{2} ∧ 1)(i_{0} ∧ 1)ξ) = E(2)_{∗}((ĩ_{1} ∧ 1)ξ) = ĩ_{1}_{∗}(v_{2}^{k}) = v_{2}^{k}. Note that i_{0}^{∗} is a monomorphism, and we see the lemma is proved. □

For computing the homotopy groups π_{∗}(L_{2}W) for a spectrum W, we use the E(2)-based Adams spectral sequence E_{r}^{∗}(W) converging to π_{∗}(L_{2}W). Lemma 1.6. v_{2}^{9t} ∈ E_{2}^{∗}(L_{2}V (1)_{3}) for t ∈ Z is a permanent cycle .

Proof. Recall [12] the spectrum C, which is defined to be a cofiber of the localization map V (0) → α^{−1}V (0) = colim_{α}V (0). Then E(2)_{∗}(C) = E(2)_{∗}/(3, v_{1}^{∞}). Since we have a commutative diagram

V (1)_{3}  V (1)_{j}  V (1)_{j−3}  V (1)_{3} V (1)_{3}  V (1)_{j+1}  V (1)_{j−2}  V (1)_{3}^{v_{1}^{j−3}} w  ^{π′_{j}} w_{_{ι_{j}} u}  ^{ι′_{j}} w_{u_{ι_{j−3}}}_{^{v_{1}^{j−2}} w}  _{^{π′_{j+1}} w}  _{^{ι′_{j+1}} w}

of cofiber sequences for j > 3, we obtain a cofiber sequence V (1)_{3} →^{f} C →^{v_{1}^{3}} C by taking homotopy colimits. It is shown in [12] that v_{2}^{9t}/v_{1}^{3} ∈ E_{2}^{∗}(C) is a permanent cycle. Furthermore, we read off from [12] that v_{1}^{3}(v_{2}^{9t}/v_{1}^{3}) = 0 ∈ π_{144t}(L_{2}C), since E_{9}^{s,144t+s}(L_{2}C) = 0 for s > 3. Therefore, v_{2}^{9t}/v_{1}^{3} is pulled back to v_{2}^{9t} ∈ π_{144t}(V (1)_{3}). □

Proof of Theorem B. By Lemma 1.6, we have a homotopy element v_{2}^{9} ∈ π_{144}(L_{2}V (1)_{2}) as the image of v_{2}^{9} ∈ π_{∗}(L_{2}V (1)_{3}) under the map ĩ_{2} : L_{2}V (1)_{3} → L_{2}V (1)_{2} of (1.3). Then this induces the desired self map by Lemma 1.5, which induces an isomorphism on E(2)_{∗}-homology. □

As an application, we consider the β-elements in the homotopy groups π_{∗}(L_{2}S^{0}). In [7], the β-elements β_{s} of the E_{2}-term E_{2}^{2,16s−4}(S^{0}) is defined as the image of v_{2}^{s} ∈ E_{2}^{0,16s}(V (1)) under the composite of the connecting homomorphisms E_{2}^{0,16s}(V (1)) → E_{2}^{1,16s−4}(V (0)) and E_{2}^{1,16s−4}(V (0)) → E_{2}^{2,16s−4}(S^{0}) associated to the cofiber sequences of (1.1). It is shown [13] that the β-element β_{s} ∈ E_{2}^{2,16s−4}(S^{0}) survives to a homotopy element of π_{16s−6}(L_{2}S^{0}) if s ≡ 0, 1, 2, 3, 5, 6 mod 9, which corresponds to the one of Ravenel’s conjectures in π_{∗}(S^{0}). Here we give another proof, which is what Oka showed in [9].

Proof of Corollary C. Since v_{2} and v_{2}^{5} are homotopy elements of π_{∗}(V (1)) [9], we define β-elements as follows:

\begin{matrix}β_{9t} = j_{0}(v_{2}^{9})^{t}, β_{9t+1} = j_{0}(v_{2}^{9})^{t}v_{2}, β_{9t+2} = v_{2}^{∗}(v_{2}^{9})^{t}v_{2}, \\ β_{9t+5} = j_{0}(v_{2}^{9})^{t}v_{2}^{5} and β_{9t+6} = v_{2}^{∗}(v_{2}^{9})^{t}v_{2}^{5}, \end{matrix}

where v_{2}^{9} is the one in Theorem B, j_{0} : V (1) → S^{6} is the projection to the top cell, and v_{2}^{∗}: Σ^{10}V (2) → S^{0} is the Spanier-Whitehead dual of v_{2}. It follows from the Geometric Boundary Theorem (cf. [10, Th. 2.3.4]) that each β-element β_{s} in the E_{2}-term survives to the homotopy element β_{s} defined above.

