The homotopy groups π_{∗}(L_{n}T (m) ∧ V (n − 2))

Katsumi Shimomura

Abstract. Let V_{T (m)}(n) denote the spectrum such that BP_{∗}(V_{T (m)}(n)) = BP_{∗}/I_{n+1}[t_{1}, …, t_{m}] for the ideal I_{n+1} = (p, v_{1}, …, v_{n}). In the title, we write T (m) ∧ V (n − 2) as V_{T (m)}(n − 2). Ravenel determined the structure of the Adams-Novikov E_{2}-term for the homotopy groups π_{∗}(L_{n}V_{T (m)}(n − 1)) for n ≤ m + 2 and n < m + 2 if p = 2, where L_{n} denotes the Bousfield-Ravenel localization functor with respect to v_{n}^{−1}BP. In this note, we show the existence of V_{T (m)}(n) for n ≤ m + 2, and determine the Adam-Novikov E_{2}-term E_{2}^{s,t} converging to π_{∗}(L_{n}V_{T (m)}(n − 2)) for m ≥ n^{2} − n − 1. This is a generalization of [2] in which π_{∗}(L_{2}T (m) ∧ V (0)) is determined.

Introduction

In this paper, we assume that everything is localized away from a prime number p. Let L_{n} denote the Bousfield-Ravenel localization functor with respect to the Johnson-Wilson spectrum E(n) [5]. Here, E(n)_{∗} = Z_{(p)}[v_{1}, …, v_{n−1}, v_{n}, v_{n}^{−1}]. Ravenel introduced the spectrum T (m) for an integer m ≥ 0, which is characterized by the Brown-Peterson homology

BP_{∗}(T (m)) = BP_{∗}[t_{1}, t_{2}, …, t_{m}] ⊂ BP_{∗}BP = BP_{∗}[t_{1}, t_{2}, …],

which is an inclusion of BP_{∗}BP -comodules. Here we study the E_{2}-term

Ext_{E(n)_{∗}E(n)}^{∗}(E(n)_{∗}, E(n)_{∗}/(p, v_{1}, …, v_{n−2})[t_{1}, …, t_{m}])

of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{n}T (m) ∧ V (n − 2)), if V (n − 2) exists. First, we give a proof of the existence of V_{T (m)}(n) such that BP_{∗}(V_{T (m)}(n)) = BP_{∗}/(p, v_{1}, …, v_{n})[t_{1}, …, t_{m}] if n ≤ m + 2. So L_{n}T (m) ∧ V (n − 2) in the title means L_{n}V_{T (m)}(n − 2). Ravenel determined the Adams-Novikov E_{2}-term for π_{∗}(L_{n}V_{T (m)}(n − 1)) (cf. [6]). In [2], Kamiya and the author determined the homotopy groups π_{∗}(L_{2}T (m) ∧ V (0)). (Note that in [2], we did not consider the case where m = 1 and p = 2, which was studied in [8].) The results of [2] show good natures of the E_{2}-term, which is expanded to this paper. One of them is: if we know the 0-th line, then we get all lines. Under this motto, we determine the Adam-Novikov E_{2}-term for the homotopy groups π_{∗}(V_{T (m)}(n − 2)) from the E_{2}-term for π_{∗}(L_{n}V_{T (m)}(n − 1)) by using Bockstein spectral sequence. Actually we determine first the 0-th line of the chromatic E_{1}- term H(m)^{0}M_{n−1}^{1} = Ext_{Σ(m)}^{∗}(B, M_{n−1}^{1}[v_{n+m}]), which is the homology groups of the cobar complex on the Σ(m)-comodule M_{n−1}^{1}[v_{n+m}] over the Hopf algebroid (B, Σ(m)) = (E(n)_{∗}, E(n)_{∗}[t_{m+1}, t_{m+2}, …]⨂_{BP_{∗}}E(n)). Then by equations of the differentials used for the 0-th line, we determine the whole E_{2}-term. To make argument simple, we do this all under the assumption m ≥ n^{2} − n − 1.

2000 Mathematics Subject Classification. Primary 55Q10; Secondary 55Q51, 55Q45.

Key words and phrases. Bousfield-Ravenel localization, Adams-Novikov spectral sequence, Ravenel spectrum.

In the next section, we show the existence of the spectrum V_{T (m)}(n). We study the right unit map η_{R} of the Hopf algebroid and then prove the change of rings theorem along the line of [4, Th.2.1] in section 2. In section 3, we determine the 0-th line H(m)^{0}M_{n−1}^{1} (Theorem 3.9). We determine the structure of the whole line H(m)^{∗}M_{n−1}^{1} in section 4 (Theorem 4.4) and show the E_{2}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{n}V_{T (m)}(n − 2)) (Theorem 5.1), and then homotopy groups itself (Corollary 5.2) in the last section.

1. The existence of a spectrum

Let I_{n+1} be the ideal of BP_{∗} = Z_{(p)}[v_{1}, v_{2}, …] generated by p, v_{1}, …, v_{n}. Then the Smith-Toda spectrum V (n) is characterized by the BP_{∗}-homology BP_{∗}(V (n)) = BP_{∗}/I_{n+1}, and is known to exist if n ≤ 3 and 2n < p. Ravenel constructed a spectrum T (m) such that BP_{∗}(T (m)) = BP_{∗}[t_{1}, …, t_{m}]. We fix an integer m > 0 and denote the spectrum T (m) by T. Then we have an existence theorem.

Lemma 1.1. Suppose that n < m + 3. Then there exists a spectrum V_{T} (n) such that

BP_{∗}(V_{T} (n)) = BP_{∗}/I_{n+1}[t_{1}, …, t_{m}].

