A RELATION BETWEEN THE PICARD GROUP OF THE E(n)-LOCAL HOMOTOPY CATEGORY AND E(n)-BASED ADAMS SPECTRAL SEQUENCE

YOUSUKE KAMIYA AND KATSUMI SHIMOMURA

Abstract. Let E(n) be the n-th Johnson-Wilson spectrum at the prime p, and Pic(L_{n}) denotes the abelian group consisting of invertible spectra of the E(n)-local stable homotopy category L_{n} with multiplication defined by the smash product. Hovey and Sadofsky showed in [5] that Pic(L_{n}) is a set, the decomposition Pic(L_{n}) = Z ⊕ Pic(L_{n})^{0}, that Pic(L_{n})^{0} = 0 if 2p − 2 > n^{2} + n, and that Pic(L_{1})^{0} = Z/2 at the prime 2. In this paper, we generalize their results to #Pic(L_{n})^{0} ≤ #(⨁_{s>2} E_{s}^{s,s−1}(S^{0})), where E_{s}^{s,s−1}(S^{0}) denotes the E_{s}-term of the E(n)-based Adams spectral sequence converging to the homotopy groups π_{∗}(L_{n}S^{0}) of the E(n)-local sphere L_{n}S^{0}. As an example, we consider the case n = 2 and p = 3, and show that Z/3 ⊂ Pic(L_{2})^{0} ⊂ E_{5}^{5,4} = Z/3 ⊕ Z/3. This is shown by constructing an invertible spectrum that generates Z/3.

1. Introduction

Let S_{(p)} denote the stable homotopy category of spectra localized away from the prime p, and E(n) ∈ S_{(p)} be the n-th Johnson-Wilson spectrum. The spectrum E(n) is characterized by the homotopy groups π_{∗}E(n)(= E(n)_{∗}) = Z_{(p)}[v_{1}, v_{2}, ⋯, v_{n}, v_{n}^{−1}] ⊂ v_{n}^{−1}BP_{∗}, where BP denotes the Brown-Peterson spectrum with BP_{∗} = Z_{(p)}[v_{1}, v_{2}, ⋯]. We call a spectrum X E(n)-local if the homotopy groups [C, X]_{∗} = 0 for any C with E(n)_{∗}(C) = 0. Let L_{n} denote the full subcategory of S_{(p)} consisting of the E(n)-local spectra. Then we 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} invertible if there exists a spectrum Y such that X ∧ Y = L_{n}S^{0}, and denote the collection of isomorphism classes of invertible spectra by Pic(L_{n}). In [5], Hovey and Sadofsky showed that Pic(L_{n}) is a set, and is an abelian group with multiplication given by the smash product. Furthermore, they showed the natural splitting Pic(L_{n}) = Z ⊕ Pic(L_{n})^{0}, where Pic(L_{n})^{0} denotes the subgroup consisting of isomorphism classes of invertible spectra whose HQ_{∗}-homology is Q at dimension 0, and determined the group Pic(L_{n})^{0} to be trivial if 2p − 2 > n^{2} + n, and be Z/2 if n = 1 and p = 2.

In this paper, we generalize these results. Hovey and Sadofsky also showed in [5] that a spectrum X ∈ L_{n} is invertible if and only if it is finite and there is an isomorphism E(n)_{∗}(X) = E(n)_{∗} as an E(n)_{∗}(E(n))-comodule. Theorem 1.1. A spectrum X ∈ L_{n} is invertible if and only if there is an isomorphism E(n)_{∗}(X) = E(n)_{∗} as an E(n)_{∗}(E(n))-comodule.

2000 Mathematics Subject Classification. Primary 55Q99.

Key words and phrases. Johnson-Wilson spectrum, Bousfield localization, Picard group.

Let E(n)_{r}^{s,t}(X) denote the E_{r}-term of the E(n)-based Adams spectral sequence converging to π_{∗}(L_{n}X). Then the theorem implies an isomorphism E(n)_{2}^{∗,∗}(X) = E(n)_{2}^{∗,∗}(S^{0}) for an invertible spectrum X. In particular, 1 ∈ Z_{(p)} = E_{2}^{0,0}(X) for X ∈ Pic(L_{n})^{0}. For each k > 0, Pic(L_{n})_{k}^{0} denotes a semigroup consisting of invertible spectra X ∈ Pic(L_{n})^{0} such that d_{s}(1) = 0 ∈ E(n)_{s}^{s,s−1}(X) for s < kq +1, where q = 2p − 2. Note that Pic(L_{n})_{k}^{0} ⊃ Pic(L_{n})_{k+1}^{0}. Then we define its associated graded group

GPic(L_{n})^{0} = ⨁_{k>0}(Pic(L_{n})_{k}^{0}/Pic(L_{n})_{k+1}^{0})

(see Lemma 2.8).

Theorem 1.2. The associated graded group GPic(L_{n})^{0} is isomorphic to a subgroup of ⨁_{r>2} E(n)_{r}^{r,r−1}(S^{0}).

The E_{r}-terms E(n)_{r}^{∗,∗}(S^{0}) are determined in the cases (a) n^{2} + n < 2p − 2, (b) n = 1, (c) p > 2, n = 2 (cf. [9], [6], [14]) ：

(1.3) The structure of the group ⨁_{r>2} E(n)_{r}^{r,r−1}(S^{0}) is determined as follows:

(a) ⨁_{r>2} E(n)_{r}^{r,r−1}(S^{0}) = 0 if n^{2} + n < 2p − 2.

(b) ⨁_{r>2} E(1)_{r}^{r,r−1}(S^{0}) = \{ \begin{matrix}0 & p > 2 \\ E_{3}^{3,2}(S^{0}) = Z/2 & p = 2 \end{matrix}

(c) ⨁_{r>2} E(2)_{r}^{r,r−1}(S^{0}) = \{ \begin{matrix}0 & p > 3 \\ E_{5}^{5,4}(S^{0}) = Z/3 ⊕ Z/3 & p = 3 \end{matrix}

This together with Theorem 1.2 imply the following:

Corollary 1.4. The structure of the group Pic(L_{n}) is as follows :

(a) ([5]) Pic(L_{n}) = Z if n^{2} + n < 2p − 2

(b) ([5]) Pic(L_{1}) ⊂ Z ⊕ Z/2 if p = 2
(c) Pic(L_{2}) ⊂ Z ⊕ Z/3 ⊕ Z/3 if p = 3

In [5], it is shown that the inclusion of (b) is an isomorphism and Pic(L_{1})^{0} is generated by the E(1)-localization of the so-called “question mark” complex L_{1}QM(≄ L_{1}S^{0}). Now suppose that p = 3 and n = 2. Then E_{2}^{5,4}(S^{0}) = Z/3 ⊕ Z/3 is generated by χ_{1} = v_{2}^{−2}h_{11}b_{10}^{2} and χ_{2} = v_{2}^{−1}ξb_{10}ζ_{2} ([16], cf. [14]).

Proposition 1.5. There exists an invertible spectrum X ∈ L_{2} such that d_{5}(1) = χ_{1} ∈ E(2)_{5}^{5,4}(X) = E(2)_{5}^{5,4}(S^{0}).

