ON THE HOMOTOPY GROUPS OF E(n)-LOCAL SPECTRA WITH UNUSUAL INVARIANT IDEALS

HIROFUMI NAKAI AND KATSUMI SHIMOMURA

Abstract. Let E(n) and T (m) for non-negative integers n and m denote the Johnson-Wilson and the Ravenel spectra. Then we have a spectrum whose E(n)-homology is E(n)_{∗}(T (m))/(v_{1}, ..., v_{n−1}). The homotopy groups of the spectrum estimate of the order of an element of π_{∗}(L_{n}T (m)). We thought that the homotopy groups are easier to be determined than π_{∗}(L_{n}T (m)). We present here an example which is true, and some examples which are more difficult unexpectedly.

1. Introduction

In [5] Ravenel has defined the homotopy associative commutative ring spectrum T (m), which is constructed as a summand of p-component of the Thom spectrum induced by the map ΩSU(p^{m}) → BU. It is extensively used in [5, section 7] to compute the homotopy groups of spheres in terms of “the method of infinite descent”. The Adams-Novikov E_{2}-term converging to the stable homotopy groups π_{∗}(T (m)) is described by use of the Hopf algebroid (BP_{∗}, Γ(m+1)) (cf. [5, Definition 7.1.1]). In particular, the 0-th line is

Ext_{Γ(m+1)}^{0}(BP_{∗}, BP_{∗}) = Z_{(p)}[v_{1}, …, v_{m}] ⊂ BP_{∗} = Z_{(p)}[v_{1}, …],

and the more the value of m, the more primitives we obtain. Since the multiplication by v_{k} (1 < = k < = m) on BP_{∗}(T (m)) is a Γ(m + 1)-comodule map and it makes possible to consider various self maps on the spectrum T (m). We denote T (m)/I for the spectrum whose BP_{∗}-homology is isomorphic to BP_{∗}(T (m))/I for an ideal I of BP_{∗}. In this notation, we encounter unusual ideals. For example, Kamiya and the second author ([2], [7]) determined homotopy groups π_{∗}(L_{n}T (m)/I_{n−1}^{1}) for I_{n−1}^{1} = (p, v_{1}, …, v_{n−2}, v_{n−1}^{∞}). We read off that v_{n+k} acts monomorphically on the homotopy groups. Thus we meet the ideal (I_{n−1}^{1}, v_{n+k}) and obtain

π_{∗}(L_{n}T (m)/(I_{n−1}^{1}, v_{n+k})) = π_{∗}(L_{n}T (m)/I_{n−1}^{1})/(v_{n+k})

for n < = m and 0 < k < m.

In this paper, we mainly study the localized cases with respect to the Johnson-Wilson spectrum E(n), and consider the ideal J_{n} = (v_{1}, …, v_{n−1}) of E(n)_{∗}. We compute the homotopy groups of a spectrum X by the E(n)-based Adams spectral sequence E_{r}^{s,t}(X) ⇒ π_{∗}(L_{n}X). The E_{2}-term is given by E_{2}^{∗}(X) = Ext_{E(n)_{∗}E(n)}^{∗} (E(n)_{∗}, E(n)_{∗}(X)).

2000 Mathematics Subject Classification. 55Q99.

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

Theorem 1.1. For n < = m, the Adams-Novikov E_{2}-term E_{2}^{0}(T (m)/J_{n}) is isomorphic to Z_{(p)} and E_{2}^{1}(T (m)/J_{n}) is isomorphic to the direct sum of the cyclic

Z_{(p)}[v_{n}^{±1}, v_{n+1}, …, v_{m}]-module generated by \frac{v_{m+1}^{e_{1}} ⋯ v_{m+n}^{e_{n}}}{p^{1+ν(e_{k})}} of order p^{1+ν(e_{k})} with

ν(e_{k}) = min{ν(e_{1}), ⋯, ν(e_{n})}.

This follows immediately from Corollary 4.4. For the case where n > m, we have an example which has a similar result (cf. Proposition 4.7):

Proposition 1.2. The Adams-Novikov E_{2}-term E_{2}^{0}(T (1)/(v_{1})) is isomorphic to Z_{(p)} and E_{2}^{1}(T (1)/(v_{1})) is the direct sum of the cyclic Z_{(p)}-module generated by

v_{2}^{sp^{i}}v_{3}^{tp^{j}}/p^{1+min(i,j)} of order p^{1+min(i,j)}.

In these cases, we did not determine E_{2}^{s} for s > 1 since there is an obstruction, which comes from the generators known as b_{i,j} (see (3.2)). For p = 2, we have the relation b_{i,j} = h_{i,j}^{2}, which makes possible to compute for s > 1. Since the Adams-Novikov differentials are read off from [4], we obtain the Adams-Novikov E_{∞}-term.

Theorem 1.3. Let p = 2. The E_{∞}-term E_{∞}^{s,∗}(L_{2}T (1)/(v_{1})) is isomorphic to Z_{(2)} if s = 0 and is isomorphic to the tensor product of Λ(ρ_{2}) and the direct sum of

(1) { \atop v_{2}A[h_{20}]}, v_{3}B[h_{30}]/(h_{30}^{3}) and v_{3}Bh_{30}h_{31} whose elements are of order two; and

(2) M^{0} and M^{1}. Here the modules are given in section five.