In order to prove this, we use the Adams-Novikov spectral sequence converging to π_{∗}(T), whose E_{2}-term is E_{2}^{s,t} = Ext_{Γ}^{s,t}(A, BP_{∗}(T)). We use Ravenel’s notation Γ(m) (cf. [7]) which denotes the Hopf algebroid Γ(m) = A[t_{m+1}, t_{m+2}, …] over A. Then we have a change of rings theorem E_{2}^{s,t} = Ext_{Γ(m)}^{s,t}(A, A), since BP_{∗}(T) = Γ□_{Γ(m)}A. For the structure map, we have

Lemma 1.2. Let η_{R}: A → Γ(m) be the right unit. Then

η_{R}(v_{k}) = v_{k}

for k ≤ m.

Proof. Our computation is based on Hazewinkel’s and Quillen’s formulae:

(1.3) \begin{aligned}v_{k} & = & pl_{k} − ∑_{i=1}^{k−1} l_{i}v_{k−i}^{p^{i}} & ∈ BP_{∗}⨂Q = Z_{(p)}[l_{1}, l_{2}, …] \\ η_{R}(l_{k}) & = & ∑_{i=0}^{k} l_{i}t_{k−i}^{p^{i}} & ∈ BP_{∗}(BP)⨂Q = BP_{∗}⨂Q[t_{1}, t_{2}, …] \end{aligned}

Here l_{0} = 1. Then this follows immediate by computing it mod (t_{1}, …, t_{m}). □

Proof of Lemma 1.1. Consider the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(T) with the E_{2}-term E_{2}^{s,t} = Ext_{Γ(m)}^{s,t}(A, A). Observing the reduced cobar complex, we obtain the vanishing line such that E_{2}^{s,t} = 0 if t < 2s(p^{m+1} − 1), since the kernel \overline{Γ(m)} of the counit map ϵ: Γ(m) → A is 2(p^{m+1} − 1) − 1 connected. Therefore, v_{k} is a permanent cycle of the spectral sequence if k ≤ m + 2, and we denote the homotopy element by v_{k} as well.

Now we define a module spectrum V_{T} (k) inductively. Put V_{T} (−1) = T and suppose that a module spectrum V_{T} (k) is defined. Then we define the self map v_{k+1} by the composite v_{k+1} : V_{T} (k) T ∧ V_{T} (k) V_{T} (k) w^{v_{k+1}∧1} ^{ν_{k}} w for the structure map ν_{k} of V_{T} (k). Then we define V_{T} (k + 1) as a cofiber of v_{k+1}. The module structure ν_{k+1} is defined by the commutative diagram:

T ∧ V_{T} (k)  T ∧ V_{T} (k)  T ∧ V_{T} (k + 1) V_{T} (k)  V_{T} (k)  V_{T} (k + 1)^{1∧v_{k+1}} w_{_{ν_{k}} u}  w_{u_{ν_{k}}}  _{u_{ν_{k+1}}}_{^{v_{k+1}} w}  w

□

Remark 1.4. There is a more strict vanishing line E_{2}^{s,t} = 0 if t < 2(ps_{1} + s_{2})(p^{m+1} − 1) for s = 2s_{1} + s_{2} with s_{2} = 0, 1. In this case, the condition on n and m is replaced by n ≤ m + 3.

Remark 1.5. This proof works for the existence of spectrum MJ_{T} if n < m + 3 such that BP_{∗}(MJ_{T}) = BP_{∗}/J[t_{1}, …, t_{m}], where J is an invariant ideal of BP_{∗} generated by p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n}^{i_{n}}.

2. A change of rings theorem

Let (B, Σ) be the Hopf algebroid associated to the n-th Johnson-Wilson spectrum E(n) at the prime p. Then (B, Σ) = (E(n)_{∗}, E(n)_{∗}(E(n))) = (Z_{(p)}[v_{1}, v_{2}, …, v_{n}, v_{n}^{−1} ], B[t_{1}, t_{2}, …]⨂_{BP_{∗}}B). Here BP_{∗} acts on B by sending v_{i} for i > n to zero. We write

Ext^{∗}(M) = Ext_{Σ}^{∗}(B, M)

for a Σ-comodule M. Consider the Σ-comodules

L(m) = B[t_{1}, …, t_{m}],
Φ(m) = L(m)⨂_{BP_{∗}}B
= B[t_{1}, …, t_{m}]/(η_{R}(v_{n+i}) : i > 0) and
B(m) = B[v_{n+1}, …, v_{n+m}].

The comodule structures of these comodules are defined by

ψ_{L(m)} : L(m) y w E(n)_{∗}(BP) ^{Δ} wE(n)_{∗}(BP)⨂_{B}Σ ww L(m)⨂_{B}Σ ψ_{Φ(m)} : Φ(m) y w Σ ^{Δ} wΣ⨂_{B}Σ ww Φ(m)⨂_{B}Σ and

ψ_{B(m)} : B(m) y w v_{n}^{−1}BP_{∗} w_{η_{R}}v_{n}^{−1}BP_{∗}⨂_{BP_{∗}}BP_{∗}(BP) ww B(m)⨂_{B}Σ

We have a similar formula to Landweber’s one.

Lemma 2.1. Let η̃_{R} denote the composite BP_{∗} wBP_{∗}(BP) ww BP_{∗}[t_{m}]. Then we have

η̃_{R}(v_{m+n}) ≡ v_{m+n} + v_{n}t_{m}^{p^{k}} − v_{n}^{p^{m}}t_{m} mod I_{n}.

Proof. We compute η_{R}(v_{m+n}) in BP_{∗}⨂Q[t_{m}]/(l_{1}, …, l_{n−1}) by the formulae (1.3). Inductively we see that η_{R}(v_{k}) = v_{k} for k < n + m. Then

η_{R}(v_{m+n}) = p(l_{m+n} + l_{n}t_{m}^{p^{m}}) − ∑_{i=1}^{m−1} l_{i}v_{m+n−1}^{p^{i}} − (l_{m} + t_{m})v_{n}^{p^{m}} − ∑_{i=m+1}^{m+n−1} l_{i}v_{m+n−1}^{p^{i}}

and we have the lemma. □

We recall the chromatic comodules defined inductively by N_{j}^{0} = B/I_{j}, M_{j}^{i} = v_{i+j}^{−1}N_{j}^{0} and the short exact sequence

(2.2)  0 wN_{j}^{i} y w M_{j}^{i} wN_{j}^{i+1} w0

for i + j ≤ n. Note that N_{j}^{n−j} = M_{j}^{n−j}.