The invertible spectrum X is closely related to Toda’s V (1\frac{1}{2}) given in [17]. Indeed, there is a map X → Σ^{−21}L_{2}V (1\frac{1}{2}) that induces the projection E(2)_{∗} → E(2)_{∗}/(3). Note that E(2)_{∗}(Σ^{−21}V (1\frac{1}{2})) = E(2)_{∗}/(3), while L_{2}V (1\frac{1}{2}) is not even a V (0)-module spectrum (cf. [18, Th.6.4]), where V (0) denotes the mod 3 Moore spectrum. We construct X by lifting the E(2)-based Adams tower for Σ^{−21}V (1\frac{1}{2}) to that for X along the projection E(2)_{∗} → E(2)_{∗}/(3) = E(2)_{∗}(Σ^{−21}V (1\frac{1}{2})).

By this proposition and Theorem 1.2, we obtain

Theorem 1.6. For p = 3 and n = 2, Z ⊕ Z/3 ⊂ Pic(L_{2}) ⊂ Z ⊕ Z/3 ⊕ Z/3.

We conjecture that there is no invertible spectrum corresponding to χ_{2}, because if such a spectrum U exists, then the structure of the homotopy groups π_{∗}(V (1)∧U) would be ugly in a sense. But, by contrast, the the authors of [2] seem to believe that the corresponding element exists in Pic(L_{K(2)}) at the prime 3.

PICARD GROUP AND E(n)-ADAMS SPECTRAL SEQUENCE 3

This paper is organized as follows: In the next section, we show Theorems 1.1 and 1.2. We recollect some properties of v_{n}^{−1}BP including a change of rings theorem obtained from the results of Hovey and Sadofsky in section three. In section four, we introduce v_{2}^{−1}BP -based Adams m-tower in order to show Proposition 1.5, which is proved in section five.

The authors would like to thank Professors Hans-Werner Henn and Paul Goerss for a thorough reading of a previous draft of the this paper, and for several helpful communications.

2. Picard group Pic(L_{n}) and E(n)-based Adams spectral sequence

In this section, we fix a non-negative integer n. Consider the E(n)-based Adams spectral sequence E_{r}^{s,t}(X) converging to the homotopy groups π_{∗}(L_{n}X), whose E_{2}- term is given by E_{2}^{s,t}(X) = Ext_{E(n)_{∗}E(n)} (E(n)_{∗}, E(n)_{∗}(X)). Hopkins and Ravenel showed that there exist integers r_{0}, s_{0} such that E_{r_{0}}^{s,∗}(S^{0}) = 0 for s > s_{0} (cf. [11]).

Consider a collection of the isomorphism classes of spectra:

U(= U(p, n)) = {[L_{n}X] | E(n)_{∗}(X) = E(n)_{∗} as an E(n)_{∗}(E(n))-comodule},

and the subcollection

U^{0} = {[L_{n}X] ∈ U | E_{2}^{0,0}(X) = Z_{(p)}}.

Here [X] denotes the isomorphism class of X. In the following, we write X ∈ U^{0} instead of [X] ∈ U^{0} for the sake of simplicity. For X ∈ U^{0}, the E_{2}-term E_{2}^{∗,∗}(X) is isomorphic to E_{2}^{∗,∗}(S^{0}) by definition, and we denote the generator of E_{2}^{0,0}(X)(= Z_{(p)}) by g_{X}. The element g_{X} is also a generator of E_{2}^{∗,∗}(X) as an E_{2}^{∗,∗}(S^{0})-module. We define subcollections U_{k}’s of U^{0} for k ≥ 0 by

U_{k} = {X ∈ U^{0} | d_{s}(g_{X}) = 0 ∈ E_{s}^{s,s−1}(X) for s < kq + 1}.

Lemma 2.1. Let X ∈ U_{k} for an integer k > 0. Then, E_{kq+1}^{∗,∗}(X) = E_{kq+1}^{∗,∗}(S^{0}).

Proof. By the identity S^{0}∧X → X, we have a natural pairing E_{r}^{∗,∗}(S^{0})⊗E_{r}^{∗,∗}(X) → E_{r}^{∗,∗}(X), which makes the Adams differential a derivation (cf. [10, 2.3.3]). Therefore, d_{s}(xg_{X}) = d_{s}(x)g_{X} + xd_{s}(g_{X}) = d_{s}(x)g_{X} for s < kq + 1 and x ∈ E_{s}^{∗,∗}(S^{0}), which shows that the Adams differentials on E_{s}^{∗,∗}(X) and E_{s}^{∗,∗}(S^{0}) agree for s < kq + 1. □

Corollary 2.2. Let k be an integer such that kq + 1 > s_{0}. Then L_{n}X ≃ L_{n}S^{0} for X ∈ U_{k}.

Proof. Lemma 2.1 implies E_{kq+1}^{∗,∗}(X) = E_{kq+1}^{∗,∗}(S^{0}) for X ∈ U_{k}. Since kq + 1 > s_{0} and E_{r_{0}}^{s,∗} = 0 for s > s_{0}, we see that d_{r}(g_{X}) = 0 for any r > 1. Therefore, g_{X} detects a map g_{X}: S^{0} → X, which induces an isomorphism (g_{X})_{∗}: E(n)_{∗} ≅ E(n)_{∗}X. □

Let k_{0} be the minimal integer such that k_{0}q + 1 > s_{0}. Then U_{k_{0}} = {L_{n}S^{0}} by Corollary 2.2, and we have the finite descending sequence

(2.3)  U^{0} ⊃ U_{1} ⊃ U_{2} ⊃ ⋯ ⊃ U_{k} ⊃ U_{k+1} ⊃ ⋯ ⊃ U_{k_{0}} = {L_{n}S^{0}}.

We will define U_{k}/U_{k+1}, and show it a group. Lemma 2.4. The filtration {U_{k}} of (2.3) satisfies the followings :

(1) If X, Y ∈ U_{k}, then X ∧ Y ∈ U_{k}.

(2) For X ∈ U_{k}, there exists X′ ∈ U_{k} such that X ∧ X′ ∈ U_{k+1}.

(3) For X, Y ∈ U_{k} and X′, Y′ ∈ U_{k} given in (2), if X ∧ Y′ ∈ U_{k+1}, then

Y ∧ X′ ∈ U_{k+1}.
(4) If X ∧ Y, Y ∈ U_{k}, then X ∈ U_{k}.

Proof. Since the Adams differential is a derivation, the first statement follows.

Recall that g_{X} denotes the generator of E_{2}^{0,0}(X), and we see that g_{X∧Y} = g_{X}g_{Y} for X, Y ∈ U^{0}. Indeed, X ∧ Y ∈ U^{0}. If X, Y ∈ U_{k}, then E_{kq+1}^{s,t}(X) = E_{kq+1}^{s,t}(Y) = E_{kq+1}^{s,t}(X ∧ Y) = E_{kq+1}^{s,t}(S^{0}) by Lemma 2.1 and the first statement. Under this equation, xg_{X} = xg_{Y} = xg_{X∧Y} for x ∈ E_{kq+1}^{s,t}(S^{0}). Since the Adams differential acts as a derivation, we see that

(2.5)  d_{kq+1}(g_{X∧Y}) = (x + y)g_{X∧Y} ,

if d_{kq+1}(g_{X}) = xg_{X} and d_{kq+1}(g_{Y}) = yg_{Y} for x, y ∈ E_{kq+1}^{s,s−1}(S^{0}).