Since V (1)_{3} is a ring spectrum and there is a map v_{2}^{9t}: Σ^{144t}S^{0} → L_{2}V (1)_{3} by Lemma 1.6, we have the self map v_{2}^{9t} as the composite Σ^{144t}V (1)_{3} = V (1)_{3} ∧

Σ^{144t}S^{0} →^{1∧v_{2}^{9t}} V (1)_{3} ∧ L_{2}V (1)_{3} → L_{2}V (1)_{3}. Oka also showed v_{1}^{2}v_{2}^{3} ∈ π_{56}(V (1)_{3}) in

[9, Lemma 4]. Then the composite Σ^{144t+42}S^{0} →^{v_{1}^{2}v_{2}^{3}} Σ^{144t−14}V (1)_{3} →^{v_{2}^{9t}} Σ^{−14}L_{2}V (1)_{3}

→^{j′_{0}} L_{2}S^{0} gives a homotopy element, which is shown to be detected by the element β_{9t+3} ∈ E_{2}^{2,144t+44}(S^{0}) by the Geometric Boundary Theorem. Here j′_{0} is the projection to the top cell. □

2. The homotopy groups π_{∗}(V (1) ∧ X)

Let E(n) denote the Johnson-Wilson spectrum, and L_{n} the category of the E(n)- local spectra. Then we have the Bousfield localization functor L_{n}: S_{p} → L_{n} with respect to E(n), where S_{p} denotes the category of (p)-local spectra. We call a spectrum U ∈ L_{n} invertible if there exists a spectrum U′ ∈ L_{n} such that U ∧ U′ = L_{n}S^{0}. Let Pic(L_{n}) denote the collection of isomorphism classes of invertible spectra. Then in [4], Hovey and Sadofsky showed that Pic(L_{n}) is a group with multiplication given by [U] · [V] = [U ∧ V] for invertible spectra U and V. Here [U] denotes the isomorphism class of U. Since [L_{n}S^{k}] ∈ Pic(L_{n}), Pic(L_{n}) is the direct sum of Z = {L_{n}S^{k}|k ∈ Z} and a subgroup Pic(L_{n})^{0}. We write E_{r}^{s,t}(W) for a spectrum W as the E_{r}-term of the E(n)-based Adams spectral sequence converging to π_{∗}(L_{n}W). It is further shown in [4] that E(n)_{∗}(U) for an invertible spectrum U is isomorphic to E(n)_{∗} as an E(n)_{∗}E(n)-comodule. It follows that the E_{2}-term E_{2}^{s,t}(U) is isomorphic to the E_{2}-term E_{2}^{s,t}(S^{0}). In [5] (cf. [6]) it is shown that there is a descending filtration {F_{r}} of Pic(L_{n})^{0} so that a monomorphism

(2.1)  φ_{r}: F_{r}/F_{r+1} → E_{r}^{r,r−1}(S^{0})

is defined for each r > 1 by assigning an isomorphism class [U] to d_{r}(g) ∈ E_{r}^{r,r−1}(U) = E_{r}^{r,r−1}(S^{0}), where g is the generator of E_{2}^{0,0}(U) = E_{2}^{0,0}(S^{0}) = Z_{(p)}.

Now turn to the case where p = 3 and n = 2. Consider the chromatic comodules N_{0}^{0} = E(2)_{∗}, M_{0}^{0} = 3^{−1}E(2)_{∗}, N_{0}^{1} = E(2)_{∗}/(3^{∞}), M_{0}^{1} = v_{1}^{−1}N_{0}^{1} and N_{0}^{2} = M_{0}^{2} = E(2)_{∗}/(3^{∞}, v_{1}^{∞}) that fit in the exact sequences

0 ⟶ N_{0}^{0} ⟶ M_{0}^{0} ⟶ N_{0}^{1} ⟶ 0 and 0 ⟶ N_{0}^{1} ⟶ M_{0}^{1} ⟶ M_{0}^{2} ⟶ 0.