Lemma 2.3. There is an isomorphism

M_{0}^{n}⨂_{B}L(m) ≅ M_{0}^{n}⨂_{B}Φ(m)⨂_{B}B(m)

of Σ-comodules.

Proof. We prove this by induction on m. For m = 0, it is trivial.

By Lemma 2.1, we see that \overline{η_{R}}(v_{n+m+1}) = v_{n}t_{m+1}^{p^{n}} − v_{n}t_{m+1} + w in L(m + 1) for some w ∈ L(m) + I_{n}BP_{∗}[t_{m+1}], where \overline{η_{R}} denotes the composition BP_{∗} ^{η_{R}}w BP_{∗}BP wL(m + 1). Using this equation finite times, we see that for any k ≥ 0 and u ∈ M_{0}^{n}, there exist elements w_{i,j} ∈ L(m) for integers i and j with 0 ≤ i ≤ n−1 uniquely such that

u ⊗ t_{m+1}^{k} = u ⊗ (∑ w_{i,j}t_{m+1}^{i}\overline{η_{R}}(v_{n+m+1})^{j}) .

Now we define L(m)-module map

L: M_{0}^{n}⨂L(m + 1) wM_{0}^{n}⨂L(m)[t_{m+1}]/(η_{R}(v_{m+n+1}))⨂_{B}B[v_{n+m+1}]

by L(u ⊗ t_{m+1}^{k}) = u ⊗ (∑ w_{i,j}t_{m+1}^{i} ⊗ v_{n+m+1}^{j}). Then we see that L is a comodule map by definition. Since M_{0}^{n}⨂L(m)[t_{m+1}]/(η_{R}(v_{m+n+1}))⨂_{B}B[v_{n+m+1}] is isomorphic to M_{0}^{n}⨂Φ(m+1)⨂_{B}B(m+1) as comodules by inductive hypothesis, we have the lemma. □

Let Σ(m) denote the Hopf algebroid

Σ(m) = E(n)_{∗}[t_{m}, t_{m+1}, …]⨂_{BP_{∗}}E(n)_{∗}

over B. Then

(2.4)  Φ(m) = Σ□_{Σ(m)}B,

and we obtain the change of rings theorem.

Lemma 2.5. Suppose that a Σ-comodule M is a B/I_{n−j+1}-module and each element x of M is killed by I_{n}^{k} for some k = k(x). Then there is an isomorphism

Ext^{∗}(M⨂_{B}L(m)/I_{n−j+1})

≅ Ext_{Σ(m+1)}^{∗}(B, M⨂_{B}(B/I_{n−j+1})[v_{m+n−j+2}, …, v_{m+n}])⨂_{B}B(m − j + 1).

Proof. The assumption on M gives us a comodule M′ such that M = M′⨂_{B}M_{0}^{n}. By (2.4), we have

Ext^{∗}(M⨂_{B}L(m)/I_{n−j+1}) ≅ Ext_{Σ(m+1)}^{∗}(B, M⨂_{B}B(m)/I_{n−j+1}).

The lemma now follows from the equation B(m)/I_{n−j+1} = (B/I_{n−j+1})[v_{m+n−j+2}, …, v_{m+n}])⨂_{B}B(m − j + 1) and the fact that the elements of B(m − j + 1) are all primitive. □

In this paper, we study the module Ext^{∗}(M_{n−1}^{1}⨂_{B}L(m)) which is isomorphic to Ext_{Σ(m+1)}^{∗}(B, M_{n−1}^{1}[v_{m+n}])⨂_{B}B(m − 1) by Lemma 2.5 for j = 2. We also use the notation

H(m)^{∗}M = Ext_{Σ(m+1)}(B, M[v_{m+n}])

for a Σ-comodule M, and compute it by the cobar complex Ω^{∗}M[v_{m+n}] associated to the Hopf algebroid (B/I_{n−1}, Σ(m + 1)/I_{n−1}). Hereafter we write M[v_{m+n}] for M⨂_{B}B[v_{m+n}]. Then,

Ext^{∗}(M_{n−1}^{1}⨂_{B}L(m)) ≅ H(m)^{∗}M_{n−1}^{1}⨂_{B}B(m − 1).

3. H(m)^{0}M_{n−1}^{1}

Here we compute the cobar complex Ω^{∗}M for the Hopf algebroid (B′, Σ′) = (B/I_{n−1}, Σ(m + 1)/I_{n−1}).

Lemma 3.1. Let η_{R} denote the structure map B′[v_{m+n}] wB′[v_{m+n}]⨂_{B}Σ′. Then we have

η_{R}(v_{m+n}) = v_{m+n} + v_{n−1}t_{m+1}^{p^{n−1}} − v_{n−1}^{p^{m+1}}t_{m+1}

for non-negative integers m and n. Furthermore, we have relations

t_{m+i}^{p^{n}} = v_{n}^{p^{m+i}−1}t_{m+i} − v_{n−1}v_{n}^{−1}t_{m+i+1}^{p^{n−1}} + v_{n−1}^{p^{m+i+1}}v_{n}^{−1}t_{m+i+1} if 0 < i < n − 1,
t_{m+n−1}^{p^{n}} = v_{n}^{p^{m+n−1}−1}t_{m+n−1} − v_{n−1}v_{n}^{−1}t_{m+n}^{p^{n−1}} + v_{n−1}^{p^{m+n}}v_{n}^{−1}t_{m+n}
−v_{n−1}v_{n}^{−1}C_{n−1}(n + m) and
t_{m+n}^{p^{n}} ≡ v_{n}^{p^{m+n}−1}t_{m+n} mod I_{n}.

Here

pC_{k}(n + m) = η_{R}(v_{m+n}^{p^{k}}) − (v_{m+n}^{p^{k}} + v_{n−1}^{p^{k}}t_{m+1}^{p^{n+k−1}} − v_{n−1}^{p^{m+k+1}}t_{m+1}^{p^{k}}).

