Picard groups of some local categories

By

Yousuke Kamiya^{∗}, Katsumi Shimomura^{∗∗}

Abstract

For each p-local spectrum E, L_{E} denotes the full subcategory consisting of E-local spectra of the category of p-local spectra. The Picard group Pic(L_{E}) is the collection of isomorphism classes of invertible spectra in L_{E}. If this is a set, it is a group with multiplication defined by the smash product. We show that if a spectrum E satisfies a relation ⟨E⟩ ≥ ⟨HZ/p⟩ of the Bousfield classes, then Pic(L_{E}) = Z. In particular, Pic(L_{E}) = Z if E is connective.

§1. Introduction

Throughout this paper, we work in the homotopy category S_{p} of plocal spectra for a fixed prime number p. For a spectrum E ∈ S_{p}, a spectrum X ∈ S_{p} is called E-acyclic if X ∧ E = pt, and called E-local if [C, X]_{∗} = 0 for any E-acyclic spectrum C. Let L_{E} denote the full subcategory of S_{p} consisting of E-local spectra. Then we have the Bousfield localization functor L_{E}: S_{p} → L_{E} [1]. We call a spectrum X ∈ L_{E} invertible if there is a spectrum Y such that L_{E}(X ∧ Y) = L_{E}S^{0}. Let Pic(L_{E}) denote the collection of the isomorphism classes of invertible spectra. If Pic(L_{E}) is a set, then it is a group whose multiplication is given by [X] · [Y] = [L_{E}(X ∧ Y)] and the unit element is [L_{E}S^{0}]. Here [X] denotes the isomorphism class of X. In the following, we

Communicated by K. Saito. Received August 29, 2001. Revised March 13, 2002, February 17, 2003, July 15, 2004, January 16, 2006,

June 8, 2006 2000 Mathematics Subject Classification: Primary 55Q99; Secondary 55P60. Key words: Picard Group, Stable homotopy category, E-local category. ∗101 Sejour Nishinomiya-Imazu, 2-17, Imazuyama-Naka, Nishinomiya, Hyogo,

663-8211, Japan email: kamiya yousuke@ybb.ne.jp ∗∗Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-

8520, Japan email: katsumi@math.kochi-u.ac.jp

write X ∈ Pic(L_{E}) instead of [X] ∈ Pic(L_{E}). It is well known that Pic(L_{S^{0}}) = Pic(S_{p}) = {S^{n} | n ∈ Z} ≅ Z. Consider the Morava K-theories K(n) and the Johnson-Wilson spectra E(n). In [5], Hopkins, Mahowald and Sadofsky showed that the Picard group Pic(L_{K(n)}) for each K(n) contains Z_{p} and determined it for n < 2. Note that Pic(L_{K(n)}) is often written by Pic_{n}. For E(n), Hovey and Sadofsky showed Pic(L_{E(n)}) = Z in [9] if 2p − 2 > n^{2} + n. In [9], they also determined Pic(L_{E(1)}) at the prime 2, which is the direct sum of Z and Z/2. In [12], we gave an estimate of Pic(L_{E(n)}) by using E_{r}-term of the E(n)-based Adams spectral sequence converging to π_{∗}(L_{E(n)}S^{0}), and showed that Pic(L_{E(2)}) at the prime 3 is isomorphic to one of Z ⊕ Z/3 and Z ⊕ Z/3 ⊕ Z/3.

Recall that the Bousfield class ⟨E⟩ denotes the collection of E-acyclic spectra, and we write ⟨F ⟩ ≥ ⟨E⟩ if ⟨F ⟩ ⊂ ⟨E⟩. Let HG denote the Eilenberg-MacLane spectrum for a group G. Then one of our results is the following:

Theorem 1.1. If ⟨E⟩ ≥ ⟨HZ/p⟩, then Pic(L_{E}) = Z.

It is shown in [1, Lemma 3.2] that if E is connective, then ⟨E⟩ ≥ ⟨HZ/p⟩ or ⟨E⟩ = ⟨HQ⟩. It is well known (cf. [5]) that Pic(L_{HQ}) = Z.