These associate the long exact sequences

(2.2)  \begin{matrix}0 ⟶ H^{0}N_{0}^{0} ⟶ H^{0}M_{0}^{0} ⟶ H^{0}N_{0}^{1} ⟶^{δ} H^{1}N_{0}^{0} ⟶ ⋯ and \\ 0 ⟶ H^{0}N_{0}^{1} ⟶ H^{0}M_{0}^{1} ⟶ H^{0}M_{0}^{2} ⟶^{δ′} H^{1}N_{0}^{1} ⟶ ⋯, \end{matrix}

where H^{k}M = Ext_{E(2)_{∗}E(2)}^{k}(E(2)_{∗}, M) for an E(2)_{∗}E(2)-comodule M. Then the universal Greek letter map η: H^{k}M_{0}^{2} → H^{k+2}N_{0}^{0} = E_{2}^{∗}(S^{0}) is defined as the composite η = δδ′. The E_{2}-term E_{2}^{∗}(S^{0}) = H^{∗}N_{0}^{0} is given in [14] (cf. [13]) by using (2.2). In particular, E_{2}^{5,4}(S^{0}) = Z/3{η(v_{2}^{−1}h_{11}b_{10}/3v_{1}), η(v_{2}^{−1}ξζ_{2}/3v_{1})}. In [5], we show that there is an invertible spectrum X ∈ L_{2} such that φ_{5}([X]) = cη(v_{2}^{−1}h_{11}b_{10}/3v_{1}). In other words,

(2.3)  d_{5}(g) = cη(v_{2}^{−1}h_{11}b_{10}/3v_{1})g ∈ E_{5}^{5,4}(X)

for the generator g ∈ E_{2}^{0,0}(X). Here c is the non-zero element of Z/3 that appears in the Toda differential

(2.4)  d_{5}(β_{3/3}) = cα_{1}β_{1}^{3},

where the elements β_{3/3}, α_{1} and β_{1} are defined by

β_{3/3} = η(v_{2}^{3}/3v_{1}^{3}), α_{1} = δ(v_{1}/3) and β_{1} = η(v_{2}/3v_{1}).

The E_{2}-term E_{2}^{∗,∗}(V (1)) of the E(2)-based Adams spectral sequence converging to the homotopy groups π_{∗}(L_{2}V (1)) is isomorphic to

(2.5)  (F ⨁F^{∗})⨂K(2)_{∗}[b_{10}]⨂Λ(ζ_{2})

as a K(2)_{∗}-module. Here, K(2)_{∗} = Z/3[v_{2}^{±1}],

F = Z/3{1, h_{10}, h_{11}, b_{11}}, F^{∗} = Z/3{ξ, ψ_{0}, ψ_{1}, b_{11}ξ},

and the degrees of the generators are:

\begin{matrix}|1| = 0, |h_{10}| = 3, |h_{11}| = 11, |b_{10}| = 10, |b_{11}| = 34, \\ |ξ| = 6, |ψ_{0}| = 13, |ψ_{1}| = 21, and |ζ_{2}| = −1. \end{matrix}

Let (i_{0})_{∗}: E_{2}^{∗}(S^{0}) → E_{2}^{∗}(V (1)) be the induced map from the inclusion i_{0} : S^{0} → V (1). Then

(i_{0})_{∗}(β_{3/3}) = b_{11}, (i_{0})_{∗}(α_{1}) = h_{10} and (i_{0})_{∗}(β_{1}) = b_{10}.

Since E(2)∧X = E(2), the generator g ∈ E_{2}^{0}(X) induces an isomorphism g: E_{2}^{∗}(V (1)) ≅ E_{2}^{∗}(V (1) ∧ X). By the structure (2.5), we see the following Lemma 2.6. v_{2}^{3}g induces an isomorphism E_{2}^{s,t}(V (1)) ≅ E_{2}^{s,t+48}(V (1) ∧ X).

We will show that the map v_{2}^{3}g induces an isomorphism of the differential modules on E_{5}- and E_{9}-terms. Here E_{2}^{∗}(W) = E_{5}^{∗}(W) and E_{6}^{∗}(W) = E_{9}^{∗}(W) for a spectrum W are differential modules with differentials d_{5} and d_{9} of the E(2)-based Adams spectral sequence. Lemma 2.7. β_{3/3}^{2}g(≠ 0) ∈ E_{9}^{4}(X).

Proof. Recall [11, Prop. 5.9] the relations in the E_{2}-term E_{2}^{∗}(V (1)):

(2.8)  v_{2}h_{11}b_{10} = −h_{10}b_{11} and b_{11}^{2} = −v_{2}^{3}b_{10}^{2}.

