THE COHOMOLOGY OF S(n, n − 1) RELEVANT TO THE MORAVA STABILIZER ALGEBRA AT AN ODD PRIME

KATSUMI SHIMOMURA AND SHINGO TOKASHIKI Dedicated to Professor Takao Matumoto on his sixtieth birthday

Abstract. In [3], Ravenel introduced Hopf algebras S(n, k) relevant to the Morava stabilizer algebra S_{n}, and determined H^{∗}S(n, k) for k ≥ n at an odd prime p. In this paper, we introduce a spectral sequence analogous to Ravenel’s infinite descent one and determine H^{∗}S(n, n−1) at an odd prime p. Its structure is unexpectedly complex, though it has a horizontal vanishing line: H^{s}S(n, n − 1) = 0 for s > n^{2}.

1. Introduction

In the stable homotopy theory, the chromatic argument plays an important role. The argument bases on the structure of the cohomology of the Morava stabilizer algebra S(n) at each prime number p. This is a Hopf algebra defined as S(n) = 𝔽_{p} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} 𝔽_{p} for the Hopf algebroid

(BP_{∗}, BP_{∗}BP) = (ℤ_{(p)}[v_{1}, v_{2}, …], BP_{∗}[t_{1}, t_{2}, …])

associated with the Brown-Peterson spectrum BP. Here, BP_{∗} acts on 𝔽_{p} by sending v_{n} to 1 and v_{i} (i ≠ n) to zero. One of the basic result of the cohomology is the Morava structure theorem, a statement of which says that H^{s}S(n) is trivial if (p − 1) |̸ n and s > n^{2}. Ravenel ([2]) determined the structure of H^{∗}S(n) for n ≤ 2 and for n = 3 and p > 3. In order to study the structure of H^{∗}S(n) for a large n, one way to understand the cohomology is to study the cohomology of S(n, m). Here we write

S(n, m) = S(n)/(t_{j} : j < m).

Ravenel determined the cohomology of it for n ≤ m and p > 2, or for n < m and p = 2. In this paper we determine the cohomology H^{∗}S(n, n − 1) for an odd prime p. Let V (n) and T (m) for n, m ≥ 0 denote the Smith-Toda and the Ravenel spectra, respectively, characterized by BP_{∗}(V (n)) = BP_{∗}/(p, v_{1}, …, v_{n}) and BP_{∗}(T (n)) = BP_{∗}[t_{1}, …, t_{m}]. Then the cohomology gives the E_{2}-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_{∗}(L_{n}V (n − 1) ∧ T (n − 1)) if L_{n}V (n − 1) exists.

2000 Mathematics Subject Classification. Primary 55Q99; Secondary 55Q51, 20J06.

2 KATSUMI SHIMOMURA AND SHINGO TOKASHIKI

Though the Smith-Toda spectrum V (n) is not known to exist, the spectrum L_{n}V (n − 1) exists if n^{2} + n < 2p by [5].

In order to state our result, recall the result of Ravenel’s:

H^{∗}S(n, n) = Λ(h_{i,j} : n ≤ i ≤ 2n − 1, j ∈ ℤ/n).

We consider the Hopf algebra S(n, n − 1) over 𝔽_{p}, and use an abbreviation Ext^{s}(M) = Ext_{S(n,n−1)}^{s}(𝔽_{p}, M) for an S(n, n − 1)-comodule M. Let P_{k} = 𝔽_{p}[α]/(α^{p^{k}}) for k ≤ n denote the S(n, n − 1)-comodule with structure map η given by η(α) = α ⊗ 1 + 1 ⊗ t_{n−1}. Then we have a spectral sequence

(1.1)  Ext^{∗}(P_{k}) ⊗ Λ(h_{n−1,k−1}) ⊗ 𝔽_{p}[b_{n−1,k−1}] ⟹ Ext^{∗}P_{k−1},

which is an analogy of Ravenel’s infinite descent one. In particular, Ext^{∗}(P_{n}) = H^{∗}S(n, n) and Ext^{∗}(P_{0}) = H^{∗}S(n, n − 1). We introduce modules

T_{1} = h_{n−1,1}h_{2n−2,2}Λ(c_{1}, h_{2n−1,1}) ⊕ h_{n−1,2}c_{1}Λ(h_{2n−1,1})
⊕𝔽_{p}{h_{n−1,2}, c_{1}h_{2n−2,2}}
D_{1}^{0} = 𝔽_{p}{1, h_{n−1,1}, c_{1}, h_{n−1,1}h_{2n−1,1}, c_{1}h_{2n−1,1}, h_{n−1,1}c_{1}h_{2n−1,1}}

for c_{k} = h_{n,k} − h_{n,k−1}, and modules M_{k} defined inductively by

M_{n−1} = 0, M_{n−2} = 𝔽_{p},
M_{k} = Λ(c_{k+2}h_{2n−2,k+2}) ⊗ M_{k+1}

⊕Λ(h_{2n−2,k+2}) ⊗ 𝔽_{p}{h_{n−1,k+3}, h_{n−1,k+2}h_{2n−2,k+3}} ⊗ M_{k+2}.

We first state the structure of Ext^{∗}(P_{1}), which is less complex than that of Ext^{∗}(P_{0}):

Theorem 1.2. The Ext-groups Ext^{∗}(P_{1}) are isomorphic to the tensor product of Λ(c_{2}, c_{3}, …, c_{n−1}, h_{n,0}, h_{2n−2,0}, h_{2n−2,1}), Λ(h_{i,j} : n < i < 2n − 2, j ∈ ℤ/n) and T_{1}⊗M_{1}⊕D_{1}^{0}⊗M_{0}. In particular, for an integer n = 2s+1+ε ≥ 3 with odd s and ε = 0, 1, the Poincaré series of the E_{2}-term is

\begin{matrix}(1 + t)^{n^{2}−2n+1}( ∑_{α=0}^{s−1} \binom{s − 1 + ε + α}{2α + ε}t^{s−α}(1 + t^{2})^{2α+ε}(1 + t)^{2s−2α+1} \\ + ∑_{α=0}^{s−1+ε} \binom{s + α}{2α + 1 − ε}t^{s−1+ε−α}(1 + t^{2})^{2α+1−ε}(1 + t)^{2s−1−2α+2ε}(1 + t + t^{2})). \end{matrix}

Moreover, the E_{2}-term has the horizontal vanishing line : E_{2}^{s,∗} = 0 if s > n^{2}.