Turn to the second. Since X ∈ U_{k}, we may put d_{kq+1}(g_{X}) = xg_{X} ∈ E_{kq+1}^{kq+1,kq}(X) for an element x ∈ E_{kq+1}^{kq+1,kq}(S^{0}). We see that E_{2}^{s,∗}(S^{0}) is a torsion group at s > 0 by observing the chromatic spectral sequence. It follows that the element x has a finite order, which we write m. Let X^{m−1} denote the (m − 1)-fold smash product of X with X^{0} = S^{0}. Then X^{m−1} ∈ U_{k} by the first statement, and d_{kq+1}(g_{X^{m−1}}) = (m − 1)xg_{X^{m−1}} = −xg_{X^{m−1}} by (2.5). Now put X′ = X^{m−1}, and we compute d_{kq+1}(g_{X∧X′}) = (x − x)g_{X∧X′} = 0 by (2.5) as desired. For the later use, we write down

(2.6) If d_{kq+1}(g_{X}) = xg_{X} for X ∈ U_{k}, then d_{kq+1}(g_{X′}) = −xg_{X′}.

Suppose that d_{kq+1}(g_{X}) = xg_{X} and d_{kq+1}(g_{Y}) = yg_{Y}. Then d_{kq+1}(g_{X∧Y′}) = (x−y)g_{X∧Y} by (2.5) and (2.6). The hypothesis X ∧Y′ ∈ U_{k+1} shows d_{kq+1}(g_{X∧Y′}) = 0, and we have x = y. Therefore, d_{kq+1}(g_{X′∧Y}) = (−x + y)g_{X∧Y} = 0, and we obtain the third statement.

For the last statement, we note that X ∈ U^{0}. Indeed, E(n)_{∗} = E(n)_{∗}(X ∧ Y) = E(n)_{∗}(X) ⊗_{E(n)_{∗}} E(n)_{∗}(Y) = E(n)_{∗}(X) as a comodule, and Z_{(p)} = E_{2}^{0,0}(X ∧ Y) = E_{2}^{0,0}(X) ⊗ E_{2}^{0,0}(Y) = E_{2}^{0,0}(X). Suppose that d_{sq+1}(g_{X}) = xg_{X} for s < k and x ∈ E_{sq+1}^{sq+1,sq}(S^{0}). Since d_{sq+1}(g_{X∧Y}) = 0 and d_{sq+1}(g_{Y}) = 0 by the hypothesis, we see 0 = d_{sq+1}(g_{X∧Y}) = xg_{X∧Y} by (2.5), and so x = 0 by Lemma 2.1 as desired. □

Proof of Theorem 1.1. It suffices to show that if X ∈ U^{0}, then X is invertible, since if X ∈ U and HQ_{k}(X) = Q, then Σ^{−k}X ∈ U^{0}. Indeed, X^{−1} = Σ^{−k}(Σ^{−k}X)^{−1}.

For X ∈ U^{0}, put X_{0} = X. Inductively we define X_{i+1} = X_{i} ∧ X′_{i} ∈ U_{i+1} for X_{i} ∈ U_{i} for X′_{i} of Lemma 2.4(2).

Now put X^{−1} = X′_{0} ∧ X′_{1} ∧ ⋯ X′_{k_{0}−1}, and we see that

X ∧ X^{−1} = X_{k_{0}} = L_{n}S^{0}. □

Remark: The spectrum X^{−1} in this proof is the Spanier-Whitehead dual DX = F (X, L_{n}S^{0}) of X ([4, Prop. A.2.8]).

The first statement of Lemma 2.4 shows that U_{k} is a semigroup. The second enables us to define a relation ∼ in U_{k} ：

For X, Y ∈ U_{k}, X ∼ Y if X ∧ Y′ ∈ U_{k+1} for Y′ ∈ U_{k} with Y ∧ Y′ ∈ U_{k+1}.

Lemma 2.7. The relation ∼ defines an equivalence one in U_{k}.

Proof. We first show that the relation is well defined. It suffices to show that the definition does not depend on the choice of Y′ such that Y ∧ Y′ ∈ U_{k+1} given in Lemma 2.4(2). Suppose that Y′ and Y′′ be such spectra. If X ∧ Y′ ∈ U_{k+1}, then X ∧ Y′ ∧ Y ∧ Y′′ ∈ U_{k+1} by Lemma 2.4(1). Therefore, X ∧ Y′′ ∧ Y ∧ Y′ ∈ U_{k+1}, and we see that X ∧ Y′′ ∈ U_{k+1} by Lemma 2.4(4).

By the definition of X′, we see X ∧ X′ ∈ U_{k+1}, and so X ∼ X.

If X ∼ Y, then X ∧ Y′ ∈ U_{k+1}, which implies Y ∧ X′ ∈ U_{k+1} by Lemma 2.4(3), and Y ∼ X.

Suppose that X ∼ Y and Y ∼ Z. Then X ∧ Y′, Y ∧ Z′ ∈ U_{k+1}, and U_{k+1} ∋ X ∧ Y′ ∧ Y ∧ Z′ = X ∧ Z′ ∧ Y ∧ Y′ by Lemma 2.4(1). Since Y ∧ Y′ ∈ U_{k+1}, we see X ∼ Z by Lemma 2.4(4). □

We define U_{k}/U_{k+1} = U_{k}/ ∼. Lemma 2.8. For each k > 0, U_{k}/U_{k+1} is an abelian group with multiplication given by the smash product. Furthermore, it is isomorphic to a subgroup of the E_{kq+1}-term E_{kq+1}^{kq+1,kq}(S^{0}) of the E(n)-based Adams spectral sequence.

Proof. Let [X] ∈ U_{k}/U_{k+1} denote the equivalent class of X ∈ U_{k}. We define a mapping f: U_{k}/U_{k+1} → E_{kq+1}^{kq+1,kq}(S^{0}) by f([X]) = x_{X} for x_{X} given by d_{kq+1}(g_{X}) = x_{X}g_{X}. If X ∼ Y, then X ∧Y′ ∈ U_{k+1}, and 0 = d_{kq+1}(g_{X∧Y′}) = (x_{X} +x_{Y′})g_{X∧Y′} = (x_{X} − x_{Y})g_{X∧Y′} by (2.5) and (2.6), which shows x_{X} = x_{Y}. Thus we see that f is well defined.

Suppose that f([X]) = f([Y]), which we put x. Then d_{kq+1}(Y′) = −xg_{Y′} by (2.6). Therefore, d_{kq+1}(g_{X∧Y′}) = (x − x)g_{X∧Y′} = 0, and so X ∧ Y′ ∈ U_{k+1} and [X] = [Y]. Thus, f is an injection.