Proof. Using the formulae (1.3), we compute η_{R}(v_{k}) in BQ[t_{m+1}, t_{m+2}, …]

for BQ = (BP_{∗}⨂Q)/(l_{1}, …, l_{n−2}, v_{n+1}, …, { \atop v_{m+n}}, …). Here x̂ means that x is

taken off. Note that v_{i} = 0 unless i = n − 1, n nor m + n, v_{i} = pl_{i} if i = n − 1 or n, v_{m+n} = pl_{m+n} − l_{m}v_{n}^{p^{n}} − l_{m+1}v_{n−1}^{p^{m+1}}, and pl_{n+i} = l_{i+1}v_{n−1}^{p^{i+1}} + l_{i}v_{n}^{p^{i}} for i > 0 with

i ≠ n in BQ, which in particular equals zero if i < n − 2 and equals l_{n−1}v_{n−1}^{p^{n−1}} if i = n − 2. By (1.3), we easily see that η_{R}(v_{i}) = v_{i} if i < m + n and compute

η_{R}(v_{m+n}) = p(l_{m+n} + l_{n−1}t_{m+1}^{p^{n−1}}) − l_{m}v_{n}^{p^{m}} − (l_{m+1} + t_{m+1})v_{n−1}^{p^{m+1}}

to obtain the first equation. Similarly, we compute for 0 < i ≤ n

0 = η_{R}(v_{m+n+i})

= p(l_{m+n+i} + l_{n−1}t_{m+i+1}^{p^{n−1}} + l_{n}t_{m+i}^{p^{n}} + \underline{l_{n+i−2}t_{m+2}^{p^{n+i−2}}} + \underline{l_{n+i−1}t_{m+1}^{p^{n+i−1}}}_{a})

−\underline{l_{n−1}η_{R}(v_{m+i+1})^{p^{n−1}}}_{a} − \underline{l_{n}η_{R}(v_{m+i})^{p^{n}}} − (l_{m+i} + \underline{l_{i−1}t_{m+1}^{p^{i−1}}} + t_{m+i})v_{n}^{p^{m+i}}

−(l_{m+i+1} + \underline{l_{i}t_{m+1}^{p^{i}}}_{a} + \underline{l_{i−1}t_{m+2}^{p^{i−1}}} + t_{m+i+1})v_{n−1}^{p^{m+i+1}}.

If i < n − 1, then the underlined terms are zero, and we have the relation

0 = v_{n−1}t_{m+i+1}^{p^{n−1}} + v_{n}t_{m+i}^{p^{n}} − t_{m+i}v_{n}^{p^{m+i}} − t_{m+i+1}v_{n−1}^{p^{m+i+1}},

which shows the second equation. If i = n − 1, then the underlined terms with a are pl_{n+i−1}t_{m+1}^{p^{2n−2}} = l_{n−1}v_{n−1}^{p^{n−1}}t_{m+1}^{p^{n+i−1}}, −l_{n−1}η_{R}(v_{m+i+1})^{p^{n−1}} = −l_{n−1}(v_{m+n}^{p^{n−1}} +

v_{n−1}^{p^{n−1}}t_{m+1}^{p^{2n−2}} −v_{n−1}^{p^{m+n}}t_{m+1}^{p^{n−1}} +pC_{n−1}(m+n)) and −l_{i}v_{n−1}^{p^{m+i+1}}t_{m+1}^{p^{i}} = −l_{n−1}v_{n−1}^{p^{m+n}}t_{m+1}^{p^{n−1}}, and we have the third one. The last congruence also follows from the equation. □

We now define the elements x_{i} for i ≥ 0 similar to those of [3].

x_{0} = v_{m+n}

x_{i} = \{ \begin{matrix}x_{i−1}^{p} & n |̸ i < n^{2} \\ x_{(k−1)n}^{p^{n}} − v_{n−1}^{a_{i−n}(p^{n}−1)}v_{n}^{b_{i−n}(p^{n}−1)+p^{m+kn}−p^{k(n−1)}}x_{(k−1)n} & i = kn < n^{2} \\ x_{(n−1)n}^{p^{n}} − v_{n−1}^{a_{i−n}(p^{n}−1)}v_{n}^{b_{i−n}(p^{n}−1)+p^{m+n^{2}} −p^{(n−1)n}}x_{(n−1)n} \\ +(−1)^{n}v_{n−1}^{a_{i−n}p^{n}}v_{n}^{b_{i−n}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}v_{n+m}^{p^{n^{2}−n}−p^{n^{2}−n−1}}w_{n^{2}−n−1} \\ i = n^{2} \\ x_{i−1}^{p} − v_{n−1}^{p^{j}a_{i−n}}v_{n}^{p^{j}b_{i−n}}v_{n+m}^{p^{j}c}(x_{i−n−2}^{p} − x_{i−n−1}) & i = n^{2} + j > n^{2} \end{matrix}

where w_{k} denotes the element x_{0}^{p^{k}} − x_{k} and integers a_{i}, b_{i}, c_{i}, a, b and c are defined by

a_{0} = 1

a_{i} = \{ \begin{matrix}pa_{i−1} & n |̸ i < n^{2} \\ pa_{i−1} + p^{k(n−1)} = p^{k(n−1)} \frac{p^{k+1} − 1}{p − 1} & i = kn < n^{2} \\ p^{i−n^{2}}a + a_{i−n−1} & i ≥ n^{2} \end{matrix}

b_{0} = 0

b_{i} = \{ \begin{matrix}pb_{i−1} & n |̸ i < n^{2} \\ pb_{i−1} − p^{k(n−1)} = p^{k(n−1)} \frac{p^{k} − 1}{p − 1} & i = kn < n^{2} \\ p^{i−n^{2}}b + b_{i−n−1} & i ≥ n^{2} \end{matrix}

c_{i} = \{ \begin{matrix}0 & i < n^{2} \\ p^{i−n^{2}}c + c_{i−n−1} & i ≥ n^{2} \end{matrix}
a = p^{n}a_{n^{2}−n} = p^{n^{2}−n+1} \frac{p^{n} − 1}{p − 1},
b = pb_{n^{2}−1} + p^{m+n^{2}} − p^{n(n−1)} = −p^{n^{2}−n+1} \frac{p^{n−1} − 1}{p − 1} + p^{m+n^{2}} − p^{n(n−1)},
c = (p − 1)p^{n^{2}−n−1}.