The Poincaré series is computed after solving the recurrence formula on M_{k} (see Lemma 3.7). With the spectral sequence (1.1) for k = 1,

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 3

Theorem 1.3. H^{∗}S(n, n−1) is isomorphic to the tensor product of Λ(h_{i,j} : n ≤ i < 2n − 2, j ∈ ℤ/n) and a module whose structure is given in Proposition 4.1 and the Poincaré series is: for n = 2,

(1 + t)^{2}(1 + t + t^{2})  p > 3 (Ravenel [2])
(1 + t)^{2}(1 + t^{2})/(1 − t) p = 3 (Henn [1])

for n > 2,

(1 + t)^{n^{2}−2n+2}((1 + t + 5t^{2} + t^{3} + t^{4})a(n) + t(1 + 3t + 3t^{2} + 3t^{3} + t^{4})a′(n)),

where, putting n = 2s + 1 + ε ≥ 3 with odd s and ε = 0, 1,

a(n) = ∑_{α=0}^{s−1} \binom{s − 1 + α + ε}{2α + ε}(1 + t^{2})^{2α+ε}t^{s−1−α}(1 + t)^{2s−2−2α}
a′(n) = ∑_{α=0}^{s−2+ε} \binom{s − 1 + α}{2α + 1 − ε}(1 + t^{2})^{2α+1−ε}t^{s−2+ε−α}(1 + t)^{2s−4+2ε−2α}.

If n^{2} + n < 2p, then there exists a spectrum L_{n}V (n − 1) such that BP_{∗}(L_{n}V (n − 1)) = v_{n}^{−1}BP_{∗}/(p, v_{1}, …, v_{n−1}) by [5]. The theorem shows that the Adams-Novikov E_{2}-term has the horizontal vanishing line : E_{2}^{s,∗} = 0 if s > n^{2}.

Corollary 1.4. If n^{2} + n < 2p, then the homotopy groups π_{∗}(L_{n}V (n − 1) ∧ T (n − 1)) are isomorphic to the E_{2}-term K(n)_{∗} ⊗ H^{∗}S(n, n − 1).

In the next section, we set up the spectral sequence, and then show Theorem 1.2 in section three. In the last section we determine H^{∗}S(n, n−1).

2. Construction of a spectral sequence

Let BP_{∗}BP be the Hopf algebroid over BP_{∗} defined by the Brown-Peterson ring spectrum BP. Put K(n)_{∗} = 𝔽_{p}[v_{n}^{±1}] for n ≥ 0, and K(n)_{∗} is a BP_{∗}-module by sending v_{n} to v_{n} and v_{i} for i ≠ n to zero. Then K(n)_{∗}K(n) = K(n)_{∗} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} K(n)_{∗} has a structure of a Hopf algebra over K(n)_{∗} induced from BP_{∗}BP. Now the Morava stabilizer algebra S(n) is a Hopf algebra defined by

S(n) = 𝔽_{p} ⊗_{K(n)_{∗}} K(n)_{∗}K(n) ⊗_{K(n)_{∗}} 𝔽_{p} = 𝔽_{p}[t_{1}, t_{2}, …]/(t_{i}^{p^{n}} − t_{i}),

where 𝔽_{p} is a K(n)_{∗}-module by sending v_{n} to 1. The Hopf algebra S(n, k) over 𝔽_{p} is defined by

S(n, k) = S(n)/(t_{1}, t_{2}, …, t_{k−1}) = 𝔽_{p}[t_{k}, t_{k+1}, …]/(t_{i}^{p^{n}} − t_{i}).

4 KATSUMI SHIMOMURA AND SHINGO TOKASHIKI

In this paper we determine H^{∗}S(n, n−1) = Ext_{S(n,n−1)}^{∗}(𝔽_{p}, 𝔽_{p}) from Ravenel’s result: (2.1)

H^{∗}S(n, n) = Ext_{S(n,n)}^{∗}(𝔽_{p}, 𝔽_{p}) = Λ(h_{n,i}, h_{n+1,i}, ⋯, h_{2n−1,i} : i ∈ ℤ/n).

So we put

(2.2)  Γ = S(n, n − 1) and Σ = S(n, n).

Consider Γ-comodules

P_{k} = 𝔽_{p}[α]/(α^{p^{k}})

for 0 ≤ k ≤ n, with coaction η: P_{k} → P_{k} ⊗ Γ given by

(2.3)  η(α) = α ⊗ 1 + 1 ⊗ t_{n−1}.

Note that P_{0} = 𝔽_{p} and

(2.4)  P_{n} = Γ□_{Σ}𝔽_{p}

as a Γ-comodule, since P_{n} is isomorphic to 𝔽_{p}[t_{n−1}]/(t_{n−1}^{p^{n}} − t_{n−1}) as a Γcomodule, and we have

(2.5)  Ext_{Γ}^{∗}(P_{n}) = Ext_{Σ}^{∗}(𝔽_{p}, 𝔽_{p}) = H^{∗}Σ = H^{∗}S(n, n)

by the change of rings theorem with (2.4). Hereafter, we put

Ext^{s}(M) = Ext_{Γ}^{s}(𝔽_{p}, M)

for an Γ-comodule M. The comodules P_{k}’s give the short exact sequences

0 ^{w}P_{k−1} ^{y w}^{ι} P_{k} ^{ι′} ^{w}P′_{k}  ^{w}0 and 0 ^{w}P′_{k} ^{y w}^{κ} α_{k−1}P_{k} ^{κ′} ^{w}β_{k−1}P_{k−1}  ^{w}0

of comodules. Here, P′_{k} is isomorphic to the comodule 𝔽_{p}{α^{i} : p^{k−1} ≤ i < p^{k}} with coaction induced from (2.3), the map κ assigns α^{p^{k−1}j+i} to jα_{k−1}α^{p^{k−1}(j−1)+i} for 0 < j < p and 0 ≤ i < p^{k−1}, and β_{k−1} stands for α_{k−1}α^{p^{k−1}(p−1)}. These short exact sequences yield the exact couple

Ext^{∗}(P_{k−1})  Ext^{∗}(P′_{k})  β_{k−1}Ext^{∗}(P_{k−1}) β_{k−1}Ext^{∗}(P′_{k})  ⋯ Ext^{∗}(P_{k})  α_{k−1}Ext^{∗}(P_{k}) β_{k−1}Ext^{∗}(P_{k}) β_{k−1}α_{k−1}Ext^{∗}(P_{k})  ⋯,_{_{i} u}  _{u_{j}} u  _{u_{i}} u  _{u_{j}} u  u_{’ ’ ’)}  _{’ ’)}  _{’ ’)}  _{’ ’ ’ ’)}

which gives rise to a spectral sequence

(2.6)  E_{1}^{∗} = Ext^{∗}(P_{k}) ⊗ 𝔽_{p}[b_{n−1,k−1}] ⊗ Λ(h_{n−1,k−1}) ⟹ Ext^{∗}(P_{k−1}).