Send β_{3/3}^{2} to E_{2}^{∗}(V (1)) under the map (i_{0})_{∗}, and we have (i_{0})_{∗}(β_{3/3}^{2}) = b_{11}^{2} = −v_{2}^{3}b_{10}^{2} ≠ 0 ∈ E_{2}^{∗}(V (1)). It follows that b_{11}^{2}g ≠ 0 ∈ E_{2}^{∗}(V (1) ∧ X) and so β_{3/3}^{2}g ≠ 0 ∈ E_{2}^{∗}(X).

From the Toda differential (2.4), the derivation formula on d_{5} induces

d_{5}(β_{3/3}^{2}) = −cα_{1}β_{1}^{3}β_{3/3} ∈ E_{5}^{9}(S^{0}).

By the trivial pairing S^{0} ∧ X → X, we have the derivation formula on d_{5}. Furthermore, the universal Greek letter map η is the one of E_{2}^{∗}(S^{0})-modules. Therefore, by (2.3),

d_{5}(β_{3/3}^{2}g) = −cα_{1}β_{1}^{3}β_{3/3}g + cβ_{3/3}^{2}η(v_{2}^{−1}h_{11}b_{10}/3v_{1})g
= −cα_{1}β_{1}^{3}β_{3/3}g + cη(v_{2}^{−1}h_{11}b_{10}b_{11}^{2}/3v_{1})g.

Here v_{2}^{−1}h_{11}b_{10}b_{11}^{2}/3v_{1} = v_{2}h_{10}b_{11}b_{10}^{2}/3v_{1} by (2.8). Thus,

d_{5}(β_{3/3}^{2}g) = −cα_{1}β_{1}^{3}β_{3/3}g + cη(v_{2}h_{10}b_{11}b_{10}^{2}/3v_{1})g
= −cα_{1}β_{1}^{3}β_{3/3}g + cα_{1}β_{3/3}β_{1}^{2}η(v_{2}/3v_{1})g
= 0.

Since nothing hits β_{3/3}^{2}g under the differential d_{5} by degree reason, we obtain β_{3/3}^{2}g ≠ 0 ∈ E_{9}^{4}(X). □

By the trivial pairing V (1) ∧ X → V (1) ∧ X, we obtain the derivation formula:

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

for x ∈ E_{r}^{s,t}(V (1)) and y ∈ E_{r}^{∗}(X) (cf. [10, Th.2.3.3]). Lemma 2.10. v_{2}^{3}g induces an isomorphism E_{9}^{s,t}(V (1)) ≅ E_{9}^{s,t+48}(V (1) ∧ X).

Proof. By (2.9) and Lemma 2.7, we see that d_{5}(xb_{11}^{2}g) = d_{5}(xβ_{3/3}^{2}g) = d_{5}(x)β_{3/3}^{2}g = d_{5}(x)b_{11}^{2}g for x ∈ E_{2}^{∗}(V (1)). Since b_{11}^{2} = −v_{2}^{3}b_{10}^{2}, we obtain d_{5}(xb_{10}^{2}v_{2}^{3}g) = d_{5}(x)b_{10}^{2}v_{2}^{3}g. Since b_{10} is a polynomial generator, b_{10} acts monomorphically, and so we have

d_{5}(xv_{2}^{3}g) = d_{5}(x)v_{2}^{3}g.

This shows that v_{2}^{3}g is a map of differential modules and induces the desired isomorphism. □

Lemma 2.11. v_{2}^{3}g induces an isomorphism E_{∞}^{s,t}(V (1)) ≅ E_{∞}^{s,t+48}(V (1) ∧ X).

Proof. Since E_{2}^{13,80}(V (1) ∧ X) = Z/3{v_{2}^{−1}b_{11}b_{10}^{5}ζ_{2}g} and d_{5}(v_{2}^{−1}b_{11}b_{10}^{5}ζ_{2}g) = d_{5}(v_{2}^{−4}b_{11}b_{10}^{5}ζ_{2})v_{2}^{3}g = cv_{2}^{−4}h_{10}b_{10}^{8}ζ_{2}(v_{2}^{3}g) ≠ 0 by [11, Prop.9.9, Cor. 10.4], we obtain E_{9}^{13,80}(V (1) ∧ X) = 0. Therefore, (i_{0})_{∗}(d_{9}(β_{3/3}^{2}g)) = 0. If we show that

(2.12) d_{9}(xb_{10}) = yb_{10} implies d_{9}(x) = y in E_{9}^{∗}(V (1) ∧ X),

then we see that v_{2}^{3}g induces an isomorphism E_{13}^{s,t}(V (1)) ≅ E_{13}^{s,t+48}(V (1) ∧ X) in the same way as Lemma 2.10. Since E_{13}^{s}(V (1)) = 0 if s > 12, d_{13} = 0 and so E_{13}^{∗}(V (1) ∧ X) = E_{∞}^{∗}(V (1) ∧ X).