Since E_{kq+1}^{kq+1,kq}(S^{0}) is a set, U_{k}/U_{k+1} is a set. Define the multiplication of U_{k}/U_{k+1} by [X][Y] = [X ∧Y]. Then it defines a structure of abelian group. In fact, the associative law and the commutative law follow from those of the smash product. The unit element is [L_{n}S^{0}], and the inverse element of [X] is the equivalence class of X′ such that X ∧ X′ ∈ U_{k+1}.

Since f([X ∧ Y])g_{X∧Y} = d_{kq+1}(g_{X∧Y}) = (x_{X} + x_{Y})g_{X∧Y}, we see that f is a homomorphism. Indeed, f([X][Y]) = f([X ∧ Y]) = x_{X} + x_{Y} = f([X]) + f([Y]). □

Proof of Theorem 1.2. This follows from Lemma 2.8. □

3. Recollection on the spectrum E = v_{n}^{−1}BP

From this section on, E denotes the ring spectrum v_{n}^{−1}BP for a fixed integer n ≥ 0. Lemma 3.1. There exists a set {E_{k}|k ∈ N} of skeletons of BP such that hocolim_{k}E_{k} = E, where the colimit is taken over maps \overline{v_{n}}: E_{k} → E_{k+1} factoring v_{n}: BP → BP.

Proof. We define E_{k} and \overline{v_{n}}: E_{k} → E_{k+1}, inductively. Put E_{1} = S^{0}. Suppose that a skeleton E_{k} of BP is given, and consider the composite E_{k} →^{i_{k}} BP →^{v_{n}} Σ^{−|v_{n}|}BP, where i_{k} is the inclusion. Since BP is a homotopy colimit of its skeletons, there exists a skeleton E_{k+1} and a map \overline{v_{n}}: E_{k} → E_{k+1} such that v_{n}i_{k} = i_{k+1}\overline{v_{n}} for the inclusion i_{k+1} : E_{k+1} → BP. Thus we have a direct system \{E_{k}, \overline{v_{n}}\}, and π_{∗}(hocolim_{k}(E_{k})) = colim_{k}π_{∗}(E_{k}) = E_{∗}. This isomorphism is induced from the map hocolim_{k}(E_{k}) → E, and we obtain the desired isomorphism hocolim_{k}(E_{k}) = E. □

The spectrum E defines the Hopf algebroid (E_{∗}, E_{∗}(E)), and E_{∗}(E) is flat over E_{∗} by the Landweber exact functor theorem. Note that

\begin{matrix}E_{∗}(E) = E_{∗} ⊗_{BP_{∗}} BP_{∗}(BP) ⊗_{BP_{∗}} E_{∗} and \\ E(n)_{∗}(X) = E(n)_{∗} ⊗_{BP_{∗}} BP_{∗}(X) = E(n)_{∗} ⊗_{E_{∗}} E_{∗}(X) \end{matrix}

for a spectrum X. Proposition 3.2. Let X be a spectrum. If E_{∗}(X) ≅ E_{∗} as an E_{∗}(E)-comodule, then L_{n}X is invertible.

Proof. By Theorem 1.2, it suffices to show that E(n)_{∗}(X) ≅ E(n)_{∗} as an E(n)_{∗}(E(n))- comodule. This follows from E(n)_{∗}(X) = E(n)_{∗} ⊗_{E_{∗}} E_{∗}(X) = E(n)_{∗}. □

In [5], Hovey-Sadofsky showed that BP and E(n) are K(j)-equivalent to different wedges of suspensions of E(j). Using this fact, they showed the change of rings theorem Ext_{BP_{∗}(BP)}^{∗,∗}(BP_{∗}, M) ≅ Ext_{E(n)_{∗}(E(n))}^{∗,∗}(E(n)_{∗}, E(n)_{∗} ⊗_{BP_{∗}} M) for a BP_{∗}(BP)-comodule M, on which v_{j} acts isomorphically, and n ≥ j. Their results imply another change of rings theorem. Theorem 3.3. Suppose M is an E_{∗}(E)-comodule, on which v_{j} acts isomorphically, and j ≤ n. Then the map

Ext_{E_{∗}(E)}^{∗,∗}(E_{∗}, M) ⟶ Ext_{E(n)_{∗}(E(n))}^{∗,∗}(E(n)_{∗}, E(n)_{∗} ⊗_{E_{∗}} M)

is an isomorphism.

Proof. Consider the map q: BP →^{ι∧1} BP ∧BP →^{p} ⋁_{E∈E} Σ^{|σE|}BP → ⋁_{E∈E} Σ^{|σE|}BP ⟨j⟩

for the unit map ι: S^{0} → BP and the projection p. Here E denotes the set of exponent sequences E = (e_{1}, e_{2}, …) such that e_{i} = 0 for i < j, σE = (p^{j}e_{j}, p^{j}e_{j+1}, ...) for E ∈ E, and |E| = ∑ 2e_{i}(p^{i} − 1). In [5, Th.4.1], it is shown that q induces an equivalence L_{K(j)}BP ≃ L_{K(j)} (⋁_{E∈E} Σ^{|σE|}E(j)). Then the self map v_{n} on BP induces a split monomorphism of spectra v_{n}: L_{K(j)}BP → L_{K(j)}BP. Now by [5, Lemmas 4.5, 4.6], we have an equivalence

L_{K(j)}v_{n}^{−1}BP ≃ L_{K(j)} (BP ∨ ⋁_{m} Σ^{−m|v_{n}|}BP/(v_{n})) ,

where BP/(v_{n}) denotes the cofiber of v_{n}: BP → BP. Since the self map v_{n}: L_{K(j)}BP → L_{K(j)}BP is a split monomorphism of spectra as we have seen above, we obtain an equivalence L_{K(j)}BP/(v_{n}) ≃ L_{K(j)} (⋁_{E∈E′} Σ^{|σE|}E(j)), where E′ denotes the subset of E consisting of sequences E such that e_{n} = 0. Therefore, L_{K(j)}v_{n}^{−1}BP is equivalent to a wedge of suspensions of E(j). Now we obtain the theorem by the analogous argument in the first proof of [5, Th. 3.1]. □

4. v_{n}^{−1}BP -based Adams m-tower

In this section, E denotes v_{n}^{−1}BP as above. Let i: S^{0} → E denote the unit map and consider the cofiber sequence

S^{0} ⟶^{i} E ⟶^{j} \overline{E} ⟶^{k} S^{1}.