Furthermore, we consider the integer A_{i} defined by

A_{0} = p^{m+1},

A_{i} = \{ \begin{matrix}pA_{i−1} & n |̸ i < n^{2} or i > n^{2} \\ (p^{n} − 1)a_{(k−1)n} + A_{(k−1)n} = a_{kn} − \frac{p^{(k+1)(n−1)} − 1}{p^{n−1} − 1} + p^{m+1} \\ i = kn < n^{2} \\ p^{n}a_{n^{2}−n} + 2p^{n^{2}−n−1} − p^{n^{2}−2n−1} & i = n^{2} \end{matrix}

Then we have the key lemma to determine H(m)^{0}M_{n−1}^{1}.

Proposition 3.2. Let m and n be positive integers with m ≥ n^{2} − n − 1, and d: Ω^{0}E(n)_{∗}[v_{n+m}] → Ω^{1}E(n)_{∗}[v_{n+m}] denote the defferential defined by d(x) = η_{R}(x) − x. Then d acts on x_{i} by

d(x_{i}) ≡ v_{n−1}^{a_{i}}v_{n}^{b_{i}}v_{n+m}^{c_{i}}g_{i} mod J(A_{i}).

Here J(k) is the ideal of E(n)_{∗} generated by p, v_{1}, …, v_{n−2} and v_{n−1}^{k}, and g_{i} is the element defined by

g_{i} = \{ \begin{matrix}t_{m+k+1}^{(k+1)(n−1)} & 0 ≤ i = kn < n^{2} \\ g_{i−1}^{p} & n |̸ i < n^{2} \\ g_{i−n−1} & i > n^{2}. \end{matrix}

Proof. d(x_{0}) follows immediately from the first equation of Lemma 3.1. Note that a_{i} < A_{i} by the assumption on m and n. It suffices to show the

following:

d(x_{kn}) ≡ (−1)^{l}v_{n−1}^{a_{kn}}v_{n}^{b_{kn}}t_{m+k+1}^{p^{(k+1)(n−1)}} mod J(A_{kn}) k < n,
d(x_{kn+i}) ≡ d(x_{kn})^{p^{i}} mod J(A_{kn+i}) 0 ≤ i < n, and
d(x_{n^{2}+i}) ≡ v_{n−1}^{p^{i}a}v_{n}^{p^{i}b}v_{n+m}^{p^{i}c}d(x_{n^{2}+i−n−1}) mod J(A_{n^{2}+i}) i ≥ 0.

We use the relations given in Lemma 3.1 freely to compute in the following. Suppose that

(3.3)  d(x_{(k−1)n}) ≡ (−1)^{k}v_{n−1}^{a_{i}}v_{n}^{b_{i}}t_{m+k}^{p^{k(n−1)}} mod J(A_{i})

for i = (k − 1)n. Raise it to the p^{n}-th power, and we have

d(x_{(k−1)n}^{p^{n}})
≡ (−1)^{k}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}t_{m+k}^{p^{k(n−1)+n}} mod J(p^{n}A_{i})
≡ (−1)^{k}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}(v_{n}^{p^{m+k}−1}t_{m+k} − v_{n−1}v_{n}^{−1}t_{m+k+1}^{p^{n−1}}
+v_{n−1}^{p^{m+k+1}}v_{n}^{−1}t_{m+k+1})^{p^{k(n−1)}} mod J(p^{n}A_{i})
≡ (−1)^{k}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}(v_{n}^{p^{m+kn}−p^{k(n−1)}}t_{m+k}^{p^{k(n−1)}}
−v_{n−1}^{p^{k(n−1)}}v_{n}^{−p^{k(n−1)}}t_{m+k+1}^{p^{(k+1)(n−1)}}) mod J(p^{n}A_{i})
d(−v_{n−1}^{a_{i}(p^{n}−1)}v_{n}^{b_{i}(p^{n}−1)+p^{m+kn}−p^{k(n−1)}}x_{(k−1)n})
≡ (−1)^{k+1}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+kn}−p^{k(n−1)}}t_{m+k}^{p^{k(n−1)}} mod J(a_{i}(p^{n} − 1) + A_{i})

Here a_{i}(p^{n} − 1) + A_{i} < A_{i}(p^{n} − 1) + A_{i} = p^{n}A_{i}. Therefore, we obtain

d(x_{kn}) ≡ (−1)^{k+1}v_{n−1}^{a_{i}p^{n}+p^{k(n−1)}}v_{n}^{b_{i}p^{n}−p^{k(n−1)}}t_{m+k+1}^{p^{(k+1)(n−1)}} mod J(a_{i}(p^{n} − 1) + A_{i})

Therefore, we have the case where k < n − 1.