Corollary 1.2. If E is connective, then Pic(L_{E}) = Z.

Let I denote the Brown-Comenetz dual of S^{0}. In [8] and [10], it is shown that L_{E}S^{0} = S^{0} or SZ_{p} if ⟨E⟩ ≥ ⟨I⟩. Then the above theorem follows from

Theorem 1.3. If ⟨F ⟩ ≥ ⟨E⟩ ≥ ⟨I⟩, then Pic(L_{F}) ⊂ Pic(L_{E}) ⊂ Pic(L_{I}).

Note that if Pic(L_{I}) = Z, then Pic(L_{F}) = Pic(L_{E}) = Z, though the assumption Pic(L_{I}) = Z seems strong.

In the next section, we study basic relations between the Bousfield classes and Picard groups, and then we consider the Picard groups for spectra with a finite local and prove Theorem 1.3 in the following section. In section 4, we show Pic(L_{HZ/p}) = Z, which together with results given in the previous section proves Theorem 1.1. We describe a filtration of Pic(L_{I}) in the last section as well as the proofs of some facts. For instance, we prove that the only invertible spectrum which is also a ring spectrum is the E-local sphere L_{E}S^{0}. We also discuss there the small objects in some local stable homotopy categories.

§2. Relations between Picard groups

The function spectrum F (X, Y) is defined to be a spectrum such that [W ∧X, Y]_{∗} = [W, F (X, Y)]_{∗} for any spectra W. We define the Spanier-Whitehead dual D_{E}(X) of X in L_{E} by D_{E}(X) = F (X, L_{E}S^{0}). In particular, we write D(X) = D_{S^{0}}(X) for the ordinary Spanier-Whitehead dual. It is shown in [11, Prop. A.2.8] that if X ∈ L_{E} is invertible, then D_{E}(X) is the inverse of it, that is, L_{E}(X ∧ D_{E}(X)) = L_{E}S^{0}.

Lemma 2.1. For any spectrum X ∈ S_{p}, the Spanier-Whitehead dual D_{E}(X) is E-local.

Proof. Since C ∧ X is E-acyclic if so is C, [C, D_{E}(X)]_{∗} = [C ∧ X, L_{E}S^{0}]_{∗} = 0. □

Note that there is a relation L_{E}(L_{E}(X) ∧ L_{E}(Y)) = L_{E}(X ∧ Y) of E-localization, which we often use. For example, we have L_{E}(X ∧ L_{E}S^{0}) = X for an E-local spectrum X. By definition, if ⟨F ⟩ ≥ ⟨E⟩, then L_{E}L_{F} X = L_{E}X.

Lemma 2.2. Suppose that ⟨F ⟩ ≥ ⟨E⟩. The localization functor L_{E} induces a correspondence l: Pic(L_{F}) → Pic(L_{E}).

Proof. Define l(X) = L_{E}X. Then we see that l(X) ∈ Pic(L_{E}). In fact, L_{E}(l(X)∧l(D_{F} (X))) = L_{E}(X∧D_{F} (X)) = L_{E}(L_{F} (X∧D_{F} (X))) = L_{E}L_{F} S^{0} = L_{E}S^{0}, since D_{F} (X) is the inverse of X in L_{F}. □

We call a spectrum X strongly dualizable in L_{E} if L_{E}(F (X, L_{E}S^{0}) ∧ Y) = F (X, Y) for any Y ∈ L_{E}. In [13, Chap. III], Lewis, May and Steinberger studied properties of strongly dualizable spectra. Among them, we have isomorphisms

(2.3) X = D_{E}(D_{E}(X)) and L_{E}(D_{E}(X) ∧ D_{E}(Y)) = D_{E}(X ∧ Y)

for a strongly dualizable spectrum X and a spectrum Y.

(2.4) ([11, Prop. A.2.8]) X ∈ Pic(L_{E}) is strongly dualizable.

Lemma 2.5. If ⟨F ⟩ ≥ ⟨E⟩ and L_{F} S^{0} = L_{E}S^{0}, then Pic(L_{F}) ⊂ Pic(L_{E}).