In the section two, we consider the spectrum E_{m}(n) = v_{n}^{−1}BP ⟨n + m⟩ characterized by the coefficient ring E_{m}(n)_{∗} = E(n)_{∗}[v_{n+1}, …, v_{n+m}], and consider the Hopf algebroid (E_{m}(n)_{∗}, Σ(n, m + 1)). It allows us to consider a variation of the change-of-rings theorem given in [1]. In section three, we exhibit the formulas for the structure maps (the right unit η_{R} and the diagonal maps Δ). Using these results, we prove the existence of the spectra T (m)/J whose BP_{∗}-homology has the form BP_{∗}/J[t_{1}, …, t_{m}] with unusual ideals J, and construct self maps on these spectra. Section four is devoted to prove Theorem 1.1 and Proposition 1.2. In section five, we determine the E_{∞}-term for π_{∗}(L_{2}T (1)/(2^{∞}, v_{1})). The homotopy groups π_{∗}(L_{2}T (1)) is determined easily if p is odd, and stays undetermined if p = 2. The result of this section is the first step to understand π_{∗}(L_{2}T (1)) at the prime two. Acknowledgement. We wish to thank to the organizers of Nishida Conference held on August 2004 for making arrangement of the publication for the Proceedings. We are also grateful to Ippei Ichigi for reading the draft paper carefully and for pointing out some misprints.

2. A generalized Johnson-Wilson theory

In this section we introduce a generalized Johnson-Wilson theory E_{m}(n) and a variation of change-of-rings theorem. Let BP and BP ⟨n⟩ denote the Brown-Peterson and the Jonson-Wilson spectra. Then,

π_{∗}(BP) = BP_{∗} = Z_{(p)}[v_{1}, …, v_{n}, …],
BP_{∗}(BP) = BP_{∗}[t_{1}, …, t_{n}, …],
π_{∗}(BP ⟨n⟩) = BP ⟨n⟩_{∗} = Z_{(p)}[v_{1}, …, v_{n}] ⊂ BP_{∗}

with |v_{n}| = |t_{n}| = 2(p^{n} − 1). We put

E_{m}(n) = v_{n}^{−1}BP ⟨n + m⟩

for non-negative integers n, m. Then

E_{m}(n)_{∗} = E(n)_{∗}[v_{n+1}, …, v_{n+m}] ⊂ v_{n}^{−1}BP_{∗},

We notice that E_{0}(n) is the localized Johnson-Wilson spectrum E(n). Let Γ(m + 1) (cf. [5, 7.1.1]) be the BP_{∗}(BP)-comodule defined by

Γ(m + 1) = BP_{∗}(BP)/(t_{1}, …, t_{m}) = BP_{∗}[t_{m+1}, t_{m+2}, …].

Then, the pair (BP_{∗}, Γ(m + 1)) has the structure of the Hopf algebroid inherited from (BP_{∗}, BP_{∗}(BP)). Put

Σ_{m}(n, i) = E_{m}(n)_{∗} ⊗_{BP_{∗}} Γ(i) ⊗_{BP_{∗}} E_{m}(n)_{∗}.

We abbreviate Σ_{m}(n, 1) by Σ_{m}(n), which is isomorphic to E_{m}(n)_{∗}(E_{m}(n)), and Σ_{m}(n, m + 1) by Σ(n, m + 1). The pair (E_{m}(n)_{∗}, Σ_{m}(n, i)) is a Hopf algebroid with the structure maps inherited from those of the Hopf algebroid (BP_{∗}, Γ(i)) for all i > 0. Consider the map between Hopf algebroids

(2.1)  (E_{m}(n)_{∗}, Σ_{m}(n))  (E_{m}(n)_{∗}, Σ(n, m + 1)) w

induced from the projection from BP_{∗}(BP) to Γ(m + 1). The map is normal and that

(2.2)  E_{m}(n)_{∗}(T (m)) = E_{m}(n)_{∗}□_{Σ(n,m+1)}Σ_{m}(n)

if m > 0. By [5, A1.3.12], we have

Lemma 2.3. There is an isomorphism

Ext_{Σ_{m}(n)}(E_{m}(n)_{∗}, E_{m}(n)_{∗}(T (m))) = Ext_{Σ(n,m+1)}(E_{m}(n)_{∗}, E_{m}(n)_{∗}).

Remark 2.4. In general, the equation (2.2) does not hold if we work on Σ_{0}(n)- comodules. For example, if we set (n, i) = (2, 3), then

Σ_{0}(2, 3) = E(2)_{∗}[t_{3}, t_{4}, …]/(η_{R}(v_{k}): k > 2).

In the right hand side we have the relation v_{2}t_{1}^{p^{2}} ≡ v_{2}^{p}t_{1} mod (p) because η_{R}(v_{3}) = 0. On the other hand, we do not have any relation on t_{1} in

E(2)_{∗}T (2) = E(2)_{∗}[t_{1}, t_{2}].

Hereafter we often abbreviate Ext_{Γ(m+1)}(BP_{∗}, −), Ext_{Σ_{m}(n)}(E_{m}(n)_{∗}, −) and Ext_{Σ(n,m+1)}(E_{m}(n)_{∗}, −) by Ext_{Γ(m+1)}(−), Ext_{Σ_{m}(n)}(−) and Ext_{Σ(n,m+1)}(−) respectively.

