The homotopy groups π_{∗}(L_{2}T (n) ∧ V (0))

Yousuke Kamiya and Katsumi Shimomura

Abstract. For the Ravenel spectrum T (n) at an odd prime p, we determine the homotopy groups π_{∗}(L_{2}T (n) ∧ V (0)) for the mod p Moore spectrum V (0). This is done by determining the chromatic E_{1}-term H(k)^{∗}M_{1}^{1}, which is obtained only two key lemmas: one is to define the Miller-Ravenel-Wilson elements and the other is to give a one dimensional element ζ.

1. Introduction and the statement of results

Let T (k) denote the Ravenel ring spectrum at a prime p, which is characterized by the Brown-Peterson homology BP_{∗}(T (k)) = BP_{∗}[t_{1}, t_{2}, …, t_{k}] ⊂ BP_{∗}(BP) = BP_{∗}[t_{1}, t_{2}, …], where BP_{∗} = Z_{(p)}[v_{1}, v_{2}, …]. Note that T (0) = S^{0}. Then the homotopy groups π_{∗}(T (k)) are, in a sense, an approximation of the homotopy groups π_{∗}(S^{0}) of spheres. For the Bousfield localization functor L_{n} on the stable homotopy category with respect to v_{n}^{−1}BP, the homotopy groups π_{∗}(L_{n}S^{0}) are also an approximation of π_{∗}(S^{0}). Both of the homotopy groups are considerably easier to compute than the homotopy groups of spheres. In this paper we determine the homotopy groups of L_{2}T (k) ∧ V (0) for each k > 0 at an odd prime p, where V (0) denotes the mod p Moore spectrum. These groups are computed by the Adams-Novikov spectral sequence and the chromatic spectral sequence. The E_{1}- terms of the chromatic spectral sequence are H(k)^{∗}M_{1}^{0} and H(k)^{∗}M_{1}^{1}, where H(k)^{∗}M = Ext_{BP_{∗}(BP)}^{∗}(BP_{∗}, M⨂_{BP_{∗}}BP_{∗}(T (k))). If the prime p is odd, then the Adams-Novikov spectral sequence collapses from the E_{2}-term, and so it suffices to determine the chromatic E_{1}-terms to obtain the module structure of π_{∗}(L_{2}T (k) ∧ V (0)). Ravenel determined H(k)^{∗}M_{1}^{0} (cf. [3]) and we determine H(k)^{∗}M_{1}^{1} here not only for an odd prime p but for the prime 2 if k > 1.

2000 Mathematics Subject Classification. 55Q51, 55Q10.

Key words and phrases. Homotopy groups, L_{2}-localization, Adams-Novikov spectral sequence, chromatic spectral sequence. Ravenel spectrum

Note that H(1)^{∗}M_{1}^{1} at p = 2 is determined in [6]. For the case k = 0, it is determined in [9] and [5] if p > 3, [8] if p = 3 and [7] if p = 2, which show the computation very tough. The computation is a little easier if k = 1. If k > 1, then it is much easier, and has a some kind of good nature. This will be studied in the forthcoming paper.

In order to state our main theorem, we introduce some notations: The modules k(1)_{∗}, K(1)_{∗}, k(1/j)_{∗} and C_{k} are given by

k(1)_{∗} = Z/p[v_{1}]
K(1)_{∗} = v_{1}^{−1}k(1)_{∗}
k(1/j)_{∗} = k(1)_{∗}/(v_{1}^{j})
C_{k} = Z/p[v_{3}, …, v_{k+1}].

The integers a(n) and c_{m} for each n ≥ 0 are defined by

e_{m} = (p^{3m} − 1)/(p^{3} − 1)
a(0) = 1
a(3m + 1) = p^{3m+1} + p(p^{2} + 1)e(m)
a(3m + 2) = pa(3m + 1) + p
a(3m + 3) = pa(3m + 2)
c_{m} = p(p − 1)e_{m}

for m ≥ 0. Let β_{n/j}, ξ, ξ′ and ξ_{i,m} for i = 1, 2, 3 and m ≥ 0 denote elements with dimension given by

||β_{n/j}|| = 2n(p^{k+2} − 1) − 2j(p − 1) − 1, ||ξ|| = 2(p^{k+3} − 2p + 1) − 1, ||ξ′|| = ||ξ|| − 2(p − 1)(p^{3} + 2p^{2}) − 1, ||ξ_{1,m}|| = 2c_{m}(p^{k+2} − 1) + 2(p^{k+1} − 1) − 1,

||ξ_{2,m}|| = 2(pc_{m} + 1)(p^{k+2} − 1) − 1, ||ξ_{3,m}|| = 2p(pc_{m} + 1)(p^{k+2} − 1) − 1.

We abbreviate ξ_{i,0} = ξ_{i} for i = 1, 2, 3. We also write

ξ_{1,m}^{∗} = ξ_{2,m}ξ_{3,m}, ξ_{2,m}^{∗} = ξ_{m+1}ξ_{3,m} and ξ_{3,m}^{∗} = ξ_{1,m}ξ_{2,m}.