Proof. Take any spectrum X in Pic(L_{F}). Then X = D_{F} (D_{F} (X)) by (2.4) and (2.3), which is isomorphic to D_{E}(D_{E}(X)), since D_{E}(−) = D_{F} (−) by the assumption L_{F} S^{0} = L_{E}S^{0}. Thus, X is E-local by Lemma 2.1. Furthermore, L_{E}(X ∧ D_{E}(X)) = L_{E}L_{F} (X ∧ D_{F} (X)) = L_{E}L_{F} S^{0} = L_{E}S^{0}, and hence X ∈ Pic(L_{E}). □

§3. On spectra with a finite local

We say that a spectrum E has a finite local if there exists a nontrivial finite spectrum X such that L_{E}X = X. It is shown in [8, Prop. 7.2] that E has a finite local if and only if ⟨E⟩ ≥ ⟨I⟩. Here I denotes the Brown-Comenetz dual of the sphere spectrum. Since ⟨I⟩ = ⟨I ∧ X⟩ for any non-trivial finite spectrum X by [8, Prop. 7.1 (e)], we see that

(3.1) If E has a finite local, then so does E ∧ X for a non-trivial finite spectrum X.

It is shown in [10, Th. B.6(a)] that if E has a finite local and L_{E}HQ ≠ pt (resp. L_{E}HQ = pt), then L_{E}S^{0} = S^{0} (resp. L_{E}S^{0} = SZ_{p}) (cf. [7, Th. 3.6]). Hereafter, SG for a group G denotes the Moore spectrum with HZ_{∗}(SG) = G. If ⟨F ⟩ ≥ ⟨E⟩ ≥ ⟨I⟩, then there are two cases:

1) L_{F} S^{0} = L_{E}S^{0}

2) L_{F} S^{0} = S^{0} and L_{E}S^{0} = SZ_{p} In the first case, Lemma 2.5 shows that Pic(L_{F}) ⊂ Pic(L_{E}). We

study the second case. Put E/p^{i} = E ∧ SZ/p^{i}.

Lemma 3.2. Let E have a finite local and suppose L_{E}HQ = pt. Then ⟨E/p⟩ = ⟨E⟩.

Proof. By [14, Lemma 1.34], we see that ⟨E⟩ = ⟨E ∧ SQ⟩ ∨ ⟨E/p⟩. Since E ∧ SQ = E ∧ L_{E}SQ = E ∧ L_{E}HQ = pt by assumption, we have the lemma. □

Lemma 3.3. If X is E-local, then L_{E/p}X = L_{SZ/p}X.

Proof. Consider the cofiber sequence X →^{η} L_{E/p}X → C_{E/p}X of the localization. Then SZ/p ∧ C_{E/p}X = pt, since C_{E/p}X is E-local and SZ/p ∧ C_{E/p}X is E-acyclic. Applying now L_{SZ/p} to the cofiber sequence, we see that L_{SZ/p}η is the desired isomorphism since C_{E/p}X is SZ/p-acyclic. □

Lemma 3.4. Let E be a spectrum with a finite local. Then the correspondence l: Pic(L_{E}) → Pic(L_{E/p}) given in Lemma 2.2 is an injection.

Proof. If L_{E}HQ = pt, then L_{E}S^{0} = L_{E/p}S^{0} by Lemma 3.2, and the lemma follows from Lemma 2.5. Suppose that L_{E}HQ ≠ pt. Then L_{E}S^{0} = S^{0} and so D_{E}(−) = D(−). Let X ∈ Pic(L_{E}) be an element such that l(X) = L_{E/p}S^{0}. Since l satisfies l([X] · [Y]) = l([X]) · l([Y]), it suffices to show that X = S^{0}.

Consider the cofiber sequence X ∧ D(X) →^{η} L_{E}(X ∧ D(X)) → C_{E}(X ∧ D(X)) of the localization, which is

(3.5)  X ∧ D(X) ⟶^{η} S^{0} ⟶ C

for C = C_{E}(X∧D(X)). Since L_{E}HQ ≠ pt, ⟨E⟩ ≥ ⟨E⟩∧⟨HQ⟩ = ⟨HQ⟩ by [14, Th. 2.1(h)]. Therefore, C is HQ-acyclic.