Since E_{m}(n)_{∗} is a free E(n)_{∗}-module over the bases v^{E} = v_{n+1}^{e_{1}} ⋯ v_{n+m}^{e_{m}} for E = (e_{1}, …, e_{m}) with e_{k} > = 0, there is a homotopy equivalence E_{m}(n) = ⋁_{E} Σ^{|E|}E(n). This shows that the E(n)-based and the E_{m}(n)-based Adams spectral sequences agrees from the E_{2}-term (cf. [1]).

3. Existence of some self maps and spectra

Let J_{n} denote the ideal (v_{1}, …, v_{n−1}) of BP_{∗}. The following lemma is verified by using [3, (1.1), (1.2) and (1.3)]:

Lemma 3.1. Assume that n < = m. Then, the structure maps in (BP_{∗}, Γ(m + 1)) acts as

η_{R}(v_{k}) ≡ v_{k} for n < = k < = m,
η_{R}(v_{m+k}) ≡ v_{m+k} + pt_{m+k} for 0 < k < = n,
Δ(t_{m+k}) ≡ t_{m+k} ⊗ 1 + 1 ⊗ t_{m+k} for 0 < = k < = n,
Δ(t_{m+n+1}) ≡ t_{m+n+1} ⊗ 1 + 1 ⊗ t_{m+n+1} + v_{n}b_{m+1,n−1}

mod J_{n}, where

(3.2)  b_{i,j} = \frac{t_{i}^{p^{j+1}} ⊗ 1 + 1 ⊗ t_{i}^{p^{j+1}} − (t_{i} ⊗ 1 + 1 ⊗ t_{i})^{p^{j+1}}}{p} .
□

By this lemma, we read off the behavior of the structure maps η_{R} and Δ mod J_{n} of the Hopf algebroid (E_{m}(n)_{∗}, Σ(n, m + 1)), which we need. For n > m, we only consider the case where n = 2 and m = 1.

Lemma 3.3. The structure maps in (BP_{∗}, Γ(2)) acts as

η_{R}(v_{i}) ≡ v_{i} + pt_{i} for i = 2 and 3,
η_{R}(v_{4}) ≡ v_{4} + v_{2}t_{2}^{p^{2}} + pt_{4} + v_{2}c_{21} − η_{R}(v_{2})^{p^{2}}t_{2},
η_{R}(v_{5}) ≡ v_{5} + v_{3}t_{2}^{p^{3}} + v_{2}t_{3}^{p^{2}} + pt_{5} + v_{2}c_{31} + v_{3}c_{22} − η_{R}(v_{3})^{p^{2}}t_{2} − η_{R}(v_{2})^{p^{3}}t_{3},
Δ(t_{i}) ≡ t_{i} ⊗ 1 + 1 ⊗ t_{i} for i = 2 and 3,
Δ(t_{4}) ≡ t_{4} ⊗ 1 + 1 ⊗ t_{4} + t_{2} ⊗ t_{2}^{p^{2}} + v_{2}b_{21},
Δ(t_{5}) ≡ t_{5} ⊗ 1 + 1 ⊗ t_{5} + t_{3} ⊗ t_{2}^{p^{3}} + t_{2} ⊗ t_{3}^{p^{2}} + v_{2}b_{31} + v_{3}b_{22}

mod (v_{1}), where c_{i,j} = p^{−1}(v_{i}^{p^{j+1}} − η_{R}(v_{i}^{p^{j+1}})). In particular, b_{i,j} ≡ t_{i}^{2^{j}} ⊗ t_{i}^{2^{j}} mod (2) for p = 2. □

By use of [3, (1.1), (1.2) and (1.3)], we see easily the following:

Lemma 3.4. In (BP_{∗}, Γ(m + 1)), v_{n+k} (0 < = k < = m) are invariants mod I_{n} = (p) + J_{n}, that is, we have

v_{n+k} ∈ Ext_{Γ(m+1)}^{0}(BP_{∗}/I_{n}). □

Ravenel defined the spectrum T (m) (cf. [5, 6.5.1]), which is an associative commutative ring spectrum, and is characterized by BP_{∗}(T (m)) = BP_{∗}[t_{1}, …, t_{m}]. From now on, we study the existence of a spectrum X with BP_{∗}-homology

(3.5)  BP_{∗}(X) = BP_{∗}/I[t_{1}, …, t_{m}]

for an unusual ideal I. We write T (m)/I for such X.

For a subset A of {p, v_{1}, …, v_{n−1}}, let I(A) denote the ideal generated by A. In particular, I_{n} = I(A) for A = {p, v_{1}, …, v_{n−1}}.

The existence of a spectrum with BP_{∗}-homology BP_{∗}/I_{n} is problematic and we still have little information for such a spectrum, which we usually call the ((n − 1)- st) Smith-Toda spectrum and is denoted by V (n − 1) (e.g., [8], [9] and [5]). It is shown that for n < = 3 (1) V (n) exists if and only if p > 2n,

(2) V (n) is a ring spectrum if p > 2(n + 1). Notice that the spectrum T (m)/I_{n} exists if so does V (n − 1), since T (m)/I_{n} = T (m) ∧ V (n − 1). For the Adams-Novikov spectral sequence

(3.6)  E_{2} = Ext_{Γ(m+1)}(BP_{∗}/I_{n}) ⟹ π_{∗}(T (m)/I_{n}),

we have the following vanishing line since the reduced comodule \overline{Γ(m + 1)} is (2p^{m+1} − 3)-connected:

Lemma 3.7. In (3.6) we have E_{r}^{s,t} = 0 for t < 2s(p^{m+1} − 1). In particular, d_{r}(v_{n+k}) = 0 for all r if n + k < m + 2.