The differential of the spectral sequence is deduced from the differential of the cobar complex Ω_{Γ}^{∗}P_{n} for computing Ext^{∗}(P_{n}). To compute the differential, we first observe the structure map of the Hopf algebra Γ, which is deduced from the formulas on BP_{∗}BP (cf. [3, Cor. 4.3.15]):

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 5

Lemma 2.7. The structure map Δ: Γ → Γ ⊗ Γ acts on t_{i} as follows:

Δ(t_{i}) = \begin{cases}t_{i} ⊗ 1 + 1 ⊗ t_{i} & i < 2n − 2 \\ t_{2n−2} ⊗ 1 + t_{n−1} ⊗ t_{n−1}^{p^{n−1}} + 1 ⊗ t_{2n−1} & i = 2n − 2 \\ t_{2n−1} ⊗ 1 + t_{n} ⊗ t_{n−1}^{p^{n}} + t_{n−1} ⊗ t_{n}^{p^{n−1}} \\ +b_{n−1,n−1} + 1 ⊗ t_{2n−1} & i = 2n − 1. \end{cases}

Here note that t_{n−1}^{p^{n}} = t_{n−1} in Γ, and the element b_{n−1,k} ∈ Γ ⊗ Γ is defined by

b_{n−1,k} = \frac{1}{p} (t_{n−1}^{p^{k}} ⊗ 1 + 1 ⊗ t_{n−1}^{p^{k}} − (t_{n−1} ⊗ 1 + 1 ⊗ t_{n−1})^{p^{k}}) .

In the sequel, we make use of the abbreviation:

a = t_{n−1}, c = t_{n} − t_{n}^{p^{n−1}}, y = t_{2n−2} − t_{n−1}^{p^{n−1}+1} and x = t_{2n−1} − t_{n}t_{n−1}.

The above lemma implies the following

Lemma 2.8. The elements h_{i,j} are represented by the elements of the cobar complex Ω_{Γ}^{1}P_{n} as follows :

\begin{aligned}h_{i,j} & = & [t_{i}^{p^{j}}] for i < 2n − 2, \\ h_{2n−2,j} & = & \begin{cases}[t_{2n−2} + αa^{p^{n−1}}] & j = 0 \\ [y^{p^{j}} − a^{p^{j}+p^{j−1}} + α^{p^{j−1}}a^{p^{j}}] & j > 0, \end{cases} and \\ h_{2n−1,j} & = & [x^{p^{j}} − α^{p^{j}}c^{p^{j}} − ξ_{j−1}], \end{aligned}

where ξ_{k} = \frac{1}{p} ∑_{i=1}^{p−1} \binom{p}{i}α^{p^{k}i} ⊗ a^{p^{k}(p−i)}.

Proof. It suffices to show that the representatives are cocycles. By Lemma 2.7, t_{i}^{p^{j}} is a cocycle if j < 2n − 2, which shows the first one.

Note that η(α) = α+a ∈ Ω_{Γ}^{1}P_{n} = P_{n}⊗Γ. Furthermore, Lemma 2.7 shows that d(t_{2n−2} + αt_{n−1}^{p^{n−1}}) = −t_{n−1} ⊗ t_{n−1}^{p^{n−1}} + t_{n−1} ⊗ t_{n−1}^{p^{n−1}} = 0, and the second one follows. For the third, we compute d(t_{2n−2}^{p^{j}} − t_{n−1}^{p^{j}+p^{j−1}} + α^{p^{j−1}}t_{n−1}^{p^{j}}) = −t_{n−1}^{p^{j}} ⊗ t_{n−1}^{p^{j−1}} + t_{n−1}^{p^{j−1}} ⊗ t_{n−1}^{p^{j}} + t_{n−1}^{p^{j}} ⊗ t_{n−1}^{p^{j−1}} − t_{n−1}^{p^{j−1}} ⊗ t_{n}^{p^{j}} = 0.

For the other, we notice that d(α^{p^{k}}) = a^{p^{k}} + pξ_{k−1} in the cobar complex Ω_{ℤ[a]}^{∗}ℤ/(p^{2})[α], and so d(ξ_{k−1}) = −b_{n−1,k−1}, since d(a^{p^{k}}) = pb_{n−1,k−1} by definition. Then by Lemma 2.7, d(t_{2n−1}^{p^{j}}−t_{n}^{p^{j}}t_{n−1}^{p^{j}}−α^{p^{j}}t_{n}^{p^{j}} +α^{p^{j}}t_{n}^{p^{j−1}} −ξ_{j−1}) = −t_{n}^{p^{j}} ⊗ t_{n−1}^{p^{j}} − t_{n−1}^{p^{j}} ⊗ t_{n}^{p^{j−1}} − b_{n−1,j−1} + t_{n}^{p^{j}} ⊗ t_{n−1}^{p^{j}} + t_{n−1}^{p^{j}} ⊗ t_{n}^{p^{j}} − t_{n−1}^{p^{j}} ⊗ t_{n}^{p^{j}} + t_{n−1}^{p^{j}} ⊗ t_{n}^{p^{j−1}} + b_{n−1,j−1} = 0. Here we use the relations t_{n}^{p^{n−1+j}} = t_{n}^{p^{j−1}} ∈ Ω_{Γ}^{1}P_{n} and b_{n−1,n−1}^{p^{j}} = b_{n−1,n−1+j} = b_{n−1,j−1} ∈ Ω_{Γ}^{2}P_{n}. □

More generally, by observing the change of rings theorem, we see the representatives are written as follows:

(2.9) Let w be a monomial of Ext^{∗}(P_{n}) = Λ(h_{i,j} : n ≤ i < 2n, j ∈ ℤ/n). Then, w is represented by an element of the form

t(w) + ∑_{i=0}^{n−1} α^{p^{i}} ⊗ c_{i}(w) + ∑_{i=0}^{n−1} ξ_{i} ⊗ c′_{i}(w)

in the cobar complex Ω_{Γ}^{∗}P_{n}, where t(w), c_{i}(w) and c′_{i}(w) are elements of Ω_{Γ}^{∗}𝔽_{p} with d(c_{i}(w)) = 0 and d(c′_{i}(w)) = 0.

We consider submodules of the E_{r}-terms of the spectral sequence (2.6). For the sake of simplicity, we put

(2.10)  \begin{matrix}c_{0} = h_{n,0}, c_{i} = h_{n,i} − h_{n,i−1} (i > 0), x_{i} = h_{2n−1,i}, \\ y_{i} = h_{2n−2,i}, a_{i} = h_{n−1,i} and β_{i} = −b_{n−1,i}, \end{matrix}

for n > 2. By use of the representatives in Ω_{Γ}^{∗}P_{n}, we tell the behavior of the differential of the spectral sequence (2.6).