Note that D(X) is E-local by Lemma 2.1 and that L_{SZ/p}X = L_{E/p}X = L_{SZ/p}S^{0} by Lemma 3.3 and the assumption on X. Then, SZ/p ∧ X = SZ/p ∧ L_{SZ/p}X = SZ/p ∧ L_{SZ/p}S^{0} = SZ/p and SZ/p ∧ D(X) = ΣD(SZ/p∧X) = SZ/p by (2.3), since D(SZ/p) = Σ^{−1}SZ/p. Smashing SZ/p with the cofiber sequence (3.5) yields the cofiber sequence

SZ/p ⟶^{η} SZ/p ⟶ SZ/p ∧ C.

Applying L_{E} to this cofiber sequence, we have a commutative diagram with horizontal cofiber sequences:

SZ/p  SZ/p  SZ/p ∧ C L_{E}(SZ/p)  L_{E}(SZ/p)  L_{E}(SZ/p ∧ C) = pt.^{η}  w_{_{≅} u}  w_{_{≅} u}  u_{^{L_{E}(η)} w}  w

Here, the left and the middle vertical arrows are equivalences since L_{E}(S^{0}) = S^{0}, and so is the right vertical arrow. Thus, SZ/p ∧ C = L_{E}(SZ/p ∧ C) = SZ/p ∧ L_{E}(C) = SZ/p ∧ pt = pt, which shows C is SZ/p-acyclic.

These show that C = pt by [14, Th. 2.1(i)] and η: X ∧ D(X) = S^{0}. Therefore, X ∈ Pic(L_{S^{0}}), that is, X is a suspension of S^{0}. Since SZ/p ∧ X = SZ/p, we see that X = S^{0} as desired. □

Corollary 3.6. Suppose that E has a finite local and ⟨F ⟩ ≥ ⟨E⟩. Then Pic(L_{F}) ⊂ Pic(L_{E}). In particular, if Pic(L_{E}) = Z, then Pic(L_{F}) = Z.

Proof. If L_{F} S^{0} = L_{E}S^{0}, then the corollary follows from Lemma 2.5. If L_{F} S^{0} = S^{0} and L_{E}S^{0} = SZ_{p}, then ⟨F ⟩ ≥ ⟨F/p⟩ ≥ ⟨E/p⟩ = ⟨E⟩ by Lemma 3.2. Therefore, Lemmas 2.5 and 3.4 imply the corollary. □

Corollary 3.7. If E has a finite local, then Pic(L_{E}) ⊂ Pic(L_{I}).

§4. The Picard group of L_{HZ/p}

Put D_{p}(X) = F (X, SZ_{p}) = D_{HZ/p}(X). Then, for X ∈ Pic(L_{HZ/p}), L_{HZ/p}(X∧D_{p}(X)) = L_{HZ/p}S^{0} by [11, prop. A.2.8], and so HZ/p_{∗}(X) = Z/p. Let Pic(L_{HZ/p})^{0} denote the subcollection consisting of isomorphism classes X ∈ Pic(L_{HZ/p}) such that (HZ/p)_{0}(X) = Z/p. Then any element of Pic(L_{HZ/p}) is a suspension of an element of Pic(L_{HZ/p})^{0}. Furthermore, note the relation D_{p}D_{p}(X) = X by (2.3). We also consider the cofiber sequence

(4.1)  SZ/p ⟶^{p^{r}} SZ/p^{r+1} ⟶^{j_{r}} SZ/p^{r} ⟶^{δ_{r}} ΣSZ/p.