Consider k(1)_{∗}-modules

B(n) = ( ∑_{s∈N−pN} k(1/a(n))_{∗}⟨β_{sp^{n}/a(n)}⟩) ⨂C_{k}

and we define the k(1)_{∗}-modules

B̃_{0} = B(0)⨂Λ(ξ_{1}, ξ_{2}, ξ_{3})

\overline{B}^{0} = ∑_{n>0} B(n),
\overline{B}^{1} = \overline{B}(1)\{ξ_{2}, ξ_{3}\}⨁ ∑_{m≥0} (\overline{B}(3m + 2)\{v_{k+2}^{p^{3m+1}(p−1)}ξ_{1,m}, ξ_{3,m}\}

⨁\overline{B}(3m + 3)\{ξ_{1,m+1}, v_{k+2}^{p^{3m+1}(p−1)}ξ_{2,m}\}⨁\overline{B}(3m + 4)\{ξ_{2,m+1}, v_{k+2}^{p^{3m+3}(p−1)}ξ_{3,m}\})

\overline{B}^{2} = \overline{B}(1)ξ_{1}^{∗}⨁\overline{B}(2)ξ_{2}^{∗}⨁ ∑_{m≥0} (\overline{B}(3m + 3)v_{k+2}^{p^{3m+1}(p^{2}−1)}ξ_{3,m}^{∗}
⨁\overline{B}(3m + 4)v_{k+2}^{p^{3m+2}(p^{2}−1)}ξ_{1,m}^{∗}⨁\overline{B}(3m + 5)v_{k+2}^{p^{3m+3}(p^{2}−1)}ξ_{2,m}^{∗})
\overline{B}_{3}^{0} = ∑_{n=1}^{3} \overline{B}(n)
\overline{B}_{3}^{1} = \overline{B}(1)\{ξ_{2}, ξ_{3}\}⨁\overline{B}(2)\{v_{k+2}^{p(p−1)}ξ_{1}, ξ_{3}\}⨁\overline{B}(3)\{ξ_{1,1}, v_{k+2}^{p(p−1)}ξ_{2}\}
\overline{B}_{3}^{2} = \overline{B}(1)ξ_{1}^{∗}⨁\overline{B}(2)ξ_{2}^{∗}⨁\overline{B}(3)v_{k+2}^{p(p^{2}−1)}ξ_{3}^{∗}
\overline{B}_{>3}^{∗} = K(2)_{∗}⨂k(1/p^{3} + 2p^{2})_{∗}[v_{3}^{p^{4}}]

Then we have our main theorem:

Theorem 1.1. Let p be an odd prime. If k is an integer > 1, then the homotopy groups π_{∗}(L_{2}(T (k) ∧ V (0))) are isomorphic as a k(1)_{∗}-module to the tensor product of C_{k} and the direct sum of k(1)_{∗}, ξ_{1}K(1)_{∗}, ξ′K(1)_{∗}/k(1)_{∗} and

B̃_{0}⨁(\overline{B}^{0}⨁\overline{B}^{1}⨁\overline{B}^{2})⨂Λ(ξ).

The homotopy groups π_{∗}(L_{2}(T (1) ∧ V (0)) are isomorphic to the tensor product of C_{k} and the direct sum of k(1)_{∗}, ξ_{1}K(1)_{∗} and

\begin{matrix}B̃_{0}⨁(\overline{B}^{0}⨁\overline{B}^{1}⨁\overline{B}^{2}) \\ ⨁ (ξ_{0}(\overline{B}_{3}^{0}⨁\overline{B}_{3}^{1}⨁\overline{B}_{3}^{2})⨁ξ′\overline{B}_{>3}^{∗}⨂Λ(ξ_{1}, ξ_{2})) . \end{matrix}

The proof is given in the next section, where we also determine H(k)^{∗}M_{1}^{1} assuming two key lemmas Lemmas 2.3 and 2.4, which are proved in sections 4 and 3, respectively.

The authors express their gratitude to Prof. Xiangjun Wang who pointed out that our original proof works even at the prime 2.

2. H(k)^{∗}M_{1}^{1}

Throughout this note, we fix an integer k ≥ 1 and k > 1 if p = 2. We consider the Hopf algebroid (A, Γ(k)) = (BP_{∗}, BP_{∗}[t_{k+1}, t_{k+2}, …]) whose structure maps, say, η_{R} : A → Γ(k) and Δ: Γ(k) → Γ(k)⨂Γ(k), inherit from those of the Hopf algebroid (BP_{∗}, BP_{∗}(BP)) under the projection BP_{∗}BP → Γ(k). Then H(k)^{∗}M for a comodule M is the cohomology of the cobar complex (Ω^{∗}M, d), where Ω^{s}M = M⨂_{A}Γ(k)⨂_{A} ⋯ ⨂_{A}Γ(k) (s factors of Γ(k)) and the differential d is defined inductively by d(m) = η_{R}(m) − m for m ∈ M, d(x) = 1⊗x+x⊗1−Δ(x) for x ∈ Γ(k) and d(x⊗y) = d(x)⊗y+(−1)^{s}x⊗d(y) for x ∈ Ω^{s}M and y ∈ Γ. Note that the change of rings theorem shows an isomorphism

H(k)^{∗}M = Ext_{BP_{∗}(BP)}^{∗}(BP_{∗}, M⨂_{BP_{∗}}BP_{∗}[t_{1}, …, t_{k}]).

The comodules M_{j}^{i} are defined inductively by N_{j}^{0} = A/I_{j}, M_{j}^{i} = v_{i+j}^{−1}N_{j}^{i} and the short exact sequence

(2.1)  0 ⟶ N_{j}^{i} ⟶ M_{j}^{i} ⟶ N_{j}^{i+1} ⟶ 0.

We start with Ravenel’s results (cf. [3]):

H(k)^{∗}M_{2}^{0} = K(2)_{∗}[v_{3}, …, v_{k+2}]⨂Λ(h_{k+1,0}, h_{k+1,1}, h_{k+2,0}, h_{k+2,1})
= K(2)_{∗}[v_{k+2}]⨂Λ(h_{k+1,0}, h_{k+1,1}, h_{k+2,0})⨂C_{k},

where K(2)_{∗} = (Z/p)[v_{2}, v_{2}^{−1}], C_{k} denotes the trivial BP_{∗}(BP)-comodule (Z/p)[v_{3}, …, v_{k+1}] and h_{i,j} denotes the cohomology class represented by t_{i}^{p^{j}}.