Lemma 2.11. Suppose that a monomial w of Ext^{s}(P_{k+1}) is represented by

t(w) + ∑_{i=0}^{k} α^{p^{i}} ⊗ c_{i}(w) + ∑_{i=0}^{k} ξ_{i} ⊗ c′_{i}(w),

for t(w), c_{i}(w), c′_{i}(w) ∈ Ω^{s}𝔽_{p} with c_{i}(w) and c′_{i}(w) cocycles. Then, the differential of (2.6) is given by

d_{1}(w) = a_{k−1}c_{k}(w) and d_{2}(w) = β_{k−1}c′_{k}(w) if c_{k}(w) = 0.

Besides,

d_{3}(c_{k}y_{k}) = a_{k−1}β_{k−1} and d_{4}(a_{k}c_{k}x_{k}) = β_{k−1}^{2}.

Proof. The differential d_{1} and d_{2} are obvious. y_{k}c_{k} is represented by y^{p^{k}} ⊗ c^{p^{k}} + α^{p^{k−1}} ⊗ a^{p^{k}} ⊗ c^{p^{k}}, which is homologous to y^{p^{k}} ⊗ c^{p^{k}} + α^{p^{k−1}} ⊗ b_{n−1,k−1} by d(α^{p^{k−1}} ⊗ x_{k}), which implies the third. x_{k}a_{k}c_{k} is represented by x^{p^{k}}⊗a^{p^{k}}⊗c^{p^{k}}−α^{p^{k}}⊗c^{p^{k}}⊗a^{p^{k}}⊗c^{p^{k}}−ξ_{k−1}⊗a^{p^{k}}⊗c^{p^{k}}. Since α^{p^{k}}⊗c^{p^{k}}⊗a^{p^{k}}⊗c^{p^{k}} is homologous to 0 by d(α^{p^{k}} ⊗c^{p^{k}}a^{p^{k}} ⊗c^{p^{k}} + \frac{1}{2}α^{p^{k}} ⊗a^{p^{k}} ⊗c^{2p^{k}}) and ξ_{k−1}⊗a^{p^{k}} ⊗c^{p^{k}} to b_{n−1,k−1} ⊗ x_{k} + ξ_{k−1} ⊗ b_{n−1,k−1} by d(ξ_{k−1} ⊗ x_{k}), we obtain the fourth. □

Corollary 2.12. d_{1}(y_{i}) = \begin{cases}a_{0}a_{n−1} & i = 0 \\ a_{i−1}a_{i} & i > 0 \end{cases}, d_{1}(x_{i}) = a_{i}c_{i} and

d_{2}(wx_{i+1}) = wβ_{i} for w = a_{i+1} or c_{i+1}.

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 7

Consider the modules (2.13)

B_{s} = \begin{cases}Λ(c_{i}, x_{i}, y_{i} : 0 ≤ i < n) & s = 0 \\ Λ(c_{i}, x_{j}, y_{k} : 0 ≤ i < n − s, 0 < j < n − s, 0 ≤ k ≤ n − s\} & s > 0 \end{cases}
C = Λ(h_{i,j} : n < i < 2n − 2, 0 ≤ j < n) ; and
D_{s} = Λ(a_{s}, c_{s}, x_{s}) for s ≥ 0.

Then, (2.5) is rewritten as

Ext^{∗}(P_{n}) = B_{0} ⊗ C

by (2.1). We also introduce modules \overline{D}_{s}, D̃_{s}, D_{s}^{0} and T_{s}:

(2.14)  \begin{aligned}\overline{D}_{s} & = & 𝔽_{p}\{a_{s}, c_{s}, a_{s}x_{s}, a_{s}c_{s}x_{s}\} \\ D̃_{s} & = & 𝔽_{p}\{1, c_{s}, a_{s}x_{s}, c_{s}x_{s}\} \\ D_{s}^{0} & = & 𝔽_{p}\{1, a_{s}, c_{s}, a_{s}x_{s}, c_{s}x_{s}, a_{s}c_{s}x_{s}\} \\ T_{s} & = & y_{s+1}a_{s}Λ(c_{s}, x_{s}) ⊕ a_{s+1}c_{s}Λ(x_{s}) ⊕ 𝔽_{p}\{a_{s+1}, y_{s+1}c_{s}\} \end{aligned}

for 0 < s < n. Note that

(2.15) D_{s}^{0} = Λ(c_{s}x_{s})⊕\overline{D}_{s} = a_{s}Λ(c_{s}x_{s})⊕D̃_{s} and T_{s} = y_{s+1}\overline{D}_{s} ⊕a_{s+1}D̃_{s}.

Here we use the relation y_{s+1}a_{s}c_{s} = a_{s}a_{s+1}x_{s} of homology.

The differential defined by d(x_{s}) = a_{s}c_{s} makes D_{s} a differential module. Then, we have the cohomology groups of it.

Lemma 2.16. H^{∗}(D_{s} : d) = D_{s}^{0}.

Proof. Since D_{s} = D_{s}^{0} ⊕ D_{s}^{1} for D_{s}^{1} = 𝔽_{p}{x_{s}, a_{s}c_{s}} and the differential acts trivially on D_{s}^{0}, H^{∗}(D_{s} : d) = D_{s}^{0} ⊕ H^{∗}(D_{s}^{1} : d) = D_{s}^{0}. □

Let K denote a submodule of the E_{2}-term of the spectral sequence (2.6). We use the notation

h^{∗}(K)

as the iterated cohomology groups H^{∗}(H^{∗}(H^{∗}(H^{∗}(K : d_{1}) : d_{2}) : d_{3}) : d_{4}), when they make sense. Consider the differentials d_{r} in Lemma 2.11 on the submodule

(2.17)  K_{k} = D_{k+1}^{0} ⊗ D_{k} ⊗ Λ(y_{k+1}) ⊗ 𝔽_{p}[β_{k}].

Lemma 2.18. h^{∗}(K_{k}) = (T_{k} ⊕ D_{k}^{0} ⊗ Λ(a_{k+1}y_{k+1})) ⊗ Λ(c_{k+1}).