Combining Lemma 3.4 and 3.7, we have

Lemma 3.8. Assume that the spectrum T (m)/I_{n} exists. Then, for non-negative integers k, m and n with k < = m and n + k < m + 2, there exists a map

(3.9)  v_{n+k} : S^{2(p^{n+k}−1)}  T (m)/I_{n} w

which induces the multiplication by v_{n+k} on BP_{∗}-homology.

Proposition 3.10. Let A be a subset {w_{0}, w_{1}, …, w_{ℓ}} of {p, v_{1}, …, v_{m}}. Then, there exist T (m)-module spectra T (m)/I(A) with obvious BP_{∗}-homology.

Proof. If n = 0, then the inequality in Lemma 3.7 is always satisfied, and we have the maps (3.9) for 0 < = k < = m. Let A_{k} denote the subset {w_{0}, w_{1}, …, w_{k}} of A, and T denote T (m) for short. We construct T/I(A_{k}) inductively on k starting with T/I(A_{−1}) = T. Under the assumption that we have T/(I(A_{k−1})), define T/(I(A_{k})) as the cofiber of the self map

Σ^{|w_{k}|}T/I(A_{k−1})  T ∧ T/I(A_{k−1})  T/I(A_{k−1}),^{w_{k}∧1} w  ^{μ_{k−1}} w

which we also denote by w_{k}. Then, T/(I(A_{k})) has the desired BP_{∗}-homology. The T -module structure μ_{k} of T/I(A_{k}) is obtained by the following diagram, which is homotopy commutative since T is associative:

T ∧ Σ^{2(p^{k}−1)}T/I(A_{k−1})  T ∧ T/I(A_{k−1})  T ∧ T/I(A_{k}) Σ^{2(p^{k}−1)}T/I(A_{k−1})  T/I(A_{k−1})  T/I(A_{k}).^{1∧w_{k}} w_{_{μ_{k−1}} u}  w_{_{μ_{k−1}} u}  _{_{μ_{k}} u}_{^{1∧w_{k}} w}  w

□

Example 3.11. Proposition 3.10 shows the existence of spectra T (m)/J_{n} and T (m)/I_{n} for 0 < n < = m + 1. Furthermore, we have a homotopy equivalence T (m)/J_{n} ∧ V (0) = T (m)/I_{n}.

In general, if the ring spectrum V (n − 1) exists, then there are self maps

(3.12)  v_{n+k} : Σ^{2(p^{n+k}−1)}T (m)/I_{n}  T (m)/I_{n} w

for all k satisfying 0 < = k < = m + 1 − n by Lemma 3.8, each of which induces a multiplication by v_{n+k} on BP_{∗}-homology and we have the spectrum T (m)/J with

(3.13)  J = (p, v_{1}, …, v_{n−1}, v_{n}^{e_{0}}, …, v_{m+1}^{e_{m+1}})

for e_{k} > = 0 or = ∞. Even if V (n − 1) is not a ring spectrum, T (m)/I_{n} is a ring and has self maps (3.12) in some cases. Indeed, if V (n − 1) exists, then T (m)/I_{n} is a ring spectrum for m > 0, since [V (0), T (m) ∧ V (0)] = Z/p and [V (n − 1), T (m) ∧ V (n − 1)]_{2p^{n−1}−2} = 0.

Next we discuss the localized situations. We have a spectrum L_{n}T (m)/I_{n} for n < = m by Proposition 3.10, and the spectral sequence

(3.14)  Ext_{Σ(n,m+1)}(E_{m}(n)_{∗}/I_{n}) ⟹ π_{∗}(L_{n}T (m)/I_{n})

by Lemma 2.3. As we noted above, the E_{2}-term is isomorphic to Ext_{Γ(m+1)}(M_{n}^{0}), whose structure is determined by Ravenel ([5, Corollary 6.5.6]) as

(3.15)  K(n)_{∗}[v_{n+1}, …, v_{n+m}] ⊗ Λ(h_{i,j}: m + 1 < = i < = m + n, j ∈ Z/n)

under the condition

(3.16)  n < 2(p − 1)(m + 1)/p and n < m + 2.

Since the E_{2}-term (3.15) has the horizontal vanishing line, that is, E_{2}^{s,t} = 0 for s > n^{2}, we obtain the following proposition (Compare with Lemma 3.8):

Proposition 3.17. Let k, m and n be non-negative integers satisfying (3.16) and n^{2} < = 2(p − 1). If the spectrum L_{n}T (m)/I_{n} exists, then there exists a map

S^{2(p^{n+k}−1)}  L_{n}T (m)/I_{n}^{v_{n+k}} w

which induces the multiplication by v_{n+k} on BP_{∗}-homology. □

If L_{n}T (m)/I_{n} is an associative ring spectrum, then we have self maps of L_{n}T (m)/I_{n}- modules

(3.18)  v_{n+k} : Σ^{2(p^{n+k}−1)}L_{n}T (m)/I_{n}  L_{n}T (m)/I_{n} w

for all k satisfying 0 < = k < = m. By Proposition 3.17, each of which induces a multiplication by v_{n+k} on BP_{∗}-homology and we have the spectrum L_{n}T (m)/J with

J = (p, v_{1}, …, v_{n−1}, v_{n}^{e_{0}}, …, v_{n+m}^{e_{n+m}})

for e_{k} > = 0 or = ∞.