For k = n − 1, we make similar computation under the same supposition (3.3). Here i = (n − 2)n.

d(x_{(n−2)n}^{p^{n}})

≡ (−1)^{n−1}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}t_{m+n−1}^{p^{(n−1)^{2}+n}} mod J(p^{n}A_{i})
≡ (−1)^{n−1}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}(v_{n}^{p^{m+n−1}−1}t_{m+n−1} − v_{n−1}v_{n}^{−1}t_{m+n}^{p^{n−1}}
+v_{n−1}^{p^{m+n}}v_{n}^{−1}t_{m+n} − v_{n−1}v_{n}^{−1}C_{n−1}(n + m))^{p^{(n−1)^{2}}} mod J(p^{n}A_{i})
≡ (−1)^{n−1}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}(v_{n}^{p^{m+n(n−1)}−p^{(n−1)^{2}}} t_{m+n−1}^{p^{(n−1)^{2}}} − v_{n−1}^{p^{(n−1)^{2}}} v_{n}^{−p^{(n−1)^{2}}} t_{m+n}^{p^{n(n−1)}}
−v_{n−1}^{p^{(n−1)^{2}}} v_{n}^{−p^{(n−1)^{2}}} C_{n−1}(n + m)^{p^{(n−1)^{2}}} ) mod J(p^{n}A_{i})

d(−v_{n−1}^{a_{i}(p^{n}−1)}v_{n}^{b_{i}(p^{n}−1)+p^{m+n(n−1)}−p^{(n−1)^{2}}} x_{(n−2)n})

≡ (−1)^{n}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+n(n−1)}−p^{(n−1)^{2}}} t_{m+n−1}^{p^{(n−1)^{2}}} mod J(a_{i}(p^{n} − 1) + A_{i})

Therefore,

\begin{aligned}d(x_{(n−1)n}) & ≡ & (−1)^{n} (v_{n−1}^{a_{i}p^{n}+p^{(n−1)^{2}}} v_{n}^{b_{i}p^{n}−p^{(n−1)^{2}}} t_{m+n}^{p^{n(n−1)}} \\ & & +v_{n−1}^{a_{i}p^{n}+p^{(n−1)^{2}}} v_{n}^{b_{i}p^{n}−p^{(n−1)^{2}}} C_{n−1}(n + m)^{p^{(n−1)^{2}}} ) \end{aligned}

mod J(a_{i}(p^{n} − 1) + A_{i}).

Next we consider the case where k = n. We notice that v_{n−1}^{p^{l}−p^{l−n}} divides w_{l} by definition and

d(w_{l}) ≡ d(x_{0})^{p^{l}} − d(x_{l}) ≡ v_{n−1}^{p^{l}}t_{m+1}^{p^{n−1+l}} − d(x_{l}).

We also note that

C_{n−1}(n + m)^{p^{(n−1)^{2}}} ≡ −(v_{n−1}^{p^{n−2}}v_{n+m}^{(p−1)p^{n−2}}t_{m+1}^{p^{2n−3}})^{p^{(n−1)^{2}}}

mod J(2p^{n^{2}−2n−1}). Put here i = n^{2} − n. We see that (p^{n} − 1)a_{i} + A_{i} ≥ a_{i}p^{n} + 2p^{n^{2}−n−1} by the assumption on m and n, and we compute

d(x_{(n−1)n}^{p^{n}})

≡ (−1)^{n}(v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}t_{m+n}^{p^{n^{2}}} + v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}C_{n−1}(n + m)^{p^{n^{2}−n+1}}) mod J(p^{n}A_{i})
≡ (−1)^{n}(v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}t_{m+n}^{p^{n^{2}}}) mod J(a_{i}p^{n} + p^{n^{2}−n+1})
≡ (−1)^{n}(v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}}(v_{n}^{p^{m+n}−1}t_{m+n})^{p^{n^{2}−n}}) mod J(a_{i}p^{n} + p^{n^{2}−n})

d(−v_{n−1}^{a_{i}(p^{n}−1)}v_{n}^{b_{i}(p^{n}−1)+p^{m+n^{2}} −p^{n(n−1)}}x_{(n−1)n})

≡ (−1)^{n+1}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}t_{m+n}^{p^{n(n−1)}}
+(−1)^{n+1}v_{n−1}^{a_{i}p^{n}+p^{n^{2}−n−1}}v_{n}^{b_{i}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}v_{n+m}^{(p−1)p^{n^{2}−n−1}}t_{m+1}^{p^{n^{2}−2}}

THE HOMOTOPY GROUPS π_{∗}(L_{n}T (m) ∧ V (n − 2)) 9

mod J(a_{i}p^{n} + 2p^{n^{2}−n−1})

d((−1)^{n}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}v_{n+m}^{(p−1)p^{n^{2}−n−1}}w_{n^{2}−n−1})

≡ (−1)^{n}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}v_{n+m}^{(p−1)p^{n^{2}−n−1}}(d(x_{0})^{p^{n^{2}−n−1}} − d(x_{n^{2}−n−1}))
≡ (−1)^{n}v_{n−1}^{a_{i}p^{n}}v_{n}^{b_{i}p^{n}+p^{m+n^{2}} −p^{n(n−1)}}v_{n+m}^{(p−1)p^{n^{2}−n−1}}(v_{n−1}^{p^{n^{2}−n−1}}t_{m+1}^{p^{n^{2}−2}}
−d(x_{n^{2}−n−1}))

mod J(p^{n}a_{i} + 2p^{n^{2}−n−1} − p^{n^{2}−2n−1}). Therefore,

d(x_{n^{2}}) ≡ (−1)^{n+1}v_{n−1}^{a}v_{n}^{b}v_{n+m}^{c}d(x_{n^{2}−n−1}) mod J(A_{n^{2}}). Inductively suppose that d(x_{n^{2}+i}) ≡ (−1)^{n+1}v_{n−1}^{p^{i}a}v_{n}^{p^{i}b}v_{n+m}^{p^{i}c}d(x_{n^{2}+i−n−1}) mod J(A_{n^{2}+i}) i ≥ 0.

Then noticing that v_{n−1}^{p^{i}} divides x_{i−1}^{p} − x_{i}, we compute

d(x_{n^{2}+i}^{p}) ≡ (−1)^{n+1}v_{n−1}^{p^{i+1}a}v_{n}^{p^{i+1}b}v_{n+m}^{p^{i+1}c}d(x_{n^{2}+i−n−1})^{p} and
(−1)^{n}d(v_{n−1}^{p^{i+1}a}v_{n}^{p^{i+1}b}v_{n+m}^{p^{i+1}c}(x_{n^{2}+i−n−1}^{p} − x_{n^{2}+i−n}))
≡ (−1)^{n}v_{n−1}^{p^{i+1}a}v_{n}^{p^{i+1}b}v_{n+m}^{p^{i+1}c}(d(x_{n^{2}+i−n−1}^{p}) − d(x_{n^{2}+i−n}))

mod J(pA_{n^{2}+i}) and the induction completes. □

We have the short exact sequence

0 wM_{n}^{0} ^{φ} wM_{n−1}^{1} w^{v_{n−1}}M_{n−1}^{1} w0,

where φ(x) = x/v_{n−1}. This yields the long exact sequence

(3.4) 0 wH(m)^{0}M_{n}^{0} ^{φ} wH(m)^{0}M_{n−1}^{1} w^{v_{n−1}}H(m)^{0}M_{n−1}^{1} ^{δ} wH(m)^{1}M_{n}^{0} w· · · .