Proof. We first consider K_{k}^{0} = D_{k+1}^{0} ⊗ D_{k} ⊗ 𝔽_{p}[β_{k}]. Then H^{∗}(K_{k}^{0} : d_{1}) = D_{k+1}^{0} ⊗ D_{k}^{0} ⊗ 𝔽_{p}[β_{k}] by Lemma 2.16. Since the inclusion K_{k}^{0} → K_{k} is a map of differential modules, it suffices to compute the cohomologies h^{∗}(H^{∗}(K_{k}^{0} : d_{1}) ⊗ Λ(y_{k+1})). In this proof, we use the abbreviation: ξ′ = ξ_{k+1} and ξ = ξ_{k}

8 KATSUMI SHIMOMURA AND SHINGO TOKASHIKI

for ξ = a, c, x, y, \overline{D} and D̃. We decompose D^{0}′⊗D^{0}⊗Λ(y_{k+1}) into three summands U_{1} = 𝔽_{p}{a′, y′, a′x′, a′c′x′y′} ⊗ D^{0}, U_{2} = 𝔽_{p}{1, c′y′, a′c′x′, c′x′y′} ⊗ D^{0} and U_{3} = 𝔽_{p}{c′, a′y′, c′x′, a′x′y′} ⊗ D^{0}. In the last summand, the differential is given by d_{2}(c′x′) = c′β and d_{2}(a′x′y′) = a′y′β, and so, h^{∗}(U_{3} ⊗ 𝔽_{p}[β]) = 𝔽_{p}{c′, a′y′} ⊗ D^{0}. For the other summands, noticing that D_{s}^{0} = Λ(c_{s}x_{s}) ⊕ \overline{D}_{s} = a_{s}Λ(c_{s}x_{s}) ⊕ D̃_{s}, we see that the differentials act as follows:

a′c′x′y′(Λ(cx) ⊕ \overline{D}) a′x′(aΛ(cx) ⊕ D̃)  y′(Λ(cx) ⊕ \overline{D}) a′(aΛ(cx) ⊕ D̃)_{’ ’ ’ ’ ’ *_{d_{3}}}  _{h h h h hh j}_{d_{4}} _{h h h hh j}_{d_{2}}  ’ ’ ’ ’ ’ ’ *_{d_{1}}

c′x′y′(Λ(cx) ⊕ \overline{D}) a′c′x′(aΛ(cx) ⊕ D̃)  c′y′(Λ(cx) ⊕ \overline{D}) (aΛ(cx) ⊕ D̃)._{’ ’ ’ ’ ’ *_{d_{1}}}  _{h h h h h j}_{d_{2}} _{h h h h j}_{d_{4}}  _{’ ’ ’ ’ ’ *_{d_{3}}}

These pictures show that h^{∗}(U_{1}⊗𝔽_{p}[β]) = y′\overline{D}⊗Λ(β)⊕a′x′aΛ(cx)⊕a′D̃ and h^{∗}(U_{2} ⊗ 𝔽_{p}[β]) = D̃ ⊗ Λ(β) ⊕ aΛ(cx) ⊕ c′y′\overline{D}. The lemma now follows as the direct sum of these modules. Here we use the relation y′β\overline{D} ⊕ a′x′aΛ(cx) = a′c′y′D^{0}. □

Lemma 2.19. h^{∗}(T_{k} ⊗ 𝔽_{p}[β_{k−1}]) = 𝔽_{p}{a_{k+1}, y_{k+1}a_{k}} ⊗ Λ(c_{k}).

Proof. Note that d_{2}(y_{k+1}a_{k}c_{k}^{ε}x_{k}) = y_{k+1}a_{k}c_{k}^{ε}β_{k−1} for ε = 0, 1, and we have h^{∗}(y_{k+1}a_{k}Λ(c_{k}, x_{k})⊗𝔽_{p}[β_{k−1}]) = y_{k+1}a_{k}Λ(c_{k}). We also note that d_{2}(y_{k+1}c_{k}+ a_{k+1}x_{k}) = a_{k+1}β_{k−1}, and then h^{∗}(𝔽_{p}{a_{k+1}, y_{k+1}c_{k}} ⊗ 𝔽_{p}[β_{k−1}]) = 𝔽_{p}{a_{k+1}}. We further see that h^{∗}(a_{k+1}c_{k}Λ(x_{k}) ⊗ 𝔽_{p}[β_{k−1}]) = 𝔽_{p}{a_{k+1}c_{k}}, The direct sum of these modules shows the lemma. □

Lemma 2.20. h^{∗}(D_{1}^{0}⊗𝔽_{p}[β_{0}]) = Λ(a_{1}, c_{1}) and h^{∗}((a_{0}D_{1}^{0}⊕y_{1}D_{1}^{0})⊗𝔽_{p}[β_{0}]) = Λ(c_{1}) ⊗ 𝔽_{p}{a_{0}, a_{1}y_{1}}.

Proof. It is easy to see that h^{∗}(D_{1}^{0} ⊗ 𝔽_{p}[β_{0}]) = 𝔽_{p}{1, β_{0}, a_{1}, c_{1}}. Since β_{0} is homologous to a_{1}c_{1}, we have the first result.

Note that a_{0}D_{1}^{0} = a_{0}a_{1}Λ(c_{1}x_{1})⊕D̃_{1} and y_{1}D_{1}^{0} = y_{1}Λ(c_{1}x_{1})⊕\overline{D}_{1} by (2.15). Then, H^{∗}(a_{0}D_{1}^{0}⊕y_{1}D_{1}^{0} : d_{1}) = a_{0}D̃_{1}⊕y_{1}\overline{D}_{1} = 𝔽_{p}\{a_{0}, a_{0}c_{1}, a_{0}a_{1}x_{1}, a_{0}c_{1}x_{1}, a_{1}y_{1},

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 9

c_{1}y_{1}, a_{1}x_{1}y_{1}, a_{1}c_{1}x_{1}y_{1}}. The differentials on the module act as

\begin{matrix}d_{2}(a_{1}x_{1}y_{1}) = a_{1}y_{1}β_{0}, d_{3}(c_{1}y_{1}) = a_{0}β_{0}, d_{3}(a_{0}c_{1}x_{1}) = a_{0}c_{1}β_{0} and \\ d_{3}(a_{1}c_{1}x_{1}y_{1}) = a_{0}a_{1}x_{1}β_{0}, \end{matrix}

since we have the relation a_{0}a_{1}x_{1} = y_{1}a_{1}c_{1}. These show the second half. □

Lemma 2.21. h^{∗}(D_{1}^{0} ⊗ Λ(a_{0}, y_{1}) ⊗ 𝔽_{p}[β_{0}]) = Λ(c_{1}) ⊗ W for W = Λ(a_{1}, a_{0}y_{1}) ⊕ 𝔽_{p}{a_{0}, a_{1}y_{1}} = Λ(a_{0}, a_{1}y_{1}) ⊕ 𝔽_{p}{a_{1}, a_{0}y_{1}}.

Proof. We decompose D_{1}^{0} ⊗ Λ(a_{0}, y_{1}) into the direct sum of D_{1}^{0} ⊗ Λ(a_{0}y_{1}), a_{0}D_{1}^{0}, y_{1}D_{1}^{0}. The lemma follows from Lemma 2.20. □

3. Computation of Ext^{∗}(P_{1})

In this section, we compute the spectral sequences (2.6) for k > 0 with the abbreviation (2.10). First consider the spectral sequence for k = n − 1.

Lemma 3.1. Ext^{∗}(P_{n−1}) = D_{n−1}^{0} ⊗ B_{1} ⊗ C for n > 2.