The cofiber sequence (3.18) induces the short exact sequence of E_{2}-terms rather than the long exact sequence because v_{n+k} acts on (3.15) monomorphically. So, the E_{2}-term of E_{m}(n)-based Adams spectral sequence converging to π_{∗}(L_{n}T (m)/J) is isomorphic to the quotient of (3.15) by J. Moreover, this is isomorphic to π_{∗}(L_{n}T (m)/J).

4. Ext_{Σ(n,m+1)}^{s}(E_{m}(n)_{∗}/J_{n}) for small s

Put J_{n} denote the ideal (v_{1}, …, v_{n−1}) of E_{m}(n)_{∗} as above, and assume that n < = m (cf. Proposition 3.10). Applying Ext to the short exact sequences

\begin{matrix}0 ⟶ E_{m}(n)_{∗}/J_{n} ⟶ p^{−1}E_{m}(n)_{∗}/J_{n} ⟶ E_{m}(n)_{∗}/(p^{∞}, J_{n}) ⟶ 0, \\ 0 ⟶ E_{m}(n)_{∗}/(p, J_{n}) ⟶^{1/p} E_{m}(n)_{∗}/(p^{∞}, J_{n}) ⟶^{p} E_{m}(n)_{∗}/(p^{∞}, J_{n}) ⟶ 0 \end{matrix}

we have the long exact sequences of Ext groups. Since we know the structure of Ext(E_{m}(n)_{∗}/(p, J_{n})) as in (3.15), Ext(E_{m}(n)_{∗}/(J_{n})) is a computable object by these long exact sequences.

To compute Ext(E_{m}(n)_{∗}/(p^{∞}, J_{n})), we re-define the class h_{m+k,0} (0 < k < = n) by

(4.1)  h_{i,0} = [\frac{log(1 + pv_{i}^{−1}t_{i})}{p}] = [∑_{n>0}(−1)^{n−1} \frac{(pv_{i}^{−1}t_{i})^{n}}{pn} ] .

Lemma 4.2. For 0 < k < = n, δ(h_{m+k,0}/p^{ℓ}) = 0 for all ℓ.

Proof. It suffices to prove that ph_{m+k,0} = d(log(v_{m+k})). Since η_{R}(v_{m+k}) = v_{m+k} + pt_{m+k} for 0 < k < = n by Lemma 3.1, log(1 + pv_{m+k}^{−1}t_{m+k}) = log(η_{R}(v_{m+k})) − log(v_{m+k}) = d(log(v_{m+k})). □

Although the representative x = log(1 + pv_{m+k}^{−1}t_{m+k})/p of h_{m+k,0} has negative exponents of v_{m+k} in the coefficient, v_{m+k}^{k+1}x is a well-defined element in Σ(n, m + 1)/(p^{k}).

An easy computation with Lemma 3.1 shows the following:

Lemma 4.3. Put ν(e_{k}) = min{ν(e_{1}), ⋯, ν(e_{n})}. Then, we have

δ (\frac{v_{m+1}^{e_{1}} ⋯ v_{m+n}^{e_{n}}}{p^{1+ν(e_{k})}} ) = v_{m+1}^{e_{1}} ⋯ v_{m+n}^{e_{n}}h_{m+k,0} + ⋯

in Ext^{1}(E_{m}(n)_{∗}/(p, J_{n})) up to unit.

Corollary 4.4. For n < = m, Ext^{0}(E_{m}(n)_{∗}/(p^{∞}, J_{n})) is the direct sum of

(1) the cyclic Z_{(p)}[v_{n}^{±1}, v_{n+1}, …, v_{m}]-module generated by \frac{v_{m+1}^{e_{1}} ⋯ v_{m+n}^{e_{n}}}{p^{1+ν(e_{k})}} of or-

der p^{1+ν(e_{k})} with ν(e_{k}) = min{ν(e_{1}), ⋯, ν(e_{n})}; and

(2) Q/Z_{(p)}[v_{n}^{±1}, v_{n+1}, …, v_{m}].

Example 4.5. For m = n = 2, we have

δ (\frac{v_{3}^{sp^{i}}v_{4}^{tp^{j}}}{p^{1+min(i,j)}} ) = \{ \begin{matrix}v_{3}^{sp^{i}}v_{4}^{tp^{j}}h_{40} & for i > j \\ v_{3}^{sp^{i}}v_{4}^{tp^{j}}h_{30} & for i < j \\ v_{3}^{sp^{i}}v_{4}^{tp^{j}}(h_{30} + ah_{40}) & for i = j \end{matrix}

in Ext^{1}(E_{2}(2)_{∗}/(p, v_{1})) up to unit (where a ∈ (Z/(p))^{×}), and Ext^{0}(E_{2}(2)_{∗}/(p^{∞}, v_{1})) is the direct sum of

(1) the cyclic Z_{(p)}[v_{2}^{±}]-module generated by \frac{v_{3}^{sp^{i}}v_{4}^{tp^{j}}}{p^{1+min(i,j)}} of order p^{1+min(i,j)}; and

(2) Q/Z_{(p)}[v_{2}^{±1}].