Turn to (2.12). If we assume that d_{9}(xb_{10}) = yb_{10}, then there is an element z ∈ Ker (b_{10} : E_{9}^{s}(V (1) ∧ X) → E_{9}^{s+2}(V (1) ∧ X)) such that d_{9}(x) = y + z. Note that s ≥ 9. Since zb_{10} = 0 ∈ E_{9}^{s+2}(V (1) ∧ X), there is an element w ∈ E_{5}^{s−3}(V (1) ∧ X) such that d_{5}(w) = zb_{10}. By the structure (2.5) of E_{5}(= E_{2})-term, we see that there is an element w′ ∈ E_{5}^{s−5}(V (1) ∧ X) such that w = w′b_{10}, and d_{5}(w′) = z, since b_{10} is a monomorphism on E_{5}-terms. It follows that z = 0 in the E_{9}-term, and we have d_{9}(x) = y as desired. □

Since π_{∗}(V (1) ∧ X) is a Z/3-vector space, there is no extension problem in the spectral sequence. Corollary 2.13. The homotopy groups π_{∗}(V (1) ∧ X) are isomorphic to the E_{∞}- terms for them.

3. Invertible spectra and the Smith-Toda spectra

First we consider the case p = 2 and n = 1. Then it is shown in [4] (cf. [3]) that Pic(L_{1})^{0} = Z/2, whose generator is represented by the E(1)-localization of the question mark complex QM = V (0) ∪_{η} e^{3}, where V (0) = S^{0} ∪_{2} e^{1} is the mod 2 Moore spectrum. Let E_{r}^{∗}(W) for a spectrum W denote the E_{r}-term of the E(1)-based Adams spectral sequence for π_{∗}(L_{1}W). Since L_{1}QM is invertible, the E_{2}-term for π_{∗}(L_{1}V (0) ∧ QM) is isomorphic to that for π_{∗}(L_{1}V (0)):

(3.1)  K(1)_{∗}[h_{10}]⨂Λ(ρ_{1}),

where K(1)_{∗} = Z/2[v_{1}^{±1}]. The map φ_{3} : Pic(L_{1})^{0} = F_{3} → E_{3}^{3,2}(S^{0}) = Z/2{α_{−1}α_{1}^{2}} of (2.1) is an isomorphism such that φ_{3}([L_{1}QM]) = α_{−1}α_{1}^{2}. Here α_{k} for k ∈ Z denotes δ(v_{1}^{k}) for the connecting homomorphism δ: E_{2}^{∗}(V (0)) → E_{2}^{∗+1}(S^{0}) associated to the cofiber sequence S^{0} →^{2} S^{0} →^{i} V (0) and for v_{1}^{k} ∈ E_{2}^{0}(V (0)) = K(1)_{∗}. Note that i_{∗}(α_{−1}) = v_{1}^{−2}h_{10} and i_{∗}(α_{1}) = h_{10} for the induced map i_{∗}: E_{2}^{∗}(S^{0}) → E_{2}(V (0)) from the inclusion i. The E_{2}-term for QM is isomorphic to that for the sphere L_{1}S^{0}, and d_{3}(g) = α_{−1}α_{1}^{2}g ∈ E_{3}^{3}(QM) = E_{2}^{3}(QM) by the definition of φ_{3}. Therefore, the Adams differentials on E_{2}^{∗}(V (0) ∧ QM) are computed by d_{3}(i_{∗}(g)) = v_{1}^{−2}h_{10}^{3}g ∈ E_{3}^{3,2}(V (0) ∧ QM) and the derivation formulas associated to the trivial pairing L_{1}V (0) ∧ QM → L_{1}V (0) ∧ QM. In fact, v_{1}^{−2}g induces the isomorphism E_{r}^{∗}(V (0)) ≅ E_{r}^{∗}(V (0) ∧ QM) of spectral sequences. Therefore, we have