We have the exact sequence

(2.2)  H(k)^{∗}M_{2}^{0} ⟶^{φ} H(k)^{∗}M_{1}^{1} ⟶^{v_{1}} H(k)^{∗}M_{1}^{1} ⟶^{δ} H(k)^{∗}M_{2}^{0}

of graded modules associated to the short exact sequence (2.1), where φ(x) = x/v_{1} and δ raises the dimension 1.

Now we state our key lemmas.

Lemma 2.3. If k > 1, then there exists a cocycle r_{0} of Ω^{1}v_{2}^{−1}BP_{∗}/(p, v_{1}^{j}) for any j > 0 whose leading term is v_{k+2}^{p−1}t_{k+1}^{p}. For the case k = 1 and p > 2, the corressponding cochain r_{0} satisfies the congruence d(r_{0}) ≡ v_{1}^{p^{3}+2p^{2}}v_{2}^{p^{4}−2p^{2}−p}r_{0} ⊗ t_{k+2}^{p} mod (p, v_{1}^{2p^{3}}).

We write ζ as the homology class of r_{0}.

Lemma 2.4. For the connecting homomorphism δ, we have

δ(v_{k+2}^{s}/v_{1}) = sv_{k+2}^{s−p}ζ
δ(v_{k+2}^{sp^{n}}/v_{1}^{a(n)}) = sv_{2}^{b_{n}}v_{k+2}^{p^{n}(s−1)}ζ_{i,m}

for s > 0 and n = 3m + i > 0 with i = 1, 2, 3. Here the integers b_{n} is define by

b_{3m+i} = p^{3m+i−1}(p^{k+1} − 1) + (p^{k+1} − 1)e(m)

for i = 1, 2, 3. Here,

ζ_{1,m} = [v_{k+2}^{c_{m}}t_{k+1}], ζ_{2,m} = [v_{k+2}^{pc_{m}}t_{k+2}], ζ_{3,m} = [v_{k+2}^{p^{2}c_{m}}t_{k+2}^{p}].

and we abbreviate ζ_{i,0} to ζ_{i} for i = 1, 2, 3.

These are shown in the next section.

Consider the modules

A(n) = K(2)_{∗}{v_{k+2}^{sp^{n}} | p |̸ s}

for n ≥ 0, and put A^{∗} = ∑_{n≥1} A(n)⨂Λ(h_{k+1,0}, h_{k+2,0}, h_{k+2,1})⨂C_{k} ⊂ H(k)^{∗}M_{2}^{0}. Suppose that there exists a graded module \overline{A}^{∗} fitting in the commutative diagram

A^{∗}  \overline{A}^{∗}  \overline{A}^{∗}  A^{∗} H(k)^{∗}M_{2}^{0}  H(k)^{∗}M_{1}^{1}  H(k)^{∗}M_{1}^{1}  H(k)^{∗}M_{2}^{0}^{φ}  w z u  ^{v_{1}}  w z u  ^{δ}  w z u  z u_{^{φ} w}  _{^{v_{1}} w}  _{^{δ} w}

with exact rows.

For the case k = 1, we consider also A_{3}^{∗} = ∑_{n=1}^{3} A(n)⨂Λ(h_{2,0}, h_{3,0}, h_{3,1}), and further suppose that there exists a graded module \overline{A}_{3}^{∗} fitting in the commutative diagram

A_{3}^{∗}  \overline{A}_{3}^{∗}  \overline{A}_{3}^{∗}  A^{∗} H(k)^{∗}M_{2}^{0}  H(k)^{∗}M_{1}^{1}  H(k)^{∗}M_{1}^{1}  H(k)^{∗}M_{2}^{0}^{φ}  w z u  ^{v_{1}}  w z u  ^{δ}  w z u  z u_{^{φ} w}  _{^{v_{1}} w}  _{^{δ} w}

with exact rows.

Then we have isomorphisms.

Lemma 2.5. For the comodules given above,

H(k)^{∗}M_{1}^{1} = (\overline{A(0)}⨂Λ(ζ_{1}, ζ_{2}, ζ_{3})⨂C_{k}) ⨁ (\overline{A}^{∗}⨂Λ(ζ))

for k > 1 and

H(k)^{∗}M_{1}^{1} = (\overline{A(0)}⨂Λ(ζ_{1}, ζ_{2}, ζ_{3})⨂C_{k}) ⨁\overline{A}^{∗}⨁ (ζ\overline{A}_{3}^{∗}⨁ζ\overline{A}_{>3}^{∗}⨂Λ(ζ_{1}, ζ_{2})) .

Here \overline{A(0)} = K(2)_{∗}\{v_{k+2}^{s}/v_{1} | p |̸ s\} and \overline{A}_{>3}^{∗} = k(1/p^{3}+2p^{2})_{∗}[v_{3}^{p^{4}}]\{1/v_{1}^{p^{3}+2p^{2}})⨂K(2)_{∗}.

Proof. First we show the case for k > 1. Put B^{∗} for the right hand side of the equation of the lemma. Then it suffices to show that the sequence

(2.6)  H(k)^{∗}M_{2}^{0} ⟶^{φ} B^{∗} ⟶^{v_{1}} B^{∗} ⟶^{δ} H(k)^{∗}M_{2}^{0} ⟶^{φ} B^{∗}

is exact by [1, Remark 3.11]. Since we see that δ(v_{k+2}^{s}/v_{1}) = sv_{k+2}^{s−p}ζ by Lemma 2.4, we have an exact sequence

0 ⟶ \overline{A(0)}⨂Λ(ζ_{1}, ζ_{2}, ζ_{3}) ⟶ v_{k+2}^{1−p}ζA(0)⨂Λ(ζ_{1}, ζ_{2}, ζ_{3}) ⟶ 0.