Then it defines the E-based Adams tower

S_{0}  \overline{E}  ⋯  \overline{E}^{m}  \overline{E}^{m+1}  ⋯ E  E ∧ \overline{E}  ⋯  E ∧ \overline{E}^{m}  · · · ._{u_{i}}  _{u_{i∧1}} u ^{k}  u ^{k∧1}  _{_{i∧1_{m}} u} u^{k∧1_{m−1}}  u^{k∧1_{m}}  u ^{[ [ [[]}_{j}_{^{d_{0}} w}  _{^{d_{1}} w} _{w^{d_{m−1}}}  _{^{[ [[]}^{j∧1_{m}}}_{^{d_{m}} w}

Here \overline{E}^{m} denotes the m-fold smash product of \overline{E}, 1_{m}: \overline{E}^{m} → \overline{E}^{m} is the identity, and d_{m} = (i ∧ 1_{m+1})(j ∧ 1_{m}) = d_{0} ∧ 1_{m}. Consider the spectrum \overline{E}_{m} defined by the cofiber sequence \overline{E}^{m} ⟶^{k^{m}} S^{m} → Σ\overline{E}_{m−1}, in which k^{m} denotes the composite k(k ∧ 1_{1}) ⋯ (k ∧ 1_{m−1}). This defines the other E-based Adams tower

(4.1)  pt  _{\overline{E}_{0}}  ⋯  _{\overline{E}_{m−1}}  _{\overline{E}_{m}}  ⋯ E  E ∧ \overline{E}  ⋯  E ∧ \overline{E}^{m}  · · · . u  _{u_{d_{0}=i_{1}}} u  u ^{k_{1}}  _{_{i_{m}} u} u^{k_{m−1}}  u^{k_{m}}  u_{^{[ [} ^{[ [} ^{[ [}^{[ [}}_{^{d_{0}} w}  _{^{d_{1}} w} _{w^{d_{m−1}}}  _{^{[ [[]}^{j_{m}}}_{^{d_{m}} w}

Smashing a spectrum X with this tower, we have the E-based Adams spectral sequence converging to the homotopy groups π_{∗}(L_{n}X) with E_{2}-term

E_{2}^{s,t} = Ext_{E_{∗}(E)}^{s,t}(E_{∗}, E_{∗}(X)).

(cf. [1], [10]).

Consider a similar tower

pt  X_{0}  ⋯  X_{m−1}  X_{m}  ⋯ E  E ∧ \overline{E}  ⋯  E ∧ \overline{E}^{m}  ⋯, u  _{u_{d_{0}=i_{1}^{X}}} u  u ^{k_{1}^{X}}  _{_{i_{m}^{X}} u} u^{k_{m−1}^{X}}  u^{k_{m}^{X}}  u_{^{[ [} ^{[ [} ^{[ [}^{[ [}}_{^{d_{0}} w}  _{^{d_{1}} w} _{w^{d_{m−1}}}  _{^{d_{m}} w}_{^{[ [[]}^{j_{m}^{X}}}

in which the maps d_{k}: E ∧ \overline{E}^{k} → E ∧ \overline{E}^{k+1} are the same as those of (4.1), while X_{k}’s need not be the same as \overline{E}_{k}’s. We call this an ∞-tower. The Adams tower (4.1) is an example of ∞-towers. We also consider a similar tower

pt  X_{0}  ⋯  X_{m−2}  X_{m−1}  X_{m} E  E ∧ \overline{E}  ⋯  E ∧ \overline{E}^{m−1}  E ∧ \overline{E}^{m} u  _{u_{d_{0}=i_{1}^{X}}} u  u ^{k_{1}^{X}}  _{_{i_{m−1}^{X}} u} u^{k_{m−2}^{X}}  _{_{i_{m}^{X}} u} u^{k_{m−1}^{X}}  u ^{k_{m}^{X}}_{^{[ [} ^{[ [} ^{[ [} ^{[ [}^{[ [}}_{^{d_{0}} w}  _{^{d_{1}} w} _{^{d_{m−2}} w}  _{w^{d_{m−1}}}_{^{[ [ [ []}^{j_{m−1}^{X}}}  _{^{[ [ [ []}^{j_{m}^{X}}}

of finite length. We call it an m-tower. The first m-stages of an ∞-tower is an example of m-towers.

Consider an ∞-tower {X_{k}}. Then it contains an inverse system {X_{m} ← Σ^{−1}X_{m+1}}. We write X_{∞} = holimΣ^{−m}X_{m}. In the same manner as the Adams tower (4.1), the ∞-tower defines a spectral sequence {E_{r}^{′s,t}(Y)} for a spectrum Y abutting to π_{∗}(Y ∧ X_{∞}) by smashing the ∞-tower with Y. In particular, E_{2}^{s,t}(Y) = E_{2}^{′s,t}(Y) for any ∞-tower and any spectrum Y.

Suppose that there exists a finite spectrum W and an integer s_{0} such that E_{2}^{s}(W) = 0 for s > s_{0}. Lemma 4.2. Let F be a finite spectrum such that E_{∗}(F) is E_{∗}-free. Then the spectral sequence

E_{1}^{′∗,∗}(F ∧ W) ⇒ π_{∗}(F ∧ W ∧ X_{∞})

associated to the ∞-tower converges, and E_{2}^{s,∗}(F ∧ W) = 0 for s > s_{0}.

Proof. By the assumption on W, the cobar complex

Ω^{s−1}E_{∗}(W) → Ω^{s}E_{∗}(W) → Ω^{s+1}E_{∗}(W)

is exact for s > s_{0}. Taking the tensor product of E_{∗}(F) and the complex, we have a complex {Ω^{s}(E_{∗}(F) ⊗ E_{∗}(W))}, which is isomorphic to cobar complex {Ω^{s}(E_{∗}(F ∧ W))}. Since E_{∗}(F) is E_{∗}-free, the complex is exact for s > s_{0}. Therefore, we obtain E_{2}^{s,∗}(F ∧ W) = 0 for s > s_{0}, since the E_{2}-term is the cohomology of the complex. By this finite horizontal vanishing line, we see that lim^{1} = 0, and compute π_{∗}(F ∧ W ∧ X_{∞}) = π_{∗}(holim_{m}F ∧ W ∧ Σ^{−m}X_{m}) = lim_{m} π_{∗}(F ∧ W ∧ X_{m}), since F ∧W is finite. Therefore, this spectral sequence converges to π_{∗}(F ∧W ∧X_{∞}). □

Proposition 4.3. For an ∞-tower {X_{k}}, E_{∗}(W ∧ X_{∞}) = E_{∗}(W).

Proof. Consider the spectral sequence

(4.4)  E_{r}^{′∗,∗}(E ∧ W) ⇒ π_{∗}(E ∧ W ∧ X_{∞}) = E_{∗}(W ∧ X_{∞})

associated to the ∞-tower. For each skeleton E_{k} of BP given in Lemma 3.1, the spectral sequence

E_{r}^{′∗,∗}(E_{k} ∧ W) ⇒ π_{∗}(E_{k} ∧ W ∧ X_{∞})

associated to the ∞-tower converges by Lemma 4.2, since E_{∗}(E_{k}) is E_{∗}-free.

Lemma 3.1 gives us the map ι_{k}: E_{k} → E such that ι_{k+1}\overline{v_{n}} = ι_{k}. The map ι_{k} induces a map

{E_{r}^{′∗,∗}(E_{k} ∧ W)} → E_{r}^{′∗,∗}(E ∧ W)

of spectral sequences, and these maps induce an isomorphism colim_{k}E_{r}^{′∗,∗}(E_{k} ∧ W) ≅ E_{r}^{′∗,∗}(E ∧ W) by Lemma 3.1.