E_{∞}^{∗,∗}(V (0) ∧ QM) = N⨂Z/2[v_{1}^{4}, v_{1}^{−4}]⨂Λ(ρ_{1}),

where N is the module given by

[figure: 0; s; t − s; v_{1}^{−2}; v_{1}^{−2}h_{10}; v_{1}^{−2}h_{10}^{2}; v_{1}^{−1}; v_{1}^{−1}h_{10}; v_{1}^{−1}h_{10}^{2}; (3.2)]

in which each dot denotes Z/2 generated by the indicated element. Therefore, we see that v_{1}^{−2}g ∈ π_{−4}(L_{1}V (0) ∧ QM), and 2v_{1}^{−2}g = 0. Thus we obtain the map v_{1}^{−2}g: Σ^{−4}V (0) → L_{1}V (0) ∧ QM, which induces an isomorphism on E(1)_{∗}- homology. Proposition 3.3. The element v_{1}^{−2}g ∈ π_{−4}(L_{1}V (0) ∧ QM) induces an equivalence L_{1}Σ^{−4}V (0) ≃ L_{1}V (0) ∧ QM.

We will play the same game for the case where p = 3 and n = 2. Lemma 3.4. There is a homotopy element v_{2}^{3}g ∈ π_{48}(V (1)_{i} ∧ X) for i = 1, 2 such that E(2)_{∗}(v_{2}^{3}g) = v_{2}^{3}.

Proof. We have seen that v_{2}^{3}g ∈ π_{48}(V (1) ∧ X) in the previous section. We also see that π_{43}(V (1) ∧ X) = Z/3{v_{2}^{2}h_{11}g}. Therefore, we obtain v_{2}^{3}g ∈ π_{48}(V (1)_{2} ∧ X)

from the exact sequence π_{48}(V (1)_{2} ∧ X) →^{ĩ_{1}_{∗}} π_{48}(V (1) ∧ X) →^{j̃_{1}_{∗}} π_{43}(V (1) ∧ X) associated to the cofiber sequence (1.3) with k = 1. Indeed, v_{2}^{2}h_{11}g has the Adams filtration 1, while δ(v_{2}^{3}g) = 0 for the connecting homomorphism δ corresponding to j̃_{1}. □

Now we have the similar results to Proposition 3.3. Theorem 3.5. The element v_{2}^{3}g ∈ π_{48}(V (1)∧X) induces an equivalence Σ^{48}L_{2}V (1)_{i} ≃ V (1)_{i} ∧ X for i = 1, 2.

Proof. Since V (1)_{2} is a ring spectrum, the element v_{2}^{3}g yields the self map ṽ_{2}^{3}g: V (1)_{2} → V (1)_{2}∧X, which induces an isomorphism on E(2)_{∗}-homology. Therefore, the proposition for i = 2 follows. For i = 1, Lemmas 3.4 and 1.5 show the existence of the map v_{2}^{3}g: Σ^{48}L_{2}V (1) → V (1) ∧ X, which is also an E(2)_{∗}-equivalence. □

Proof of Corollary D. In the same manner as the proof of Corollary C, the βelements are defined as follows:

β_{9t+3}g = (j_{0} ∧ 1_{X})(v_{2}^{3}g)(v_{2}^{9})^{t}, β_{9t+4}g = (j_{0} ∧ 1_{X})(v_{2}^{3}g)(v_{2}^{9})^{t}v_{2},

β_{9t+5}g = (v_{2}^{∗} ∧ 1_{X})(v_{2}^{3}g)(v_{2}^{9})^{t}v_{2}, β_{9t+8}g = (j_{0} ∧ 1_{X})(v_{2}^{3}g)(v_{2}^{9})^{t}v_{2}^{5} and
β_{9t+9}g = (v_{2}^{∗} ∧ 1_{X})(v_{2}^{3}g)(v_{2}^{9})^{t}v_{2}^{5},
□

Remark 3.6. v_{2}^{3}g ∈ E_{2}^{0,48}(V (1)_{3} ∧X) cannot be a permanent cycle. In fact, we see that d_{9}(j̃_{3}_{∗}(v_{2}^{3}g)) = d_{9}(δ(v_{2}^{3}g)) = d_{9}(v_{2}^{2}h_{10}g) = v_{2}^{−1}b_{10}^{5}g ≠ 0, where j̃_{3} is the map of (1.3) with k = 3. Therefore, we cannot tell whether or not β_{9t+6}g survives to a homotopy element of π_{∗}(X) different from Corollary C. Indeed, we do not know if v_{1}^{2}v_{2}^{6}g ∈ π_{∗}(V (1)_{3} ∧ X).

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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

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

E-mail address: katsumi@math.kochi-u.ac.jp