By assumption, we have an exact sequence

(2.7)  A^{∗}⨂Λ(ζ) ⟶ \overline{A}^{∗}⨂Λ(ζ) ⟶ \overline{A}^{∗}⨂Λ(ζ) ⟶^{δ} A^{∗}⨂Λ(ζ).

The direct sum of these exact sequences yields the desired one.

For k = 1, it is shown in the same way. The sequence (2.6) is also exact in this case, while the sequence (2.7) is not exact since r_{0} is not a cocycle by Lemma 2.3. Lemma 2.3 also shows that r_{0} is a cocycle mod (p, v_{1}^{p^{3}+p^{2}}), which implies that the sequence

ζA_{3}^{∗} ⟶ ζ\overline{A}_{3}^{∗} ⟶ ζ\overline{A}_{3}^{∗} ⟶^{δ} ζA_{3}^{∗}

is exact by the assumption on \overline{A}_{3}^{∗}. Furthermore δ(ζ/v_{1}^{p^{3}+2p^{2}}) = v_{2}^{p^{4}−2p^{2}−p}ζζ_{3} by Lemma 2.3, and we have the exact sequence

ζK(2)_{∗}[v_{3}^{p^{4}}]⨂Λ(ζ_{3}) ⟶^{φ} ζ\overline{A}_{>3}^{∗} ⟶^{v_{1}} ζ\overline{A}_{>3}^{∗} ⟶^{δ} ζK(2)_{∗}[v_{3}^{p^{4}}]⨂Λ(ζ_{3}).

q.e.d.

Consider the k(1)_{∗}-modules defined by

\overline{A}(n) = k(1)_{∗}/(v_{1}^{a(n)})⟨v_{k+2}^{sp^{n}}/v_{1}^{a(n)}⟩
\overline{A}^{0} = ∑_{n>0} \overline{A}(n)
\overline{A}^{1} = \overline{A}(1)\{ζ_{2}, ζ_{3}\}⨁ ∑_{m≥0} (\overline{A}(3m + 2)\{v_{k+2}^{p^{3m+1}(p−1)}ζ_{1,m}, ζ_{3,m}\}

⨁\overline{A}(3m + 3)\{ζ_{1,m+1}, v_{k+2}^{p^{3m+1}(p−1)}ζ_{2,m}\}⨁\overline{A}(3m + 4)\{ζ_{2,m+1}, v_{k+2}^{p^{3m+3}(p−1)}ζ_{3,m}\})

\overline{A}^{2} = \overline{A}(1)ζ_{1}^{∗}⨁\overline{A}(2)ζ_{2}^{∗}⨁ ∑_{m≥0} (\overline{A}(3m + 3)v_{k+2}^{p^{3m+1}(p^{2}−1)}ζ_{3,m}^{∗}
⨁\overline{A}(3m + 4)v_{k+2}^{p^{3m+2}(p^{2}−1)}ζ_{1,m}^{∗}⨁\overline{A}(3m + 5)v_{k+2}^{p^{3m+3}(p^{2}−1)}ζ_{2,m}^{∗})
\overline{A}^{i} = 0 if i > 2,
\overline{A}_{3}^{0} = ∑_{n=1}^{3} \overline{A}(n)
\overline{A}_{3}^{1} = \overline{A}(1)\{ζ_{2}, ζ_{3}\}⨁\overline{A}(2)\{v_{k+2}^{p(p−1)}ζ_{1}, ζ_{3}\}⨁\overline{A}(3)\{ζ_{1,1}, v_{k+2}^{p(p−1)}ζ_{2}\}
\overline{A}_{3}^{2} = \overline{A}(1)ζ_{1}^{∗}⨁\overline{A}(2)ζ_{2}^{∗}⨁\overline{A}(3)v_{k+2}^{p(p^{2}−1)}ζ_{3}^{∗}
\overline{A}_{3}^{i} = 0 if i > 2.

Here v_{k+2}^{sp^{n}}/v_{1}^{a(n)} denotes the homology class represented by a cocycle whose leading term is v_{k+2}^{sp^{n}}/v_{1}^{a(n)} in the cobar complex Ω^{0}M_{1}^{1}, and ζ_{i,m}^{∗} denotes as follows:

ζ_{1,m}^{∗} = ζ_{2,m}ζ_{3,m}, ζ_{2,m}^{∗} = ζ_{1,m+1}ζ_{3,m} ζ_{3,m}^{∗} = ζ_{1,m}ζ_{2,m}.

Put \overline{A}^{∗} = ∑_{i≥0} \overline{A}^{i} and \overline{A}_{3}^{∗} = ∑_{i≥0} \overline{A}_{3}^{i}. Then we obtain the following

Proposition 2.8. The modules \overline{A}^{∗} and \overline{A}_{3}^{∗} fit in the commutative diagram of exact sequence given above Lemma 2.5.