Consider the filtration

F^{s}(Y) = Ker((1_{Y} ∧ κ_{s})_{∗}: π_{∗}(Y ∧ X_{∞}) → π_{∗}(Y ∧ X_{s}))

associated to the ∞-tower for a spectrum Y, where κ_{s} denotes the projection X_{∞} → X_{s} such that k_{s}^{X}κ_{s} = κ_{s−1}. We will show that the spectral sequence (4.4) converges. For this sake, it suffices to show that F^{s_{0}+1}(E ∧ W) = 0. Suppose that x ∈ F^{s_{0}+1}(E ∧ W). Since E = hocolim E_{k}, there exists an element x_{k} ∈ π_{∗}(E_{k} ∧ W ∧ X_{∞}) such that (ι_{k} ∧ 1)_{∗}(x_{k}) = x. Then, (ι_{k} ∧ 1)_{∗}(1_{E_{k}∧W} ∧ κ_{s_{0}+1})_{∗}(x_{k}) = (1_{E∧W} ∧ k_{s_{0}+1})_{∗}(ι_{k} ∧ 1)_{∗}(x_{k}) = (1_{E∧W} ∧ k_{s_{0}+1})_{∗}(x) = 0. Therefore, we have an integer j(> k) such that ι_{k,j∗}(1_{E_{j}∧W} ∧ k_{s_{0}+1})_{∗}(x_{k}) = 0, where ι_{k,j}: E_{k} → E_{j} is the composite of \overline{v_{n}} in Lemma 3.1. Put x_{j} = ι_{k,j∗}(x_{k}), and we see x_{j} ∈ F^{s_{0}+1}(E_{j} ∧W), since (1_{E_{j}∧W} ∧ k_{s_{0}+1})_{∗}(x_{j}) = ι_{k,j∗}(1_{E_{j}∧W} ∧ k_{s_{0}+1})_{∗}(x_{i}) = 0. We also see that F^{s_{0}+1}(E_{j}∧W) = 0 by Lemma 4.2. Therefore, we have x_{j} = 0, and so x = ι_{j}(x_{j}) = 0 as desired.

On the other hand, the E_{2}-term of the spectral sequence (4.4) is Ext_{E_{∗}(E)}^{∗,∗}(E_{∗}, E_{∗}(E) ⊗_{E_{∗}} E_{∗}(W)) = Ext_{E_{∗}}^{∗,∗}(E_{∗}, E_{∗}(W)) = E_{∗}(W), and the spectral sequence collapses from the E_{2}-term. □

Lemma 4.5. Let {X_{i}}_{i=0}^{m} be an m-tower and M an E-module spectrum. Then we have a split short exact sequence

0 ⟶ π_{m+t−1}(M) ⟶^{φ} [X_{m−1}, M]_{t} ⟶^{(j_{m−1}^{X})^{∗}} (Im d_{m−1}^{∗})_{t} ⟶ 0,

in which φ is defined by (k_{1}^{X} ⋯ k_{m−1}^{X})^{∗}(i^{∗})^{−1} for the unit map i: S^{0} → E, and d_{m−1}^{∗}: [E ∧ \overline{E}^{m}, M]_{t} → [E ∧ \overline{E}^{m−1}, M]_{t} is the induced map.

Proof. Applying [−, M]_{∗} to the m-tower, we have a diagram of exact sequences

[E ∧ \overline{E}, M]_{m+t−1}  [X_{0}, M]_{m+t−1}  [E, M]_{m+t−1} ⋮ [E ∧ \overline{E}^{m−1}, M]_{t+1}  [X_{m−2}, M]_{t+1}  [E ∧ \overline{E}^{m−2}, M]_{t+1} [E ∧ \overline{E}^{m}, M]_{t}  [X_{m−1}, M]_{t}  [E ∧ \overline{E}^{m−1}, M]_{t} w^{(i_{1}^{X})^{∗}}  _{u_{(k_{1}^{X})^{∗}}}  ^{(j_{0}^{X})^{∗}}_{u_{(k_{m−2}^{X})^{∗}}}_{w^{(i_{m−1}^{X})^{∗}}}  _{u_{(k_{m−1}^{X})^{∗}}}_{w^{(j_{m−2}^{X})^{∗}}}_{^{(i_{m}^{X})^{∗}} w}  _{w^{(j_{m−1}^{X})^{∗}}}

We have a split short exact sequence

0 ⟶[\overline{E}^{j+1}, M]_{∗}⟶^{(j∧1)^{∗}} [E ∧ \overline{E}^{j}, M]_{∗} ⟵^{τ}  ⟶^{(i∧1)^{∗}} [\overline{E}^{j}, M]_{∗} ⟵^{σ}  0 ⟶

for j ≥ 0 (\overline{E}^{0} = S^{0}), where σ: [\overline{E}^{j}, M]_{∗} → [E∧\overline{E}^{j}, M]_{∗} is defined by σ(f) = ν(1∧f) for the module structure ν: E ∧M → M of the module spectrum M. Splicing these sequences yields a long exact sequence

0 → (Im d_{m})_{t} ⟶ [E∧\overline{E}^{m}, M]_{t} ⟶^{d_{m−1}^{∗}} ⋯ ⟶^{d_{1}^{∗}} [E∧\overline{E}, M]_{t} ⟶^{d_{0}^{∗}} [E, M]_{t} ⟶^{i^{∗}} [S^{0}, M]_{t} → 0.

We then verify the exactness of the sequence in the lemma by chasing the diagram. The splitting s: (Imd_{m−1}^{∗})_{t} → [X_{m−1}, M]_{t} is given as follows: For x ∈ Im (d_{m−1}^{∗})_{t}, there is an element y such that x = yd_{m−1}. Then put s(y) = yi_{m}^{X}. This is well defined, since Ker d_{m−1}^{∗} = Im d_{m}^{∗} and (i_{m}^{X})^{∗}d_{m}^{∗} = 0. □

Lemma 4.6. [E, E ∧ \overline{E}^{j}]_{∗} is torsion free.

Proof. Note that [BP, E ∧\overline{E}^{j}]_{∗} = H^{∗}(BP; E_{∗}(\overline{E}^{j})) by the Atiyah-Hirzebruch spectral sequence. Since [BP, E ∧ \overline{E}^{j}]_{∗} = Hom_{E_{∗}}^{∗}(E_{∗}(BP), E_{∗}(\overline{E}^{j})) by the Adams uni-

versal coefficient theorem, lim_{v_{n}^{∗}}^{1}[BP, E ∧ \overline{E}^{j}]_{∗} = lim_{v_{n}^{∗}}^{1}Hom_{E_{∗}}^{∗}(E_{∗}(BP), E_{∗}(\overline{E}^{j})) =

0. Therefore, [E, E ∧ \overline{E}^{j}]_{∗} = lim_{v_{n}^{∗}}[BP, E ∧ \overline{E}^{j}]_{∗} by the Milnor sequence, which is torsion free. □

These lemmas imply the following

Corollary 4.7. For any m-tower \{X_{s}\}, [X_{s}, E ∧ \overline{E}^{j}]_{∗} is torsion free.