In the computations for δ(h_{31}) and δ(h_{41}), the elements b_{i,j} (cf. Lemma 3.1) occur, which are hard to express in terms of generators (3.15). We observe that the specific property for p = 2 that b_{i,j} = h_{i,j}^{2} makes the computations easy.

We consider the spectrum L_{n}T (m)/(J_{n}) for (n, m) = (2, 1), which is the simplest case satisfying n > m, and compute Ext_{Σ(2,2)}^{s}(E_{1}(2)_{∗}/(v_{1})) for s < 2 for an odd prime. We consider the case for p = 2 in the next section. Since p is odd, the inequality (3.16) is always satisfied and Ext_{Σ(2,2)}(E_{1}(2)_{∗}/(p, v_{1})) is obtained as

K(2)_{∗}[v_{3}] ⊗ Λ(h_{i,j}: 2 < = i < = 3, j ∈ Z/2).

8 HIROFUMI NAKAI AND KATSUMI SHIMOMURA

Starting from this, Ext_{Σ(2,2)}^{0}(E_{1}(2)_{∗}/(p^{∞}, v_{1})) is determined by computing

(4.6) δ : Ext(E_{1}(2)_{∗}/(p^{∞}, v_{1})) Ext(E_{1}(2)_{∗}/(p, v_{1})) w

as follows, which should be compared with Corollary 4.4: Proposition 4.7. For sp^{i} ∈ Z and tp^{j} > = 0, we have

δ(v_{2}^{sp^{i}}v_{3}^{tp^{j}}/p^{1+min(i,j)}) = \{ \begin{matrix}sv_{2}^{sp^{i}−1}v_{3}^{tp^{j}}h_{20} & if i < j, \\ tv_{2}^{sp^{i}}v_{3}^{tp^{j}−1}h_{30} & if i > j, \\ v_{2}^{sp^{i}−1}v_{3}^{tp^{j}−1}(sv_{3}h_{20} + tv_{2}h_{30}) & if i = j \end{matrix}

and Ext_{Σ(2,2)}^{0}(E_{1}(2)_{∗}/(p^{∞}, v_{1})) is the direct sum of

(1) the cyclic Z_{(p)}-module generated by v_{2}^{sp^{i}}v_{3}^{tp^{j}}/p^{1+min(i,j)} of order p^{1+min(i,j)}; and

(2) Q/Z_{(p)}.

5. The homotopy groups π_{∗}(L_{2}T (1)/(v_{1})) at the prime two We begin with recalling the result of [4]:

(5.1)  Ext(E_{1}(2)_{∗}/(2, v_{1})) = K(2)_{∗}[v_{3}, h_{20}] ⊗ Λ(h_{21}, h_{30}, h_{31}, ρ_{2})

where ρ_{2} is the generator of degree 0 represented by the cocycle v_{2}^{−5}t_{4} + v_{2}^{−10}t_{4}^{2}. We see that (4.1) for p = 2 is also a cocycle with leading term v_{i}^{−2}t_{i}^{2}, and replace the representative cocycles by

(5.2)  h_{i,0} = [t_{i}] and h_{i,1} = [∑_{n>0}(−1)^{n−1} \frac{(2v_{i}^{−1}t_{i})^{n}}{2n} ] .

Setting B = Z/2[v_{2}^{±2}, v_{3}^{2}], we rewrite the right hand side of (5.1) as

B ⊗ Λ(v_{3}) ⊗ Λ(h_{21}, h_{30}, h_{31}) ⊗ Λ(v_{2}) ⊗ Z/2[h_{20}] ⊗ Λ(ρ_{2}).

Since

h_{21}h_{31} = v_{2}^{−1}v_{3}^{2}h_{20}h_{21} + v_{2}^{2}h_{30}^{2} + v_{2}h_{20}h_{31}

by [4, p.243 (1)], we replace h_{21}h_{31} (resp. h_{21}h_{30}h_{31}) with h_{30}^{2} (resp. h_{30}^{3}). Lemma 5.3. As the Z/2-module, Ext(E_{1}(2)_{∗}/(2, v_{1})) is isomorphic to

A ⊗ Λ(v_{2}) ⊗ Z/2[h_{20}] ⊗ Λ(ρ_{2})

where

A = B ⊗ Λ(v_{3}) ⊗ (Z/2[h_{30}]/(h_{30}^{4}) ⊕ Z/2{h_{21}, h_{31}} ⊗ Λ(h_{30}))
B = Z/2[a_{2}^{±1}, a_{3}] with a_{i} = v_{i}^{2}.