Proof of Theorem 1.1. By (2.1), we have the short exact sequence 0 → Ñ_{1}^{0} → M̃_{1}^{0} → M̃_{1}^{1} → 0, where we write M̃ as M⨂_{BP_{∗}}E(2)_{∗} for a BP_{∗}(BP)- comodule M. Here E(2)_{∗} = Z_{(p)}[v_{1}, v_{2}^{±1}]. Ravenel shows that H(k)^{∗}M̃_{1}^{0} = K(1)_{∗}[v_{2}, …, v_{k+1}]⨂Λ(h_{k+1,0}). The long exact sequence associated to the short sequence splits into the exact sequence

\begin{matrix}0 → H(k)^{0}Ñ_{1}^{0} ⟶ H(k)^{0}M̃_{1}^{0} ⟶ H(k)^{0}M̃_{1}^{1} ⟶^{δ} H(k)^{1}Ñ_{1}^{0} \\ ⟶ H(k)^{1}M̃_{1}^{0} ⟶ H(k)^{1}M̃_{1}^{1} ⟶^{δ} H(k)^{2}Ñ_{1}^{0} ⟶ 0 \end{matrix}

and the isomorphisms

δ: H(k)^{i}M̃_{1}^{1} = H(k)^{i+1}Ñ_{1}^{0}

for i > 1. Put β_{i/j} = δ(v_{k+2}^{i}/v_{1}^{j}), β_{i/j}ξ_{i,m} = δ(v_{k+2}^{i}ζ_{i,m}/v_{1}^{j}), β_{i/j}v_{k+2}^{a}ξ_{i,m} = δ(v_{k+2}^{i+a}ζ_{i,m}/v_{1}^{j}), β_{i/j}ξ_{i,m}^{∗} = δ(v_{k+2}^{i}ζ_{i,m}^{∗}/v_{1}^{j}) and β_{i/j}v_{k+2}^{a}ξ_{i,m}^{∗} = δ(v_{k+2}^{i+a}ζ_{i,m}^{∗}/v_{1}^{j}). Then δ(\overline{A}(n)) = \overline{B}(n). Furthermore, we put δ(ζ/v_{1}^{p^{3}+2p^{2}}) = ξ′ if k = 1. Thus we obtain the E_{2}-term H(k)^{∗}Ñ_{1}^{0} of the Adams-Novikov spectral sequence.

Since H^{s}Ñ_{1}^{0} = 0 for s > 4, we see that the Adams-Novikov spectral sequence collapses from the E_{2}-term. q.e.d.

Rest of this section is devoted to showing Proposition 2.8. First we note a property of the non-negative integers.

Remark. For any n ≥ 0,

1. There exist integers l, m with l ≢ p^{2} − p mod (p^{3}) such that

n = p^{3m}l + c_{m}.

2. For any n ≥ 0, there exist integers l, m with l ≢ p^{2} − 1 mod (p^{3}) such

that

np = p^{3m+1}l + (p + 1)c_{m}.

Consider the modules

A_{2}(m) = A(3m + 1)⨁A(3m + 2)v_{k+2}^{−p^{3m+2}}⨁A(3m + 3)v_{k+2}^{p^{3m+2}(p−1)}
A_{3}(m) = A(3m + 2)⨁A(3m + 3)v_{k+2}^{−p^{3m+3}}⨁A(3m + 4)v_{k+2}^{p^{3m+3}(p−1)}
A_{1}(m) = A(3m + 3)⨁A(3m + 4)v_{k+2}^{−p^{3m+4}}⨁A(3m + 5)v_{k+2}^{p^{3m+4}(p−1)}
A_{3}(m)^{∗} = A(3m + 1)v_{k+2}^{−p^{3m+1}}⨁A(3m + 2)v_{k+2}^{−p^{3m+1}}⨁A(3m + 3)v_{k+2}^{p^{3m+1}(p^{2}−1)}
A_{1}(m)^{∗} = A(3m + 2)v_{k+2}^{−p^{3m+2}}⨁A(3m + 3)v_{k+2}^{−p^{3m+2}}⨁A(3m + 4)v_{k+2}^{p^{3m+2}(p^{2}−1)}
A_{2}(m)^{∗} = A(3m + 3)v_{k+2}^{−p^{3m+3}}⨁A(3m + 4)v_{k+2}^{−p^{3m+3}}⨁A(3m + 5)v_{k+2}^{p^{3m+3}(p^{2}−1)}

Then we decompose Z/p[v_{k+2}^{p}] into a direct sum.

Z/p[v_{k+2}^{p}] ≅ ∑_{m≥0} A_{2}(m)v_{k+2}^{pc_{m}}
≅ A(1)⨁ ∑_{m≥0} A_{3}(m)v_{k+2}^{p^{2}c_{m}}
≅ A(1)v_{k+2}^{−p}⨁A(2)v_{k+2}^{p(p−1)}⨁ ∑_{m≥0} A_{1}(m)v_{k+2}^{c_{m}}
≅ ∑_{m≥0} A_{3}(m)^{∗}v_{k+2}^{(p+1)c_{m}}
≅ A(1)⨁ ∑_{m≥0} A_{1}(m)^{∗}v_{k+2}^{(p^{2}+p)c_{m}}
≅ A(1)v_{k+2}^{−p}⨁A(2)v_{k+2}^{p^{2}−p}⨁ ∑_{m≥0} A_{2}(m)^{∗}v_{k+2}^{p^{2}c_{m}+c_{m+1}}.

This follows immediately from the above remark. In fact, we notice that

A_{i}(m) = {v_{k+2}^{p^{3m−1+i}l} | l ≢ p^{2} − p mod (p^{3})} for i = 2, 3
A_{1}(m) = {v_{k+2}^{p^{3m+3}l} | l ≢ p^{2} − p mod (p^{3})}
A_{i}^{∗}(m) = {v_{k+2}^{p^{3m+1+i}l} | l ≢ p^{2} − 1 mod (p^{3})} for i = 1, 2
A_{3}^{∗}(m) = {v_{k+2}^{p^{3m+1}l} | l ≢ p^{2} − 1 mod (p^{3})}.

We now restate the action of the connecting homomorphism δ: H^{∗}M_{2}^{0} → H^{∗+1}M_{1}^{1} given in Lemma 2.4.