Proof. Note that Ext^{∗}(P_{n}) = D_{n−1} ⊗ B_{1} ⊗ Λ(x_{0}) ⊗ 𝔽_{p}[β_{n−1}] ⊗ C and the differential acts only on D_{n−1} ⊗ Λ(x_{0}) ⊗ 𝔽_{p}[β_{n−1}]. The action of the differential is given by

d_{1}(x_{n−1}) = a_{n−1}c_{n−1} and d_{2}(x_{0}) = β_{n−1}.

The E_{2}-term is D_{n−1}^{0} ⊗ B_{1} ⊗ Λ(x_{0}) ⊗ 𝔽_{p}[β_{n−1}] ⊗ C by Lemma 2.16, and the E_{3}-term is D_{n−1}^{0} ⊗ B_{1} ⊗ C, which is E_{∞}-term. □

We write

(3.2)  \begin{matrix}C(k, l) = Λ(c_{k}, …, c_{l}), L_{k} = Λ(a_{k}y_{k}) and \\ J_{k} = Λ(y_{k}) ⊗ 𝔽_{p}\{a_{k+1}, a_{k}y_{k+1}\} \end{matrix}

Lemma 3.3. Ext^{∗}(P_{k}) = S_{k} ⊗ B_{n−k} ⊗ C for k > 0. Here

(3.4)  S_{k} = C(k + 1, n − 1) ⊗ (T_{k} ⊗ M_{k} ⊕ D_{k}^{0} ⊗ M_{k−1})

for the modules M_{k} defined inductively by

M_{n−1} = 0, M_{n−2} = 𝔽_{p}, and M_{k−1} = J_{k+1} ⊗ M_{k+1} ⊕ L_{k+1} ⊗ M_{k}.

Proof. This holds true for k = n − 1 by Lemma 3.1. Suppose that the lemma for k holds, and consider the spectral sequence (2.6). Since B_{n−k} =

10 KATSUMI SHIMOMURA AND SHINGO TOKASHIKI

Λ(c_{k−1}, x_{k−1}, y_{k}) ⊗ B_{n−k−1}, the E_{1}-term of it is

S_{k} ⊗ B_{n−k} ⊗ C ⊗ 𝔽_{p}[β_{k−1}] ⊗ Λ(a_{k−1})

= S_{k} ⊗ D_{k−1} ⊗ Λ(y_{k}) ⊗ 𝔽_{p}[β_{k−1}] ⊗ B_{n−k−1} ⊗ C
= C(k + 1, n − 1) ⊗ (T_{k} ⊗ M_{k} ⊕ D_{k}^{0} ⊗ M_{k−1})
⊗D_{k−1} ⊗ Λ(y_{k}) ⊗ 𝔽_{p}[β_{k−1}] ⊗ B_{n−k−1} ⊗ C
= C(k + 1, n − 1) ⊗ T_{k} ⊗ M_{k} ⊗ D_{k−1} ⊗ Λ(y_{k}) ⊗ 𝔽_{p}[β_{k−1}] ⊗ B_{n−k−1} ⊗ C
⊕C(k + 1, n − 1) ⊗ K_{k−1} ⊗ M_{k−1} ⊗ B_{n−k−1} ⊗ C

for the modules in (2.13), (2.14) and (2.17). Lemmas 2.18 and 2.19 show h^{∗}(K_{k−1} ⊗ M_{k−1}) = (T_{k−1} ⊕ Λ(a_{k}y_{k}) ⊗ D_{k−1}^{0}) ⊗ Λ(c_{k}) ⊗ M_{k−1} and h^{∗}(T_{k} ⊗ M_{k} ⊗ D_{k−1} ⊗ Λ(y_{k}) ⊗ 𝔽_{p}[β_{k−1}]) = 𝔽_{p}{a_{k}, y_{k}a_{k−1}} ⊗ M_{k} ⊗ D_{k−1}^{0} ⊗ Λ(c_{k}, y_{k}) as desired. □

Corollary 3.5. Ext^{∗}(P_{1}) = C(2, n−1)⊗(T_{1} ⊗ M_{1} ⊕ D_{1}^{0} ⊗ M_{0})⊗Λ(c_{0}, y_{0}, y_{1}) ⊗C.

In order to state the structure of M_{k}, we introduce the notation [L^{α}|J^{β}]_{k}, which denotes the direct sum of the modules X_{1} ⊗ ⋯ ⊗ X_{α+β}, where each X_{i} is L_{k(i)} or J_{k(i)}, numbers of L_{j}’s are α, and the integer k(i) is defined inductively by

k(1) = k and for i > 1, k(i) = \begin{cases}k(i − 1) + 1 & if X_{i−1} = L_{k(i−1)} \\ k(i − 1) + 2 & if X_{i−1} = J_{k(i−1)}. \end{cases}

For example,

[L^{2}|J^{1}]_{k} = L_{k} ⊗ L_{k+1} ⊗ J_{k+2} ⊕ L_{k} ⊗ J_{k+1} ⊗ L_{k+3} ⊕ J_{k} ⊗ L_{k+2} ⊗ L_{k+3}.

Lemma 3.6. The modules [L^{α}|J^{β}]_{k} satisfy 1) L_{k} ⊗ [L^{α−1}|J^{β}]_{k+1} ⊕ J_{k} ⊗ [L^{α}|J^{β−1}]_{k+2} = [L^{α}|J^{β}]_{k},

2) [L^{α−1}|J^{β}]_{k} ⊗ L_{k+α+2β−2} ⊕ [L^{α}|J^{β−1}]_{k} ⊗ J_{k+α+2β−3} = [L^{α}|J^{β}]_{k}.

Proof. By definition, [L^{α}|J^{β}]_{k} = ⨁X_{1} ⊗ ⋯ ⊗ X_{n} = ⨁_{X_{1}=L_{k}}L_{k} ⊗ X_{2} ⊗ ⋯ X_{n} ⊕ ⨁_{X_{1}=J_{k}}J_{k} ⊗ X_{2} ⊗ ⋯ ⊗ X_{n}, where n = α + β. We also see that ⨁_{X_{1}=L_{k}}X_{2} ⊗ ⋯ ⊗ X_{n} = [L^{α−1}|J^{β}]_{k+1} and ⨁_{X_{1}=J_{k}}X_{2} ⊗ ⋯ ⊗ X_{n} = [L^{α}|J^{β−1}]_{k+2} by definition, and we have the first. The latter half is verified similarly. □

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 11

Lemma 3.7. For s > 0,

M_{n−2s−1} = ⨁_{α+β=s−1}[L^{2α+1}|J^{β}]_{n−2s+1}
M_{n−2s−2} = ⨁_{α+β=s}[L^{2α}|J^{β}]_{n−2s}.