5. Generators in Pic(L_{2})

From this section on, we fix p = 3 and n = 2. Note that E = v_{2}^{−1}BP. The Toda-Smith spectrum V (n) for n = −1, 0, 1 is defined as follows: V (−1) = S^{0}, V (0) is the mod 3 Moore spectrum fitting in the cofiber sequence

(5.1)  S^{0} ^{3} wS^{0} ^{i} wV (0) ^{j} wS^{1},

and V (1) denotes the cofiber of the Adams map α fitting in the cofiber sequence

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

Note that the Adams map α induces v_{1} : BP_{∗}/(3) → BP_{∗}/(3) on BP -homology. In [18], Toda gave the element B ∈ [V (0), V (1)]_{16}, which is denoted by [βi_{1}] in [18], that induces v_{2} : BP_{∗}/(3) → BP_{∗}/(3, v_{1}) on BP_{∗}-homology. We define a spectrum Y, which is denoted by Σ^{−21}V (1\frac{1}{2}) in [18], by the cofiber sequence

(5.2)  Σ^{4}Y ^{f_{Y}} wV (0) ^{B} wΣ^{−16}V (1) ^{h_{Y}} wΣ^{5}Y.

Toda further showed in [18, Th. 6.7] that Bαi = i_{0}β_{1}^{2} for i_{0} = i_{1}i. Note that i_{0}β_{1}^{2} is detected by b_{10}^{2} ∈ E_{2}^{4,24}(V (0)). Applying E_{∗} to the cofiber sequence (5.2), we

have the long exact sequence E_{∗}/(3) ⟶^{v_{2}} E_{∗}/(3, v_{1}) ⟶^{g_{Y ∗}} E_{∗}(Y) ⟶^{f_{Y ∗}} E_{∗}/(3) ⟶^{v_{2}} E_{∗}/(3, v_{1}). Since v_{2} is an epimorphism, the sequence split into the short exact sequence

(5.3)  0 wE_{∗}(Y) w^{f_{Y ∗}} E_{∗}/(3) ^{v_{2}} wE_{∗}/(3, v_{1}) w0.

We consider the E-based Adams spectral sequence {E_{r}^{∗,∗}(V)} converging to π_{∗}(L_{2}V) for V = S^{0} and V (0). By the change of rings theorem Theorem 3.3, we identify these isomorphic E_{2}-terms in the lemma.

(5.4)([13],[14]) The E_{2}-term E_{2}^{5,4}(V) is generated by the following elements :

v_{2}^{−2}h_{11}b_{10}^{2} and v_{2}^{−1}ξb_{10}ζ_{2}  if V = S^{0},
v_{2}^{−2}h_{11}b_{10}^{2}, v_{2}^{−1}ξb_{10}ζ_{2} and v_{1}v_{2}^{−3}b_{10}b_{11}ζ_{2} if V = V (0).

Proposition 5.5. The E_{∗}-homology groups E_{∗}(Y) of Y is isomorphic to E_{∗}/(3) as an E_{∗}(E)-comodule. Furthermore, d_{5}(g_{Y}) = −v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{5}^{5,4}(Y) = E_{5}^{5,4}(V (0)) in the E_{∗}-based Adams spectral sequence converging to π_{∗}(L_{2}Y). Here g_{Y} ∈ E_{2}^{0,0}(Y) = Z/(3) is the generator.

Proof. Apply the E_{∗}-homology to the cofiber sequence (5.3), and we have a commutative diagram

0  E_{∗}(Y)  E_{∗}/(3)  E_{∗}/(3, v_{1})  0 0  E_{∗}/(3)  E_{∗}/(3)  E_{∗}/(3, v_{1})  0 w  ^{f_{Y ∗}} w  ^{v_{2}} w  w w u  _{^{v_{1}} w}  _{^{i_{∗}} w}  w u_{≅ v_{2}}

of comodules with exact rows. Now the desired isomorphism is given by the dotted arrow.

The isomorphism induces the one E_{2}^{∗,∗}(Y) = E_{2}^{∗,∗}(V (0)) of E_{2}-terms of the E_{∗}- based Adams spectral sequence. By the above commutative diagram, we see that (f_{Y})_{∗}(g_{Y}) = v_{1} ∈ E_{2}^{0,0}(V (0)), which is a permanent cycle that detects αi. Since Bαi = i_{0}β_{1}^{2} in the homotopy group, we see that d_{5}(g_{Y}) = δ(b_{10}^{2}) as is described in the following chart:

[figure: Y; V (0); V (1); Y; f_{Y}; B; h_{Y}; g; v_{1}; b_{10}^{2}; g; δ(b_{10}^{2})]

Here δ: E_{2}^{∗,∗}(V (1)) → E_{2}^{∗+1,∗}(Y) denotes the connecting homomorphism associated to the short exact sequence (5.3). Note that the left and the right hand sides of Bαi = i_{0}β_{1}^{2} represent B_{∗}(αi) = (v_{2})_{∗}(v_{1}) = 0 and b_{0}^{2} in the E_{2}-term, respectively. In the cobar complex, we compute

δ(b_{10}^{2}) = [v_{1}^{−1}d(v_{2}^{−1}b_{10}^{2})]
= [v_{1}^{−1}d(v_{2}^{−1})b_{10}^{2}]
= [v_{1}^{−1}(−v_{1}v_{2}^{−2}t_{1}^{3} + ⋯)b_{10}^{2}]
= [−(v_{2}^{−2}t_{1}^{3} + ⋯)b_{10}^{2}]
= −v_{2}^{−2}h_{11}b_{10}^{2} ≠ 0.

Here, [x] denotes the cohomology class of the cocycle x. □

Since d_{5}(g_{Y}) ≠ 0 and d_{5}(g_{V (0)}) = 0, we see that

Corollary 5.6. L_{2}Y is not equivalent to L_{2}V (0).

Proposition 5.7. E(2) ∧ Y is equivalent to E(2) ∧ V (0).

Proof. The generator g(= 1) ∈ E(2)_{∗}/(3) = π_{∗}(E(2) ∧ Y)(= Z/3) extends to the map E(2) → E(2) ∧ E(2) ∧ Y → E(2) ∧ Y. Since the order of g is 3, it further extends to E(2) ∧ V (0) → E(2) ∧ Y, which gives the desired equivalence. □

Theorem 5.8. There exists a spectrum X ∈ L_{2} such that v_{2}^{−1}BP_{∗}X = v_{2}^{−1}BP_{∗} and d_{5}(g_{X}) = −v_{2}^{−2}h_{11}b_{10}^{2} ∈ E_{5}^{5,4}(X) in the E_{∗}-based Adams spectral sequence.

Proof. Consider the ∞-tower \{\overline{E}_{s} ∧ Y \} obtained by smashing Y with the Adams tower (4.1). Let f′ denote the composite E = E ∧ S^{0} ⟶^{1∧i} E ∧ V (0) = E ∧ Y for i: S^{0} → V (0) in (5.1). Then it induces maps f′_{s}: E ∧ \overline{E}^{s} → E ∧ \overline{E}^{s} ∧ Y. Note that these are the same as i_{s}: E ∧ \overline{E}^{s} → E ∧ \overline{E}^{s} ∧ V (0) induced from i, and we have a cofiber sequence

(5.9)  E ∧ \overline{E}^{s} ⟶^{3} E ∧ \overline{E}^{s} ⟶^{f′_{s}} E ∧ \overline{E}^{s} ∧ Y ⟶^{g′_{s}} ΣE ∧ \overline{E}^{s}.