Lemma 4.2. Let X ∈ Pic(L_{HZ/p})^{0}. Then we have an equivalence L_{HZ}(X ∧ SZ/p^{r}) →^{≃} SZ/p^{r} for each r > 0 such that the diagram

L_{HZ}(X ∧ SZ/p^{r+1})  L_{HZ}(X ∧ SZ/p^{r}) SZ/p^{r+1}  SZ/p^{r}_{_{≃} u}  ^{1∧j_{r}} w  _{u_{≃}} w_{j_{r}}

commutes for the map j_{r} of (4.1).

Proof. Put Y = X ∧ SZ/p and let g be a generator of (HZ)_{0}(Y) = (HZ/p)_{0}(X) = Z/p. Since HZ/p is a ring spectrum, g: S^{0} → HZ ∧Y extends to a homotopy equivalence g̃: HZ/p → HZ ∧ Y. Then we have a map f: Y ⟶^{ι∧1} HZ ∧ Y \underleftarrow{g̃} HZ/p for the unit map ι of HZ. Note that [Y, Σ^{k}HZ/p]_{0} = (HZ/p)^{k}(Y) = Hom((HZ/p)_{k}(Y), Z/p) = Hom((HZ/p)_{k}(SZ/p), Z/p) = (HZ/p)^{k}(SZ/p) = 0 unless k = 0, 1. Consider the Postnikov tower of SZ/p:

HZ/p  X_{1}  X_{2}  ⋯ ⟵ SZ/p Σ^{2}Hπ_{1}(SZ/p) Σ^{3}Hπ_{2}(SZ/p) Σ^{4}Hπ_{3}(SZ/p)_{u} _{u}  _{u} _{u}  _{u} _{u}

Since [Y, Σ^{k}Hπ_{k−1}SZ/p]_{0} = H^{k}(Y; π_{k−1}(SZ/p)) = 0 for k > 1, the map f: Y → HZ/p is lifted up to an HZ/p-equivalence f̃: Y → SZ/p. The Spanier-Whitehead dual yields an HZ/p-equivalence D_{p}(f̃): Σ^{−1}SZ/p → D_{p}(Y) = D_{p}(X) ∧ Σ^{−1}SZ/p. We do this for D_{p}(X) instead of X, and obtain an HZ/p-equivalence f′: SZ/p → Y since D_{p}D_{p}(X) = X. Since f̃f′: SZ/p → SZ/p is an HZ/p-equivalence, it is an equivalence. Therefore, we have a decomposition Y →^{≃} SZ/p ∨ A for some spectrum A. Since (HZ)_{∗}(Y) = Z/p, we have HZ_{∗}(A) = 0

and so L_{HZ}Y →^{≃} SZ/p.

Inductively suppose that we have the equivalences of the lemma up to r. Since δ_{r} ∈ [SZ/p^{r}, SZ/p]_{−1} = Z/p is a generator, so is 1 ∧ δ_{r} ∈ [L_{HZ}(X ∧ SZ/p^{r}), L_{HZ}(X ∧ SZ/p)]_{−1}. Here [L_{HZ}(X ∧ SZ/p^{r}), L_{HZ}(X ∧ SZ/p)]_{−1} is isomorphic to [SZ/p^{r}, SZ/p]_{−1} = Z/p, since X is HZ/p-invertible. Then the composite δ′_{r}: SZ/p^{r} ←^{≃} L_{HZ}(X∧

SZ/p^{r}) ⟶^{1∧δ_{r}} L_{HZ}(X ∧ SZ/p) →^{≃} SZ/p is not zero and so δ′_{r} = kδ_{r} for

some k ≠ 0 ∈ Z/p. Then we obtain the commutative diagram:

L_{HZ}(X ∧ SZ/p)  L_{HZ}X ∧ SZ/p^{r+1}  L_{HZ}(X ∧ SZ/p^{r})  ΣL_{HZ}X ∧ SZ/p SZ/p  W_{r}  SZ/p^{r}  ΣSZ/p SZ/p  SZ/p^{r+1}  SZ/p^{r}  ΣSZ/p w^{1∧p^{r}}_{_{≃} u}  w^{1∧j_{r}} u  w^{1∧δ_{r}}_{u_{≃}}  _{u_{≃}} w_{_{k^{−1}} u}  w u  _{^{δ′_{r}}  w}_{u_{=}}  _{u_{k^{−1}}}_{^{p^{r}}  w}  _{^{j_{r}}  w}  _{^{δ_{r}}  w}

Here W_{r} denotes a fiber of δ′_{r}. Now the dotted arrows give the desired equivalence. □

Theorem 4.3. Pic(L_{HZ/p}) = Z.