Lemma 5.4. The connecting homomorphism (4.6) acts as δ(v_{i}^{s}/2) = v_{i}^{s−1}h_{i,0} and δ(a_{i}^{2^{n}s}/2^{n+2}) = a_{i}^{2^{n}s}h_{i,1} (i = 2, 3) for odd s and n > = 0. Proof. It follows from

\begin{aligned}d(v_{i}^{s}) & ≡ & 2v_{i}^{s−1}t_{i} & mod (4), \\ d(a_{i}^{2^{n}s}) & ≡ & 2^{n+2}v_{i}^{2^{n+1}s}(v_{i}^{−1}t_{i} + v_{i}^{−2}t_{i}^{2}) & mod (2^{n+3}). \end{aligned}

□

Ext(E_{1}(2)_{∗}/(2, v_{1})) is decomposed into the following four summands tensoring with Λ(ρ_{2}):

(1) v_{2}A ⊕ Λ(v_{2}) ⊗ A ⊗ Z/p(h_{20})h_{20},

(2) v_{3}B ⊕ Λ(v_{3}) ⊗ B{h_{30}, h_{30}^{2}, h_{30}^{3}} ⊕ v_{3}h_{30}h_{31}B,

(3) B ⊕ B {h_{21}, h_{31}} ⊕ v_{3}h_{21}h_{30}B,

(4) Bh_{30} {h_{21}, h_{31}} ⊕ v_{3}B {h_{21}, h_{31}}.

With respect to each summand, we construct a long exact sequence in Lemmas 5.5, 5.6 and 5.7. We often use the replacement

h_{31} = [v_{3}^{−1}t_{3} + v_{3}^{−2}t_{3}^{2}] = v_{3}^{−1}h_{30} + · · · .

If we define P_{i} (i > = 0) and Q_{j} (j > 0) by

P_{i} = Z_{(2)} {a_{2}^{2^{i}s}a_{3}^{2^{j}t}: 0 < = j < = i, 0 ≠ s ∈ Z, t > = 0} ,
Q_{j} = Z_{(2)} {a_{2}^{2^{i}s}a_{3}^{2^{j}t}: 0 < = i < j, s ∈ Z, t > 0} .

then we decompose B into

B = (⨁_{i> =0} P_{i}) ⊕ (⨁_{j>0} Q_{j}) .

Define M^{0} and M^{1} by

M^{0} = (⨁_{i> =0} P_{i} \{\frac{1}{2^{i+2}} \}) ⊕ (⨁_{j>0} Q_{j} \{\frac{1}{2^{j+2}} \}) ⊕ Q/Z_{(2)},

M^{1} = (⨁_{i> =0} P_{i} \{\frac{h_{21}}{2^{i+2}} \}) ⊕ (⨁_{j>0} Q_{j} \{\frac{h_{31}}{2^{j+2}} \}) .

Then we have the following results:

Lemma 5.5. We have long exact sequences

B  M_{0}  M_{0} B{h_{21}, h_{31}}  M^{1}  M^{1} v_{3}h_{21}h_{30}B v  w  ^{2} w ` ` ` ` ` ` ` ` ` `_{˜  δ} w  _{^{2} w} _{``````````˜}_{˜  δ}

and

v_{3}B{h_{21}, h_{31}}  (v_{3}/2)B{h_{21}, h_{31}}  (v_{3}/2)B{h_{21}, h_{31}} Bh_{30}{h_{21}, h_{31}}.v  w  ^{2} w _{```````````````˜}_{˜  δ}

Proof. In the first sequence the connecting homomorphism acts as

δ(a_{2}^{2^{i}s}a_{3}^{2^{j}t}/2^{2+min (i,j)}) = \{ \begin{matrix}a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{21} & (i < j) \\ a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{31} & (i > j) \\ a_{2}^{2^{i}s}a_{3}^{2^{j}t}(h_{21} + h_{31}) & (i = j) \end{matrix}

We also see that δ(a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{31}/2^{i+2}) for i < j, δ(a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{21}/2^{j+2}) for i > j, and δ(a_{2}^{2^{i}s}a_{3}^{2^{i}t}h_{21}/2^{i+2}) are equal to a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{21}h_{31}. Replacing h_{31} with v_{3}^{−1}h_{30} + ⋯, we have the first sequence. The second sequence is obvious. □

Lemma 5.6. We have a long exact sequence

v_{3}B  (v_{3}/2)B  (v_{3}/2)B h_{30}B ⊗ Λ(v_{3})  (v_{3}h_{30}/2)B  (v_{3}h_{30}/2)B \begin{matrix}h_{30}^{2}B ⊗ Λ(v_{3}) \\ ⊕(v_{3}h_{30}h_{31})B \end{matrix}  \begin{matrix}(v_{3}h_{30}^{2}/2)B \\ ⊕(v_{3}h_{30}h_{31}/2)B \end{matrix}  \begin{matrix}(v_{3}h_{30}^{2}/2)B \\ ⊕(v_{3}h_{30}h_{31}/2)B \end{matrix} h_{30}^{3}B ⊗ Λ(v_{3}) v  w  ^{2}  w ` ` ` ` ` ` ` ` ` ` ` ` ` `_{˜  δ} w  _{^{2} w} ` ` ` ` ` ` ` ` ` ` ` ` `_{˜  δ} w  _{^{2}w} _{````````````˜}_{˜  δ}

Proof. It follows from

δ(a_{2}^{2^{i}s}a_{3}^{2^{j}t}v_{3}h_{30}^{k}/2) = a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{30}^{k+1} for 0 < = k < = 2,
δ(a_{2}^{2^{i}s}a_{3}^{2^{j}t}v_{3}h_{30}h_{31}/2) = a_{2}^{2^{i}s}a_{3}^{2^{j}t}h_{30}^{2}h_{31}
= a_{2}^{2^{i}s}a_{3}^{2^{j}t−1}v_{3}h_{30}^{3} + ⋯
□