Lemma 2.9. For m ≥ 0, The connecting homomorphism δ behaves δ(\overline{A}(n)) = A(n)v_{k+2}^{−p^{n}}ζ_{i,m} for n = 3m + i with i = 1, 2, 3.:

This implies immediately

Corollary 2.10. For m ≥ 0, the connecting homomorphism δ acts as follows:

δ(\overline{A}(1)ζ_{3}) = A(1)v_{k+2}^{−p}ζ_{2}^{∗}
δ(\overline{A}(3m)ζ_{1,m}) = A(3m)v_{k+2}^{−p^{3m}}ζ_{2,m−1}^{∗}
δ(\overline{A}(3m + 2)v_{k+2}^{p^{3m+1}(p−1)}ζ_{1,m}) = A(3m + 2)v_{k+2}^{−p^{3m+1}}ζ_{3,m}^{∗}
δ(\overline{A}(3m + 1)ζ_{2,m}) = A(3m + 1)v_{k+2}^{−p^{3m+1}}ζ_{3,m}^{∗}
δ(\overline{A}(3m + 3)v_{k+2}^{p^{3m+2}(p−1)}ζ_{2,m}) = A(3m + 3)v_{k+2}^{−p^{3m+2}}ζ_{1,m}^{∗}
δ(\overline{A}(3m + 2)ζ_{3,m}) = A(3m + 2)v_{k+2}^{−p^{3m+2}}ζ_{1,m}^{∗}
δ(\overline{A}(3m + 4)v_{k+2}^{p^{3m+3}(p−1)}ζ_{3,m}) = A(3m + 4)v_{k+2}^{−p^{3m+4}}ζ_{2,m}^{∗}
δ(\overline{A}(1)ζ_{1}^{∗}) = A(1)1^{∗}
δ(\overline{A}(2)ζ_{2}^{∗}) = A(2)1^{∗}
δ(\overline{A}(3m + 3)v_{k+2}^{p^{3m+1}(p^{2}−1)}ζ_{3,m}^{∗}) = A(3m + 3)1^{∗}
δ(\overline{A}(3m + 4)v_{k+2}^{p^{3m+2}(p^{2}−1)}ζ_{1,m}^{∗}) = A(3m + 4)1^{∗}
δ(\overline{A}(3m + 5)v_{k+2}^{p^{3m+3}(p^{2}−1)}ζ_{2,m}^{∗}) = A(3m + 5)1^{∗},

where 1^{∗} = v_{k+2}^{−p}ζ_{1}ζ_{2}ζ_{3}.

The proof of Proposition 2.8 is now an easy check. This is displayed as a following pattern.

[figure: 3; 2; 1; 3; 2; 1; 3; 2; 1; 3; 2; 1; ζ_{3}; ζ_{2}; ζ_{1}; 1; 2; 3; 1; 2; 3; 1; 2; 3; ζ_{1}^{∗}; ζ_{2}^{∗}; ζ_{3}^{∗}; 1; 2; 3]

For example, the picture on the left upper corner displays the diagram

[figure: A(3m + 1); A(3m + 1)v_{k+2}^{−p^{3m+1}}ζ_{1,m}; A(3m + 2)v_{k+2}^{p^{3m+1}(p−1)}ζ_{1,m}; A(3m + 2)v_{k+2}^{−p^{3m+1}}ζ_{3,m}^{∗}; A(3m)ζ_{1,m}; A(3m)v_{k+2}^{−p^{3m}}ζ_{2,m−1}^{∗}.]

3. The Miller-Ravenel-Wilson elements x_{i}

In this section we prove Lemma 2.4.

Now we compute the action of η_{R} : BP_{∗} → Γ(k) = BP_{∗}[t_{k+1}, t_{k+2}, …] by using Hazewinkel’s and Quillen’s formulae.

v_{n} = pm_{n} − ∑_{i=1}^{n−1} m_{i}v_{n−i}^{p^{i}} η_{R}(m_{n}) = ∑_{i+j=n} m_{i}t_{j}^{p^{i}}.

Here BP_{∗}⨂Q = Q[m_{1}, m_{2}, …]. Since t_{i} = 0 for 0 < i ≤ k, Quillen’s formula turns into

η_{R}(m_{n}) = \{ \begin{matrix}m_{n} & for n ≤ k \\ ∑_{i=k+1}^{n} m_{n−i}t_{i}^{p^{n−i}} & for n > k. \end{matrix}

To describe η_{R}(v_{n}) we introduce an element C_{i}(n) given by

(3.1)  pC_{i}(n) = η_{R}(v_{n}^{p^{i}}) − ∑_{l} w_{l}^{p^{i}}

if we have η_{R}(v_{n}) ≡ ∑_{l} w_{l}^{p^{i}} mod (p) for some monomials w_{l} of Γ(k).

Lemma 3.2. The right unit η_{R}: BP_{∗} → Γ(k) acts on v_{n} for 0 < n ≤ k + 4 as follows:

η_{R}(v_{n}) = v_{n}  for n ≤ k
η_{R}(v_{k+1}) ≡ v_{k+1} mod (p)
η_{R}(v_{k+2}) ≡ v_{k+2} + v_{1}t_{k+1}^{p} − v_{1}^{p^{k+1}}t_{k+1} mod (p)
η_{R}(v_{k+3}) ≡ v_{k+3} + v_{2}t_{k+1}^{p^{2}} + v_{1}t_{k+2}^{p} − v_{1}C_{1}(k + 2) − v_{2}^{p^{k+1}}t_{k+1} − v_{1}^{p^{k+2}}t_{k+2} mod (p)
η_{R}(v_{k+4}) ≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{2}C_{2}(k + 2) − v_{1}C_{1}(k + 3)
−v_{3}^{p^{k+1}}t_{k+1} − v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} mod (p)  for k > 1
≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{2}C_{2}(k + 2) − v_{1}C_{1}(k + 3)
−v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} − t_{k+1}(v_{k+2}^{p^{2}} + v_{1}^{p^{2}}t_{k+1}^{p^{3}} − v_{1}^{p^{k+3}}t_{k+1}^{p^{2}}) mod (p)
for k = 1
≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{2}C_{2}(k + 2) − v_{1}C_{1}(k + 3)
−v_{3}^{p^{k+1}}t_{k+1} − v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} mod (p, v_{1}^{p^{2}})

Proof. For v_{n} with n ≤ k, it follows immediately from the formulae.