The module H(m)^{∗}M_{n}^{0} is determined by Ravenel.

Theorem 3.5 (Ravenel [6, Cor. 6.5.6] ). Suppose n < m + 2 and n < m + 1 if p = 2. Then

H(m)^{∗}M_{n}^{0} = K(n)_{∗}[v_{m+n}]⨂Λ(h_{m+i,j} | i = 1, 2, …, n; j ∈ Z/n). Here K(n)_{∗} = Z/p[v_{n}^{±1}] = M_{n}^{0} and h_{ij} denotes the generator represented by t_{i}^{p^{j}}.

Furthermore, we see the following lemma from [3, Remark 3.11]:

Lemma 3.6. If a submodule D of H^{s}M_{n−1}^{1} fits into the exact sequence H(m)^{s−1}M_{n−1}^{1} ^{δ} wH(m)^{s}M_{n}^{0} ^{φ} wD w^{v_{n−1}}D ^{δ} wH(m)^{s+1}M_{n}^{0},

then H(m)^{s}M_{n−1}^{1} = D.

We have elements

x_{i}^{s}/v_{n−1}^{j} for i, s, j ≥ 0 with 0 < j ≤ a_{i}

of H(m)^{0}M_{n−1}^{1} by Proposition 3.2. Moreover it shows the behavior of δ.

Lemma 3.7. Let δ be the connecting homomorphism H(m)^{0}M_{n−1}^{1} → H(m)^{1}M_{n}^{0}. Then

δ(x_{i}^{s}/v_{n−1}^{a_{i}}) = sv_{n}^{b_{i}}v_{m+n}^{(s−1)p^{i}+c_{i}}[g_{i}]

for any i, s ≥ 0, where [g_{i}] denotes the homology class of g_{i}.

Consider the E(n)_{∗}/I_{n−1}-module
(3.8)  D = M_{n−1}^{1}⨁ ∑_{i,s≥0} E(n)_{∗}/I_{n−1}⟨x_{i}/v_{n−1}^{a_{i}}⟩.

Here E(n)_{∗}/I_{n−1}⟨ξ⟩ denotes an E(n)_{∗}/I_{n−1}-module generated by ξ, and v_{n−1}^{k} ∈ E(n)_{∗}/I_{n−1} acts on a element of the form x/v_{n−1}^{j} by v_{n−1}^{k}(x/v_{n−1}^{j}) = x/v_{n−1}^{j−k} if j > k and = 0 otherwise. Then it is easy to see that D ⊂ H(m)^{0}M_{n−1}^{1}, Im φ ⊂ D and Im v_{n−1} ⊂ D, which shows the exact sequence 0 wH(m)^{0}M_{n}^{0} ^{φ} wD w^{v_{n−1}}D.

By the relation of Lemma 3.1, we see that [g_{i}] = v_{n}^{l}h_{j,k} for some l, j, k. Then it is easy to check v_{m+n}^{(s−1)p^{i}+c_{i}}[g_{i}]’s are linearly independent over K(n)_{∗}, and we obtain the exact sequence D w^{v_{n−1}}D ^{δ} wH(m)^{1}M_{n}^{0}. Therefore, Lemma 3.6 shows

Theorem 3.9. Let m and n be positive integers with m ≥ n^{2} − n − 1. Then the E(n)_{∗}/I_{n−1}-module H(m)^{0}M_{n−1}^{1} is isomorphic to D of (3.8).

4. H(m)^{∗}M_{n−1}^{1}

In this section we fix positive integers m and n such that n ≥ 2 and m ≥ n^{2} − n − 1.

Lemma 4.1. If d(x) ≡ v_{n−1}^{a}xg mod I_{n−1} in Ω^{1}B, then there exists a cocycle g̃ such that g̃ ≡ g mod I_{n} and d(g̃) ≡ 0 mod J(k) for each k > 0. Here J(k) is the ideal given in Proposition 3.2.

Proof. Put formally g̃ = ∑_{i>0}(−1)^{i+1} \frac{1}{i}v_{n−1}^{(i−1)a}g^{i}. Then formally, we compute

v_{n−1}^{a}g̃ = log(1 + v_{n−1}^{a}g) = log(\frac{η_{R}(x)}{x})
= log(η_{R}(x)) − log x = η_{R}(log x) − log x = d(log x).

Therefore, d(g̃) = d(d(log x)) = 0. □

We introduce some submodules of H(m)^{∗}K(n)_{∗}:

M(i) = K(n)_{∗}{v_{n+m}^{sp^{i}} | p |̸ s > 0},
M(i, k) = M((n + 1)(n − 3 + k) + 1 + i),
N(0) = ∑_{i=0}^{n^{2}−n−2} M(i)⨂Λ(γ̃_{j} | 0 ≤ j < n^{2} − n − 1) and

N(k) = ∑_{i=0}^{n} M(i, k)⨂Λ(γ̃_{j,k} | 0 ≤ j ≤ n).

Here, γ̃ denotes an element represented by a cocycle g̃ which is the cocycle obtained from g by Lemma 4.1, and we put g_{i,k} = v_{n+m}^{c_{j}k−p^{j}}g_{i} for j = (n + 1)(n − 3 + k) + 1 + i.