Proof. Let X ∈ Pic(L_{HZ/p})^{0}. Then by [7, Cor. 2.2] with n = 1 and E = HZ, we see that X = lim_{r}(L_{HZ}X ∧SZ/p^{r}), which is equivalent to lim_{r} SZ/p^{r} = SZ_{p} by Lemma 4.2. Since any element X of Pic(L_{HZ/p}) is a suspension of an element of X ∈ Pic(L_{HZ/p})^{0}, X = Σ^{t}SZ_{p} for some t ∈ Z. Therefore, Pic(L_{HZ/p}) = Z whose generator is ΣSZ_{p}. □

Proof of Theorem 1.1. Since ⟨HZ/p⟩ > ⟨I⟩, any spectrum E with ⟨E⟩ ≥ ⟨HZ/p⟩ has a finite local. Theorem 1.1 is now a corollary of Corollary 3.6 and Theorem 4.3. □

§5. Some remarks

Let K(n) denote the n-th Morava K spectrum and consider the spectrum H_{n} = ⋁_{i≥n} K(i). Then H_{0}-local (resp. H_{0}-acyclic) spectrum is called harmonic (resp. dissonant). A finite spectrum X is called a type n spectrum if K(n−1)_{∗}(X) = 0 and K(n)_{∗}(X) ≠ 0. By [6] and [4], we have a finite ring spectrum MJ_{n} such that BP_{∗}(MJ_{n}) = BP_{∗}/J_{n} for an invariant ideal J_{n} = (p^{e_{0}}, v_{1}^{e_{1}}, …, v_{n−1}^{e_{n−1}}) of BP_{∗}. The spectrum MJ_{n} is a type n spectrum, that is K(i) ∧ MJ_{n} = pt for i < n. Therefore, H_{0} ∧ MJ_{n} = ⋁_{i≥0} K(i) ∧ MJ_{n} = H_{n}.

Lemma 5.1. For n ≥ 0, ⟨H_{n}⟩ = ⟨H_{0} ∧ MJ_{n}⟩ for any MJ_{n}.

Proposition 5.2. H_{n} for n ≥ 0 has a finite local.

Proof. Since a finite spectrum is harmonic [14, Cor. 4.5], H_{0} has a finite local. Therefore, H_{n} for n > 0 has a finite local by Lemma 5.1 and (3.1). □

Let F (n) denote a finite spectrum of type n. Then it admits a self map f: F (n) → F (n) that induces the multiplication of v_{n}^{a} on BP_{∗}- homology for some integer a > 0 by [6]. The cofiber of f yields F (n+1), and so ⟨F (n)⟩ ≥ ⟨F (n + 1)⟩ (cf. the class invariance theorem [15]). Furthermore, we denote a spectrum f^{−1}F (n) as Tel(n). Note that the Bousfield class of Tel(n) depends only on n (cf. [7, Lemma 1.3]). Then the Bousfield class of H_{n}^{f} = ⋁_{i≥n} Tel(i) also depends only on n. Since K(n) ∧ Tel(n) ≠ pt, we see that ⟨Tel(n)⟩ ≥ ⟨K(n)⟩ by [14, Th. 2.1(h)], and so ⟨H_{0}^{f}⟩ ≥ ⟨E(n)⟩ for each n ≥ 0. Indeed, ⟨E(n)⟩ = ⟨⋁_{0≤i≤n} K(i)⟩.

These yield relations of Bousfield classes of spectra.