We compute η_{R}(v_{k+1}) = η_{R}(pm_{k+1} − ∑_{i=1}^{k} m_{i}v_{1+k−i}^{p^{i}}) = p(m_{k+1} + t_{k+1}) − ∑_{i=1}^{k} m_{i}v_{n+k−i}^{p^{i}} = v_{k+1} + pt_{k+1}.

For v_{k+2}, we compute

η_{R}(v_{k+2}) = η_{R}(pm_{k+2} − m_{1}v_{k+1}^{p} − ∑_{i=2}^{k} m_{i}v_{2+k−i}^{p^{i}} − m_{k+1}v_{1}^{p^{k+1}})
= p(m_{k+2} + m_{1}t_{k+1}^{p} + t_{k+2}) − m_{1}(v_{k+1}^{p} + p^{2}C_{k+1})

− ∑_{i=2}^{k} m_{i}v_{2+k−i}^{p^{i}} − (m_{k+1} + t_{k+1})v_{1}^{p^{k+1}}

= v_{k+2} + v_{1}t_{k+1}^{p} + pt_{k+2} − pv_{1}C_{k+1} − v_{1}^{p^{k+1}}t_{k+1},

where C_{k+1} is defined by η_{R}(v_{k+1}^{p}) = (v_{k+1} + pt_{k+1})^{p} = v_{k+1}^{p} + p^{2}C_{k+1}. If k > 1, then

η_{R}(v_{k+3}) = η_{R}(pm_{k+3} − (m_{1}v_{k+2}^{p} + m_{2}v_{k+1}^{p^{2}}) − ∑_{i=3}^{k} m_{i}v_{3+k−i}^{p^{i}}

−(m_{k+1}v_{2}^{p^{k+1}} + m_{k+2}v_{1}^{p^{k+2}}))

= p(\underline{m_{k+3}}_{1} + \underline{m_{2}t_{k+1}^{p^{2}}}_{2} + m_{1}t_{k+2}^{p} + t_{k+3})

−m_{1}(\underline{v_{k+2}^{p}}_{1} + \underline{v_{1}^{p}t_{k+1}^{p^{2}}}_{2} − \underline{v_{1}^{p^{k+2}}t_{k+1}^{p}}_{3} + pC_{1}(k + 2))

−m_{2}(\underline{v_{k+1}^{p^{2}}}_{1} + p^{3}C_{k+2}) − \underline{∑_{i=3}^{k} m_{i}v_{3+k−i}^{p^{i}}}_{1}

−(\underline{m_{k+1}}_{1} + t_{k+1})v_{2}^{p^{k+1}} − (\underline{m_{k+2}}_{1} + \underline{m_{1}t_{k+1}^{p}}_{3} + t_{k+2})v_{1}^{p^{k+2}}

= v_{k+3} + v_{2}t_{k+1}^{p^{2}} + v_{1}t_{k+2}^{p} + pt_{k+3} − v_{1}C_{1}(k + 2)
−p^{3}m_{2}C_{k+2} − v_{2}^{p^{k+1}}t_{k+1} − v_{1}^{p^{k+2}}t_{k+2},

The underlined terms numbered 1, 2 and 3 sum up to v_{k+3}, v_{2}t_{k+1}^{p^{2}} and 0, respectively. Here note that p^{i+1}m_{i} ≡ 0 mod (p). If k = 1, then k + 1 = 2, and so

η_{R}(v_{k+3}) = η_{R}(pm_{k+3} − m_{1}v_{k+2}^{p} − m_{k+1}v_{k+1}^{p^{k+1}} − m_{k+2}v_{1}^{p^{k+2}})
= p(\underline{m_{k+3}}_{1} + \underline{m_{2}t_{k+1}^{p^{2}}}_{2} + m_{1}t_{k+2}^{p} + t_{k+3})
−m_{1}(\underline{v_{k+2}^{p}}_{1} + \underline{v_{1}^{p}t_{k+1}^{p^{2}}}_{2} − \underline{v_{1}^{p^{k+2}}t_{k+1}^{p}}_{3} + pC_{1}(k + 2))
−(\underline{m_{k+1}}_{1} + t_{k+1})(v_{k+1}^{p^{2}} + p^{3}C_{k+2}) − (\underline{m_{k+2}}_{1} + \underline{m_{1}t_{k+1}^{p}}_{3} + t_{k+2})v_{1}^{p^{k+2}}
≡ v_{k+3} + v_{2}t_{k+1}^{p^{2}} + v_{1}t_{k+2}^{p} − v_{1}C_{1}(k + 2) − v_{2}^{p^{k+1}}t_{k+1} − v_{1}^{p^{k+2}}t_{k+2} mod (p),

and we see there is no difference between the cases k > 1 and k = 1. In the same way, assume k > 2, and

η_{R}(v_{k+4})

= η_{R}(pm_{k+4} − (m_{1}v_{k+3}^{p} + m_{2}v_{k+2}^{p^{2}} + m_{3}v_{k+1}^{p^{3}}) − ∑_{i=4}^{k} m_{i}v_{4+k−i}^{p^{i}}
−(m_{k+1}v_{3}^{p^{k+1}} + m_{k+2}v_{2}^{p^{k+2}} + m_{k+3}v_{1}^{p^{k+3}}))
= p(\underline{m_{k+4}}_{1} + \underline{m_{3}t_{k+1}^{p^{3}}}_{2} + \underline{m_{2}t_{k+2}^{p^{2}}}_{3} + m_{1}t_{k+3}^{p} + t_{k+4})