THE HOMOTOPY GROUPS π_{∗}(L_{n}T (m) ∧ V (n − 2)) 11

Remark 4.2. The elements γ̃_{λ} above are not well defined themselves, but ξγ̃_{λ} for ξ ∈ M(i) are well defined. Moreover, M(i, k)γ̃_{j,k} ≠ M(i, k)γ_{l} for l = (n + 1)(n − 3 + k) + 1 + j.

Then,

H(m)^{∗}K(n)_{∗} = K(n)_{∗}⨁ ∑_{k≥0} N(k)_{∗}.

In order to state our main theorem, we further introduce submodules of H(m)^{∗}M_{n−1}^{1}:

M̃(i) = E(n)_{∗}/I_{n−1}{x_{i}^{s}/v_{n−1}^{a_{i}} | p |̸ s}
M̃(i, k) = M̃((n + 1)(n − 3 + k) + 1 + i),
Ñ(0) = ∑_{i=0}^{n^{2}−n−2} M̃(i)⨂Λ(γ̃_{j} | 0 ≤ j < n^{2} − n − 1, j ≠ i)
Ñ(k) = ∑_{i=0}^{n} M̃(i, k)⨂Λ(γ̃_{j,k} | 0 ≤ j ≤ n, j ≠ i) for k > 0 .

By Lemma 3.7, we have the exact sequences

0 wK(n)_{∗} ^{φ} wM_{n−1}^{1} w^{v_{n−1}}M_{n−1}^{1} w0,

N(0) ^{φ} wÑ(0) w^{v_{n−1}}Ñ(0) ^{δ} wN(0)

and

N(k) ^{φ} wÑ(k) w^{v_{n−1}}Ñ(k) ^{δ} wN(k).

Putting these together yields the exact sequence

H^{∗}K(n)_{∗} ^{φ} wD w^{v_{n−1}}D ^{δ} wH^{∗}K(n)_{∗},

where

(4.3)  \begin{aligned}D & = & M_{n−1}^{1}⨁Ñ(0)⨂Λ(γ_{j} | n^{2} − n − 1 ≤ j < n^{2}) \\ & & ⨁ ∑_{i≥0} Ñ(k)⨂Λ(γ_{j} | 0 ≤ j < n^{2} − n − 1), \end{aligned}

and Lemma 3.6 shows the following

Theorem 4.4. Let n and m be positive integers with n ≥ 2 and m ≥ n^{2} − n. Then H^{∗}M_{n−1}^{1} is isomorphic to D of (4.3).

Here we also notice a similar remark to Remark 4.2. Ravenel’s Theorem 3.5 shows that H(m)^{s}M_{n}^{0} = 0 if s > n^{2}, and we have

Corollary 4.5. Let m ≥ n^{2} − n − 1 with n ≥ 2. Then H(m)^{s}M_{n−1}^{1} = 0 if s ≥ n. In particular, H(m)^{n^{2}}M_{n−1}^{1} = 0.

5. The homotopy groups π_{∗}(L_{n}V_{T (m)}(n − 2))

Consider the cofiber sequence

L_{n}V_{T} (n − 2) wL_{n−1}V_{T} (n − 2) wL_{n}W,

where T = T (m). Then this induces the long exact sequence

⋯ wExt^{∗}(E(n)_{∗}(V_{T} (n − 2))) wExt^{∗}(E(n)_{∗}(L_{n−1}V_{T} (n − 2)))

wExt^{∗}(E(n)_{∗}(W)) w· · · .

By definition, we see that

Ext^{∗}(E(n)_{∗}(W)) = H(m)^{∗}M_{n−1}^{1}⨂_{B}B(m − 1).

We have a change of rings theorem (cf. [1])

Ext^{∗}(E(n)_{∗}(L_{n−1}V_{T} (n−2))) = Ext_{E(n−1)_{∗}(E(n−1))}^{∗}(E(n−1)_{∗}, E(n−1)_{∗}(V_{T} (n−2))),

whose structure is

K(n − 1)_{∗}[v_{n}, …, v_{n+m−1}]⨂Λ(h_{m+i,j} | i = 1, …, n − 1; j ∈ Z/(n − 1))

by Ravenel’s Theorem 3.5. Therefore, we have

Theorem 5.1. The E_{2}-term of the Adams-Novikov spectral sequence for π_{∗}(L_{n}V_{T} (n − 1)) is isomorphic to the direct sum of Z/p[v_{n−1}, v_{n}, …, v_{n+m−1}], K(n − 1)_{∗}[v_{n}, …, v_{n+m−1}]⨂E(h_{m+i,j} | i = 1, …, n − 1; j ∈ Z/(n − 1)) and

Σ(Ñ(0)⨂Λ(γ_{j} | n^{2} − n − 1 ≤ j < n^{2})⨁ ∑_{i>0} Ñ(i)⨂Λ(γ_{j} | 0 ≤ j < n^{2} − n − 1).

Here Σ denotes an operation that raises degree one, E(x_{i}) denotes a Z/p-vector space isomorphic to the quotient Λ(x_{i})/(Z/p).

By Corollary 4.5, the E_{2}-term has a horizontal vanishing line and we have

Corollary 5.2. If n^{2} < 2p − 1 and n^{2} − n − 1 ≤ m, then the homotopy groups π_{∗}(L_{n}V_{T} (n − 1)) are isomorphic to the E_{2}-term.

References

[1] M. Hovey and H. Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, J. London Math. Soc. 60 (1999), 284–302.

[2] Y. Kamiya and K. Shimomura, The homotopy groups π_{∗}(L_{2}T (n) ∧ V (0)), preprint.

[3] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469–516.

[4] M. Mahowald and K. Shimomura, The Adams-Novikov spectral sequence for the L_{2} localization of a v_{2} spectrum, Contemporary Math. 146 (1993), 237–250.

[5] D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351–414.

[6] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986.

[7] D. C. Ravenel, The microstable Adams-Novikov spectral sequence, preprint.

[8] K. Shimomura, The homotopy groups of the L_{2}-localized Mahowald spectrum X⟨1⟩, Forum Math. 7 (1995), 685–707.

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

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