Proof. Since M_{n−1} = 0 and M_{n−2} = 𝔽_{p}, we see that M_{n−3} = L_{n−1} = [L^{0+1}|J^{0}]_{n−1} and M_{n−4} = L_{n−2}⊗M_{n−3}⊕J_{n−2}⊗M_{n−2} = L_{n−2}⊗L_{n−1}⊕J_{n−2} = [L^{2}|J^{0}]_{n−2} ⊕ [L^{0}|J^{1}]_{n−2}, which shows the case for s = 1.

Suppose that the lemma holds true for 2s. Then,

M_{n−2s−1} = L_{n−2s+1} ⊗ M_{n−2s} ⊕ J_{n−2s+1} ⊗ M_{n−2s+1}
= ⨁_{α+β=s−1}L_{n−2s+1} ⊗ [L^{2α}|J^{β}]_{n−2s+2}
⊕⨁_{α+β=s−2}J_{n−2s+1} ⊗ [L^{2α+1}|J^{β}]_{n−2s+3}
= ⨁_{α=0}^{s−1}L_{n−2s+1} ⊗ [L^{2α}|J^{s−1−α}]_{n−2s+2}
⊕⨁_{α=0}^{s−2}J_{n−2s+1} ⊗ [L^{2α+1}|J^{s−2−α}]_{n−2s+3}
= L_{n−2s+1} ⊗ [L^{2s−2}|J^{0}]_{n−2s+2} ⊕ ⨁_{α=0}^{s−2}[L^{2α+1}|J^{s−1−α}]_{n−2s+1}

by Lemma 3.6 1), which shows the case for 2s + 1.

M_{n−2s−2} = L_{n−2s} ⊗ M_{n−2s−1} ⊕ J_{n−2s} ⊗ M_{n−2s}
= ⨁_{α+β=s−1}L_{n−2s} ⊗ [L^{2α+1}|J^{β}]_{n−2s+1}
⊕⨁_{α+β=s−1}J_{n−2s} ⊗ [L^{2α}|J^{β}]_{n−2s+2}
= ⨁_{β=0}^{s−1}L_{n−2s} ⊗ [L^{2s−1−2β}|J^{β}]_{n−2s+1}
⊕⨁_{β=0}^{s−1}J_{n−2s} ⊗ [L^{2s−2−2β}|J^{β}]_{n−2s+2}
= ⨁_{β=0}^{s−1}L_{n−2s} ⊗ [L^{2s−1−2β}|J^{β}]_{n−2s+1}
⊕⨁_{β=1}^{s}J_{n−2s} ⊗ [L^{2s−2β}|J^{β−1}]_{n−2s+2}
= L_{n−2s} ⊗ [L^{2s−1}|J^{0}]_{n−2s+1} ⊕ ⨁_{β=1}^{s−1}L_{n−2s} ⊗ [L^{2s−1−2β}|J^{β}]_{n−2s+1}

⊕⨁_{β=1}^{s−1}J_{n−2s} ⊗ [L^{2s−2β}|J^{β−1}]_{n−2s+2} ⊕ J_{n−2s} ⊗ [L^{0}|J^{s−1}]_{n−2s+2},

and Lemma 3.6 1) shows the case for 2s + 2, which completes induction as desired. □

This together with Corollary 3.5 shows the following:

Corollary 3.8. Ext^{∗}(P_{1}) is isomorphic additively to the tensor product of C(2, n − 1) ⊗ Λ(c_{0}, y_{0}, y_{1}) ⊗ C and the module

⨁_{α+β=s−1}T_{1} ⊗ [L^{2α+ε}|J^{β}]_{3} ⊕ ⨁_{α+β=s−1+ε}D_{1}^{0} ⊗ [L^{2α+1−ε}|J^{β}]_{2}

for n = 2s + 1 + ε ≥ 3.

4. Ext^{∗}(P_{0}) = H^{∗}S(n, n − 1)

Since S(n, k) is defined for k ≥ 1, the smallest n is two. H^{∗}S(2, 1) = H^{∗}S(2) is given in [2] (cf. [3]). For n > 2, consider the spectral sequence (2.6) for k = 1. Here we also use the abbreviation (2.10). Let W_{1} denote a module fitting in Ext^{∗}(P_{1}) = W_{1} ⊗ Λ(y_{0}). Then, the E_{1}-term of (2.6) for k = 1 is W_{1} ⊗ Λ(y_{0}) ⊗ Λ(a_{0}) ⊗ 𝔽_{p}[β_{0}]. We first determine h^{∗}(W_{1} ⊗ Λ(a_{0}) ⊗ 𝔽_{p}[β_{0}]) ⊗ Λ(y_{0}), and then we determine the E_{∞}-term of (2.6) as a cohomology with differential d(y_{0}) = a_{0}a_{n−1}.

h^{∗}(W_{1} ⊗ Λ(a_{0}) ⊗ 𝔽_{p}[β]) ⊗ Λ(y_{0})
= h^{∗}(C(2, n − 1) ⊗ (T_{1} ⊗ M_{1} ⊕ D_{1}^{0} ⊗ M_{0})
⊗Λ(c_{0}, y_{0}, y_{1}) ⊗ C ⊗ Λ(a_{0}) ⊗ 𝔽_{p}[β_{0}])
= C(0, n − 1) ⊗ (𝔽_{p}{a_{2}, a_{1}y_{2}} ⊗ Λ(a_{0}, y_{1}) ⊗ M_{1}
⊕W ⊗ M_{0}) ⊗ Λ(y_{0}) ⊗ C by Lemmas 2.19 and 2.21
= C(0, n − 1) ⊗ (J_{1} ⊗ Λ(a_{0}, y_{0}) ⊗ M_{1}
⊕L_{1} ⊗ M_{0} ⊗ Λ(a_{0}, y_{0}) ⊕ J_{0} ⊗ M_{0}) ⊗ C
= C(0, n − 1) ⊗ (M_{−1} ⊗ Λ(a_{0}, y_{0}) ⊕ J_{0} ⊗ M_{0}) ⊗ C.

For n = 2s + 1 + ε ≥ 3 with ε = 0, 1,

M_{−1} = ⨁_{α+β=s}[L^{2α+ε}|J^{β}]_{1}
= ⨁_{α+β=s−1+ε}[L^{2α+1−ε}|J^{β}]_{1} ⊗ L_{n−1}
⊕⨁_{α+β=s−1}[L^{2α+ε}|J^{β}]_{1} ⊗ J_{n−2} by Lemma 3.6 2).