Inductively, we will construct an s-tower \{X_{t}\}_{t=0}^{s} and a map φ: \{X_{t}\} → \{\overline{E} ∧Y \} of towers for each s. Here the map φ of towers means a set of maps \{f_{t} : X_{t} → \overline{E}∧Y \} with f_{t−1}k_{t}^{X} = k_{t}f_{t}. Note that if every map f_{t} satisfies f_{t}j_{t}^{X} = (j_{t} ∧ 1_{Y})f′_{t}, then φ is a map of towers.

Put X_{0} = E, f_{0} = f′_{0} : X_{0} = E → \overline{E}_{0} ∧ Y = E ∧ Y. Then we see that f_{0}j_{0}^{X} = (j_{0} ∧ 1_{Y})f′_{0}, since both of j_{0} and j_{0}^{X} are the identity maps.

Now suppose that s-tower \{X_{t}\}_{t=0}^{s} is constructed and each map f_{t} : X_{t} → \overline{E}^{t}∧Y for 0 ≤ t ≤ s satisfies f_{t}j_{t}^{X} = (j_{t} ∧ 1_{Y})f′_{t}. Since the groups [U, E ∧ \overline{E}^{s+1}]_{∗} for U = E ∧ \overline{E}^{s} and U = X_{s} are torsion free by Lemma 4.6 and Corollary 4.7, we have the commutative diagram

[X_{s}, E ∧ \overline{E}^{s+1}]_{0}  [X_{s}, E ∧ \overline{E}^{s+1}]_{0}  [X_{s}, E ∧ \overline{E}^{s+1} ∧ Y]_{0} [E ∧ \overline{E}^{s}, E ∧ \overline{E}^{s+1}]_{0}  [E ∧ \overline{E}^{s}, E ∧ \overline{E}^{s+1}]_{0}  [E ∧ \overline{E}^{s}, E ∧ \overline{E}^{s+1} ∧ Y]_{0} [X_{s−1}, E ∧ \overline{E}^{s+1}]_{0}  [X_{s−1}, E ∧ \overline{E}^{s+1}]_{0}  [X_{s−1}, E ∧ \overline{E}^{s+1} ∧ Y]_{0}_{_{(j_{s}^{X})^{∗}} u}  y  w^{3}  _{u_{(j_{s}^{X})^{∗}}}  ^{f′_{s+1∗}} ww  _{u_{(j_{s}^{X})^{∗}}}_{_{(i_{s}^{X})^{∗}} u}  _{y  w^{3}}  _{u_{(i_{s}^{X})^{∗}}}  _{ww^{f′_{s+1∗}}}  _{u_{(i_{s}^{X})^{∗}}}_{y  w^{3}}  _{^{f′_{s+1∗}} ww}

of the short exact sequences associated to the cofiber sequence (5.9).

Since the top f′_{s+1∗} is an epimorphism, we have an element i_{s+1}^{′X} ∈ [X_{s}, E ∧\overline{E}^{s+1}] such that f′_{s+1}i_{s+1}^{′X} = (i_{s+1} ∧ 1_{Y})f_{s} ∈ [X_{s}, E ∧ \overline{E}^{s+1} ∧ Y]_{0}. Then f′_{s+1}i_{s+1}^{′X}j_{s}^{X} = (i_{s+1} ∧ 1_{Y})f_{s}j_{s}^{X} = (i_{s+1} ∧ 1_{Y})(j_{s} ∧ 1_{Y})f′_{s} = (d_{s} ∧ 1_{Y})f′_{s} = f′_{s+1}d_{s}^{X} by the inductive hypothesis, and there is an element w ∈ [E ∧ \overline{E}^{s}, E ∧ \overline{E}^{s+1}]_{0} such that 3w = i_{s+1}^{′X}j_{s}^{X} − d_{s}^{X}. On the other hand, 3wi_{s}^{X} = i_{s+1}^{′X}j_{s}^{X}i_{s}^{X} − d_{s}^{X}i_{s}^{X} = 0 ∈ [X_{s−1}, E ∧ \overline{E}^{s+1}]_{0}. Since the bottom 3 is monomorphic, wi_{s}^{X} = 0, and so we have an element \overline{w} ∈ [X_{s}, E ∧ \overline{E}^{s+1}]_{0} such that w = \overline{w}j_{s+1}^{X}. Put i_{s+1}^{X} = i_{s+1}^{′X} − 3\overline{w}, and we see f′_{s+1}i_{s+1}^{X} = f′_{s+1}(i_{s+1}^{′X} − 3\overline{w}) = (i_{s+1} ∧ 1_{Y})f_{s} and i_{s+1}^{X}j_{s}^{X} = (i_{s+1}^{′X} − 3\overline{w})j_{s}^{X} = d_{s}^{X}. Now we define X_{s+1} to be the cofiber of i_{s+1}^{X} to see the existence of a map f_{s+1} : X_{s+1} → E ∧\overline{E}^{s+1} ∧Y such that f_{s+1}j_{s+1}^{X} = (j_{s+1} ∧1_{Y})f′_{s+1}, since f′_{s+1}i_{s+1}^{X} = (i_{s+1} ∧ 1_{Y})f_{s}. Thus we complete the induction, and obtain an ∞-tower {X_{s}}. Put X_{∞} = holim X_{s}.

Consider the finite spectrum W = S^{0} ∪_{α_{1}} e^{4} ∪_{−α_{1}} e^{8}, which is the 8-skeleton of BP, where α_{1} ∈ π_{3}(S^{0}) denotes the generator. Then there is an integer s_{0} such that E_{2}^{s,∗}(W) = 0 for s > s_{0} by [10, 7.4.2] and [16]. Therefore, E_{∗}(W ∧ X_{∞}) = E_{∗}(W) by Lemma 4.3. Note that E ∧ W = E ∨ Σ^{4}E ∨ Σ^{8}E, since E_{∗}(α_{1}) = 0. Therefore, we see E_{∗}(X_{∞}) = E_{∗} by ⨁_{3} E_{∗}(X_{∞}) = E_{∗}(W ∧ X_{∞}) = E_{∗}(W) = ⨁_{3} E_{∗}.

The map φ defines the map f: X_{∞} → L_{2}Y such that E_{∗} = E_{∗}(X_{∞}) → E_{∗}(Y) = E_{∗}/(3) is the projection. Then it further induces the map f_{∗}: E_{r}^{∗,∗}(X_{∞}) → E_{r}^{∗,∗}(Y) of the E-based Adams spectral sequences. By Lemma 5.5, f_{∗}d_{5}(g_{X_{∞}}) = d_{5}(g_{Y}) = −v_{2}^{−2}h_{11}b_{10}^{2}, and we see that d_{5}(g_{X_{∞}}) = −v_{2}^{−2}h_{11}b_{10}^{2} by (5.4). Thus X_{∞} is the desired spectrum. □

Proof of Proposition 1.5. The spectrum X in Theorem 5.8 is invertible by Proposition 3.2. By Theorem 3.3, we see d_{5}(g_{X}) = −χ_{1} in the E(2)-based Adams spectral sequence. The inverse spectrum X^{−1} (see the proof of Theorem 1.1) is the desired spectrum. □

References

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Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

E-mail address: Yousuke.Kamiya@ma7.seikyou.ne.jp

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

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