⟨S^{0}⟩ ≥ ⟨F (1)⟩ ≥ ⟨F (2)⟩ ≥ ⋯ ≥ ⟨HZ/p⟩ ≥ ⟨I⟩

⟨S^{0}⟩ ≥ ⟨X(1)⟩ ≥ ⟨X(2)⟩ ≥ ⋯ ≥ ⟨BP ⟩ ≥ ⟨HZ⟩ ≥ ⟨HZ/p⟩ ≥ ⟨I⟩ ⟨S^{0}⟩ ≥ ⟨H_{0}^{f}⟩ ≥ ⟨H_{1}^{f}⟩ ≥ ⋯ ≥ ⟨I⟩

⟨BP ⟩ ≥ ⟨H_{0}⟩ ≥ ⟨H_{1}⟩ ≥ ⋯ ≥ ⟨I⟩

⟨BP ⟩ ≥ ⟨H_{0}⟩ ≥ ⋯ ≥ ⟨I ∨ E(n)⟩ ≥ ⟨I ∨ E(n − 1)⟩ ≥ ⋯

⋯ ≥ ⟨I ∨ E(0)⟩ = ⟨I ∨ HQ⟩ ≥ ⟨I⟩

⟨HZ/p⟩ ≥ ⟨IBP ⟩ ≥ ⋯ ≥ ⟨IX(n)⟩ ≥ ⋯ ≥ ⟨IX(1)⟩ ≥ ⟨I⟩

Here X(i) is the spectrum given in [14], and IY is the Brown-Comenetz dual of Y. The second sequence is given in [14, p.369] and the bottom sequence is given in [8, Th.8.4]. The Picard groups of the spectra appearing in the first two sequences except for I are all isomorphic to Z by Corollary 3.6 and Theorem 4.3. The other sequences yield the filtrations of Pic(L_{I})

Z = Pic(L_{S^{0}}) ⊂ Pic(L_{H_{0}^{f}}) ⊂ Pic(L_{H_{1}^{f}}) ⊂ ⋯ ⊂ Pic(L_{I})
Z = Pic(L_{BP}) ⊂ Pic(L_{H_{0}}) ⊂ Pic(L_{H_{1}}) ⊂ ⋯ ⊂ Pic(L_{I})

Z = Pic(L_{BP}) ⊂ Pic(L_{H_{0}}) ⊂ ⋯ ⊂ Pic(L_{I∨E(n)}) ⊂ Pic(L_{I∨E(n−1)}) ⊂ ⋯

⋯ ⊂ Pic(L_{I∨E(0)}) ⊂ Pic(L_{I})

Z = Pic(L_{HZ/p}) ⊂ Pic(L_{IBP}) ⊂ ⋯ ⊂ Pic(L_{IX(n)}) ⊂ ⋯
⋯ ⊂ Pic(L_{IX(1)}) ⊂ Pic(L_{I})

by Corollary 3.6. We have a problem

Problem. Is Pic(L_{I}) = Z?

If this has an affirmative answer, then for all spectra E with a finite local, Pic(L_{E}) = Z by Corollary 3.6. For each X ∈ Pic(L_{I}), put W_{X} = X ∧ D_{p}(X) and let F_{X} be the fiber of the localization map W_{X} → L_{I}W_{X} = SZ_{p}. For a spectrum E, let aE denote the spectrum given by Bousfield in [1, Lemma 1.13] (see also [2] for its proof). Then, we have ⟨aE⟩ ≥ ⟨C⟩ for any E-acyclic spectrum C.

Proposition 5.3. The collection of X ∈ Pic(L_{I}) such that Pic(L_{aF_{X}}) = Z consists of suspensions of SZ_{p}. In particular, if Pic(L_{aF_{X}}) = Z for each X ∈ Pic(L_{I}), then Pic(L_{I}) = Z.

Proof. By definition, F_{X} is I-acyclic, which implies that ⟨aF_{X}⟩ ≥ ⟨I⟩. Since F_{X} is also aF_{X}-acyclic, it follows that L_{aF_{X}}W_{X} = SZ_{p}. Therefore, X ∈ Pic(L_{aF_{X}}) = Z and so X is a suspension of SZ_{p}. □

Since ⟨C⟩ ≤ ⟨aE⟩ for an E-acyclic spectrum C as noted above, we see that HZ/p∧F_{X} = pt implies ⟨aF_{X}⟩ ≥ ⟨HZ/p⟩, and so Pic(L_{aF_{X}}) = Z by Theorem 1.1.