Lemma 5.7. We have a long exact sequence

v_{2}A  (v_{2}/2)A  (v_{2}/2)A h_{20}A ⊗ Λ(v_{2})  (v_{2}h_{20}/2)A  (v_{2}h_{20}/2)A h_{20}^{2}A ⊗ Λ(v_{2})  (v_{2}h_{20}^{2}/2)A  ⋯ v  w  ^{2} w ` ` ` ` ` ` ` ` ` ` ` ` `_{˜  δ} w  _{^{2} w} ` ` ` ` ` ` ` ` ` ` ` `_{˜  δ} w  _{^{2}  w}

Proof. Notice that each exponent of v_{2} in (v_{2}h_{20}^{k}/2)A is odd. Since we have d(x) = 0 for x ∈ A in the cobar complex, we have

d(v_{2}^{2s+1}v_{3}^{t}x) = d(v_{2}^{2s+1}v_{3}^{t}) ⊗ x.

We see that

d(v_{2}^{2s+1}v_{3}^{t}) = \{ \begin{matrix}2v_{2}^{2s}v_{3}^{2n}t_{2} + ⋯ & for t = 2n, \\ d(v_{2}^{2s+1}v_{3}^{t}) = 2v_{2}^{2s}v_{3}^{2n}(v_{3}t_{2} + v_{2}t_{3}) + ⋯ & for t = 2n + 1. \end{matrix}

In both cases we obtain

δ (\frac{v_{2}^{2s+1}v_{3}^{t}x}{2}) = v_{2}^{2s}v_{3}^{t}h_{20}x

replacing v_{3}h_{20} by v_{3}h_{20} = [v_{3}t_{2} + v_{2}t_{3}] only for the case t = 2n + 1. □

By the above three lemmas, we obtain the chart of differentials

v_{3}B  h_{30}B v_{3}h_{30}B  h_{30}^{2}B v_{3}h_{30}^{2}B  h_{30}^{3}B v_{3}h_{30}h_{31}B  v_{3}h_{30}^{3}B w  w  w w

B  h_{21}B h_{31}B  v_{3}h_{21}h_{30}B v_{3}B{h_{21}, h_{31}}  h_{30}B{h_{21}, h_{31}} w_{```````}_{ˆ```````}ˆ w w

v_{2}A  h_{20}A v_{2}h_{20}A  h_{20}^{2}A v_{2}h_{20}^{2}A  h_{20}^{3}A ⋮ w  w  w

Thus we conclude that

Lemma 5.8. Ext_{Σ(2,2)}(E_{1}(2)_{∗}, E_{1}(2)_{∗}/(2^{∞}, v_{1})) is the tensor product of Λ(ρ_{2}) and the direct sum of

(1) v_{2}A[h_{20}], v_{3}B[h_{30}]/(h_{30}^{3}) and v_{3}Bh_{30}h_{31} whose elements are of order two; and

(2) M^{0} and M^{1}. □

Let E_{∞}^{∗}(X) for a spectrum X denote the E_{∞}-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{2}X).

Theorem 5.9. The E_{∞}-term E_{∞}^{∗}(L_{2}T (1)/(2^{∞}, v_{1}) is the tensor product of Λ(ρ_{2}) and the direct sum of

(1) { \atop v_{2}A[h_{20}]}, v_{3}B[h_{30}]/(h_{30}^{3}) and v_{3}Bh_{30}h_{31} whose elements are of order two; and

(2) M^{0} and M^{1}.

where { \atop v_{2}A[h_{20}]} denote the module

(Z/2[v_{2}^{±2}, v_{3}^{4}] ⊗ Λ(v_{3}) ⊗ (Z/2[h_{30}]/(h_{30}^{4}) ⊕ Z/2{h_{21}, h_{31}} ⊗ Λ(h_{30}))) [h_{20}]/(h_{20}^{3}).

Proof. In [4], the Adams-Novikov differentials are determined in the Adams-Novikov spectral sequence for L_{2}T (1)/I_{2} (written as D in [4]) as

d_{3}(v_{3}) = 0 and d_{3}(v_{3}^{k}) = v_{2}^{2}v_{3}^{k−2}h_{20}^{3} for 2 < = k < = 3.

and d_{3}(v_{3}^{k}x) = d_{3}(v_{3}^{k})x for x = h_{20}, h_{21}, h_{30} and h_{31}. Note that for each element wa_{3}^{2t+1} ∈ v_{2}A[h_{20}], we see that

d_{3}(wa_{3}^{2t+1}/2) = wa_{3}^{2t}h_{20}^{3}/2 ∈ v_{2}A[h_{20}].

This shows the structure of π_{∗}(L_{2}T (1)/(2^{∞}, v_{1})), since it has a horizontal vanishing line. □

Proof of Theorem 1.3. Consider the cofiber sequence

T (1)/(v_{1})  T (1)/(v_{1}) ∧ SQ  T (1)/(2^{∞}, v_{1}). w  w

Then, the homotopy groups of T (m)/(v_{1}) ∧ SQ and T (1)/(2^{∞}, v_{1}) are determined in [5, Corollary 6.5.6] and Theorem 5.9, respectively. □

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General Education Division, Oshima National College of Maritime Technology, Yamaguchi 742-2193, Japan

  E-mail address: nakai@oshima-k.ac.jp

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

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