Put A_{0} = ⨁_{α+β=s−1}[L^{2α+ε}|J^{β}]_{1} and A_{1} = ⨁_{α+β=s−2+ε}[L^{2α+1−ε}|J^{β}]_{1}. Then,

M_{−1} = A_{0} ⊗ L_{n−2} ⊗ L_{n−1} ⊕ A_{1} ⊗ J_{n−3} ⊗ L_{n−1} ⊕ A_{0} ⊗ J_{n−2}
= A_{0} ⊗ (L_{n−2} ⊗ L_{n−1} ⊕ J_{n−2}) ⊕ A_{1} ⊗ J_{n−3} ⊗ L_{n−1}
= A_{0} ⊗ 𝔽_{p}{1, a_{n−2}y_{n−2}, a_{n−1}y_{n−1}, a_{n−2}a_{n−1}y_{n−2}y_{n−1}, a_{n−1}, a_{n−2}y_{n−1},
a_{n−1}y_{n−2}, a_{n−2}y_{n−2}y_{n−1}}

⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}, a_{n−3}y_{n−2}, a_{n−2}y_{n−3}, a_{n−3}y_{n−3}y_{n−2}, a_{n−2}a_{n−1}y_{n−1}, a_{n−3}a_{n−1}y_{n−2}y_{n−1}, a_{n−2}a_{n−1}y_{n−3}y_{n−1}, a_{n−3}a_{n−1}y_{n−3}y_{n−2}y_{n−1}}

THE COHOMOLOGY OF S(n, n − 1) AT AN ODD PRIME 13

= A_{0} ⊗ 𝔽_{p}{1, a_{n−2}y_{n−2}, a_{n−2}y_{n−1}, a_{n−2}y_{n−2}y_{n−1}}
⊕a_{n−1}A_{0} ⊗ 𝔽_{p}{y_{n−1}, a_{n−2}y_{n−2}y_{n−1}, 1, y_{n−2}}
⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}, a_{n−3}y_{n−2}, a_{n−2}y_{n−3}, a_{n−3}y_{n−3}y_{n−2}}
⊕a_{n−1}A_{1} ⊗ 𝔽_{p}{a_{n−2}y_{n−1}, a_{n−3}y_{n−2}y_{n−1},
a_{n−2}y_{n−3}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}y_{n−1}}
= A_{0} ⊗ Λ(a_{n−1}, a_{n−2}y_{n−2}y_{n−1})
⊕A_{0} ⊗ 𝔽_{p}{a_{n−2}y_{n−2}, a_{n−2}y_{n−1}, a_{n−1}y_{n−2}, vy_{n−1}}
⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}, a_{n−3}y_{n−2}, a_{n−2}y_{n−3}, a_{n−3}y_{n−3}y_{n−2}}
⊕a_{n−1}A_{1} ⊗ 𝔽_{p}{a_{n−2}y_{n−1}, a_{n−3}y_{n−2}y_{n−1},
a_{n−2}y_{n−3}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}y_{n−1}}

For n = 3, A_{0} = 𝔽_{p} and A_{1} = 0, and so

M_{−1} = Λ(a_{2}, a_{1}y_{1}y_{2}) ⊕ 𝔽_{p}\{a_{1}, a_{2}\} ⊗ 𝔽_{p}\{y_{1}, y_{2}\} = \overline{M}_{−1} ⊗ Λ(a_{2}) ⊕ M̃_{−1}. For n > 3, we make more computation. Since A_{0} = A_{1} ⊗ L_{n−3} ⊕ A′_{0} ⊗ J_{n−4} for A′_{0} = ⨁_{α+β=s−2}[L^{2α+ε}|J^{β}]_{1}, we see that

M_{−1} = A_{0} ⊗ Λ(a_{n−1}, a_{n−2}y_{n−2}y_{n−1}) ⊕ A_{0} ⊗ 𝔽_{p}{a_{n−2}y_{n−2}, a_{n−1}y_{n−1}}

⊕a_{n−1}y_{n−2}(A_{1} ⊕ A′_{0} ⊗ J_{n−4}) ⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}, a_{n−3}y_{n−2}, a_{n−2}y_{n−3}} ⊕a_{n−1}A_{1} ⊗ 𝔽_{p}{a_{n−3}y_{n−2}y_{n−1}, a_{n−2}y_{n−3}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}y_{n−1}} ⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}} ⊗ Λ(a_{n−1}) ⊕a_{n−3}a_{n−2}y_{n−3}y_{n−1}A_{1} ⊕ a_{n−2}y_{n−2}A′_{0} ⊗ J_{n−4}

= \overline{M}_{−1} ⊗ Λ(a_{n−1}) ⊕ M̃_{−1},

where \overline{M}_{−1} = A_{0} ⊗Λ(a_{n−2}y_{n−2}y_{n−1})⊕A_{1} ⊗𝔽_{p}\{a_{n−2}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}\} and

M̃_{−1} = A_{0} ⊗ 𝔽_{p}{a_{n−2}y_{n−2}, a_{n−1}y_{n−1}} ⊕ a_{n−1}y_{n−2}(A_{1} ⊕ A′_{0} ⊗ J_{n−4})

⊕A_{1} ⊗ 𝔽_{p}{a_{n−2}, a_{n−3}y_{n−2}, a_{n−2}y_{n−3}} ⊕a_{n−1}A_{1} ⊗ 𝔽_{p}{a_{n−3}y_{n−2}y_{n−1}, a_{n−2}y_{n−3}y_{n−1}, a_{n−3}y_{n−3}y_{n−2}y_{n−1}} ⊕a_{n−3}a_{n−2}y_{n−3}y_{n−1}A_{1} ⊕ a_{n−2}y_{n−1}A′_{0} ⊗ J_{n−4}.

Note here that A′_{0} = 0 if n = 4. We further introduce a module Y_{n}: Y_{n} = 𝔽_{p}{1, a_{0}, a_{n−1}, a_{0}y_{0}, a_{n−1}y_{0}, a_{0}a_{n−1}y_{0}}, which is H^{∗}(Λ(a_{0}, a_{n−1}, y_{0}) : d). Then, we obtain

H^{∗}(M_{−1} ⊗ Λ(a_{0}, y_{0}) : d) = Y_{n} ⊗ \overline{M}_{−1} ⊕ M̃_{−1} ⊗ Λ(a_{0}, y_{0}),

and the following:

Proposition 4.1. For n ≥ 3, Ext^{∗}(P_{0}) = C(0, n − 1) ⊗ C ⊗ (Y_{n} ⊗ \overline{M}_{−1} ⊕

M̃_{−1} ⊗ Λ(a_{0}, y_{0}) ⊕ J_{0} ⊗ M_{0}).

References

[1] H. -W. Henn, Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups, Duke Math. J. 91 (1998), 561–585.

[2] D. C. Ravenel, The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977), 287–297.

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

[4] D. C. Ravenel, The method of infinite descent in stable homotopy theory I, Contemp. Math. 293 (2002), 251-284.

[5] K. Shimomura and Z. Yosimura, BP -Hopf module spectrum and BP_{∗}-Adams spectral sequence, Publ. RIMS, Kyoto Univ. 21 (1986), 925–947.

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

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

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

 E-mail address: shin-toka.nov-17.ito-oki@docomo.ne.jp