Corollary 5.4. Let X be an element of Pic(L_{I}). If (HZ/p)_{∗}(F_{X}) = 0, then X is a suspension of SZ_{p}.

To study the filtration of Pic(L_{I}), the next problem seems interesting.

Problem. Determine Pic(L_{I∨K(n)}).

Next we consider an invertible spectrum that is also a ring spectrum. We deduce the following by the similar argument given in the previous sections.

Proposition 5.5. An invertible ring spectrum is the sphere spectrum in L_{E} for any E.

Proof. Let X be an invertible ring spectrum in L_{E}. Then X ∧ X = X ∨ (X ∧ \overline{X}) for a cofiber \overline{X} of the unit map S^{0} → X. Let D_{E} denote the Spanier-Whitehead dual functor on L_{E}. Since D_{E}(X) is an inverse of X, we see that

\begin{matrix}X = L_{E}(D_{E}(X) ∧ X ∧ X) = L_{E}(D_{E}(X) ∧ X) ∨ L_{E}(D_{E}(X) ∧ X ∧ \overline{X}) \\ = L_{E}S^{0} ∨ L_{E}\overline{X}. \end{matrix}

Thus [S^{0}, L_{E}S^{0}]_{∗} is a direct summand of [X, X]_{∗}. On the other hand, [X, X]_{∗} is isomorphic to [S^{0}, L_{E}S^{0}]_{∗} under the map assigning f: X → X to f ∧1: L_{E}(X∧D_{E}(X)) → L_{E}(X∧D_{E}(X)). Since the isomorphism is the one of [S^{0}, L_{E}S^{0}]_{∗}-modules and it sends the identity to identity, the other direct summands of [X, X]_{∗} are zero. In particular, we obtain L_{E}\overline{X} = pt as desired. □

In the category L_{E}, a spectrum A is small if [A, L_{E}(⋁_{α} X_{α})]_{∗} = ⨁_{α}[A, X_{α}]_{∗} for any wedge sum L_{E}(⋁_{α} X_{α}) in L_{E}. In [10, Appendix B], Hovey and Strickland showed that if E has a finite local and is I-acyclic, then the category L_{E} does not have non-zero small objects, and that BP, H_{0}, HZ, HZ/p and I are examples for E and F (n) is not. Here, we give other examples. Since I and K(n) are I-acyclic, we see the following:

Proposition 5.6. For n ≥ 0, L_{H_{n}}, L_{I∨E(n)} and L_{I∨K(n)} do not have non-zero small objects.

Since the sphere spectrum S^{0} has a finite local, so does the finite spectrum F (n) by (3.1). Furthermore, we see Tel(n) ∧ F (n + 1) = pt by definition. Thus Tel(n) is F (n+1)-acyclic, and so is I-acyclic. Thus we have the following:

Proposition 5.7. For n ≥ 0, L_{H_{n}^{f}} does not have non-zero small objects.

This proof indicates the following fact on spectra with a finite acyclic. Here a spectrum E has a finite acyclic if there is a finite spectrum X ≠ pt such that L_{E}X = pt.

Proposition 5.8. If E has a finite acyclic, then E is I-acyclic.

This follows from

Proposition 5.9. If E admits an E-acyclic spectrum that has a finite local, then E is I-acyclic.

Proof. Let X denote the E-acyclic spectrum of the proposition. Then E ∧ X = pt. Since ⟨X⟩ ≥ ⟨I⟩, we obtain E ∧ I = pt. □

Proposition 5.10. Neither L_{X(n)} nor L_{IX(n)} has non-zero small objects.

Proof. By [8, Th. 8.4], ⟨I⟩ ≤ ⟨IX(n)⟩ ≤ ⟨X(n)⟩, and so both of X(n) and IX(n) have finite locals. It is also shown in [8, Lemma 7.1] that X(n) is I-acyclic, and so is IX(n). □

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