−m_{1}(\underline{v_{k+3}^{p}}_{1} + \underline{v_{2}^{p}t_{k+1}^{p^{3}}}_{2} + \underline{v_{1}^{p}t_{k+2}^{p^{2}}}_{3} − \underline{v_{1}^{p}C_{2}(k + 2)}_{7} − \underline{v_{2}^{p^{k+2}}t_{k+1}^{p}}_{4}

−\underline{v_{1}^{p^{k+3}}t_{k+2}^{p}}_{6} + pC_{1}(k + 3))

−m_{2}(\underline{v_{k+2}^{p^{2}}}_{1} + \underline{v_{1}^{p^{2}}t_{k+1}^{p^{3}}}_{2} − \underline{v_{1}^{p^{k+3}}t_{k+1}^{p^{2}}}_{5} + \underline{pC_{2}(k + 2))}_{7} − m_{3}(\underline{v_{k+1}^{p^{3}}}_{1} + p^{4}C_{k+3})

−\underline{∑_{i=4}^{k} m_{i}v_{4+k−i}^{p^{i}}}_{1} − (\underline{m_{k+1}}_{1} + t_{k+1})v_{3}^{p^{k+1}} − (\underline{m_{k+2}}_{1} + \underline{m_{1}t_{k+1}^{p}}_{4} + t_{k+2})v_{2}^{p^{k+2}}

−(\underline{m_{k+3}}_{1} + \underline{m_{2}t_{k+1}^{p^{2}}}_{5} + \underline{m_{1}t_{k+2}^{p}}_{6} + t_{k+3})v_{1}^{p^{k+3}}
≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{1}C_{1}(k + 3)
−v_{3}^{p^{k+1}}t_{k+1} − v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} − v_{2}C_{2}(k + 2) mod (p)

If k = 2, then

η_{R}(v_{k+4})

= η_{R}(pm_{k+4} − (m_{1}v_{k+3}^{p} + m_{2}v_{k+2}^{p^{2}}) − m_{k+1}v_{k+1}^{p^{k+1}} − (m_{k+2}v_{2}^{p^{k+2}} + m_{k+3}v_{1}^{p^{k+3}}))

= p(\underline{m_{k+4}}_{1} + \underline{m_{3}t_{k+1}^{p^{3}}}_{2} + \underline{m_{2}t_{k+2}^{p^{2}}}_{3} + m_{1}t_{k+3}^{p} + t_{k+4})

−m_{1}(\underline{v_{k+3}^{p}}_{1} + \underline{v_{2}^{p}t_{k+1}^{p^{3}}}_{2} + \underline{v_{1}^{p}t_{k+2}^{p^{2}}}_{3} − \underline{v_{1}^{p}C_{2}(k + 2)}_{7} − \underline{v_{2}^{p^{k+2}}t_{k+1}^{p}}_{4} − \underline{v_{1}^{p^{k+3}}t_{k+2}^{p}}_{6} + pC_{1}(k + 3)) −m_{2}(\underline{v_{k+2}^{p^{2}}}_{1} + \underline{v_{1}^{p^{2}}t_{k+1}^{p^{3}}}_{2} − \underline{v_{1}^{p^{k+3}}t_{k+1}^{p^{2}}}_{5} + \underline{pC_{2}(k + 2))}_{7} − (\underline{m_{k+1}}_{1} + t_{k+1})(v_{k+1}^{p^{k+1}} + p^{4}C_{k+3})

−(\underline{m_{k+2}}_{1} + \underline{m_{1}t_{k+1}^{p}}_{4} + t_{k+2})v_{2}^{p^{k+2}} − (\underline{m_{k+3}}_{1} + \underline{m_{2}t_{k+1}^{p^{2}}}_{5} + \underline{m_{1}t_{k+2}^{p}}_{6} + t_{k+3})v_{1}^{p^{k+3}}
≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{1}C_{1}(k + 3)
−v_{3}^{p^{k+1}}t_{k+1} − v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} − v_{2}C_{2}(k + 2) mod (p)

Last we compute the case k = 1, η_{R}(v_{k+4})

= η_{R}(pm_{k+4} − m_{1}v_{k+3}^{p} − m_{k+1}v_{k+2}^{p^{k+1}} − m_{k+2}v_{k+1}^{p^{k+2}} − m_{k+3}v_{1}^{p^{k+3}})
= p(\underline{m_{k+4}}_{1} + \underline{m_{3}t_{k+1}^{p^{3}}}_{2} + \underline{m_{2}t_{k+2}^{p^{2}}}_{3} + m_{1}t_{k+3}^{p} + t_{k+4})

−m_{1}(\underline{v_{k+3}^{p}}_{1} + \underline{v_{2}^{p}t_{k+1}^{p^{3}}}_{2} + \underline{v_{1}^{p}t_{k+2}^{p^{2}}}_{3} − \underline{v_{1}^{p}C_{2}(k + 2)}_{7} − \underline{v_{2}^{p^{k+2}}t_{k+1}^{p}}_{4} − \underline{v_{1}^{p^{k+3}}t_{k+2}^{p}}_{6} + pC_{1}(k + 3))

−m_{k+1}(\underline{v_{k+2}^{p^{2}}}_{1} + \underline{v_{1}^{p^{2}}t_{k+1}^{p^{3}}}_{2} − \underline{v_{1}^{p^{k+3}}t_{k+1}^{p^{2}}}_{5} + \underline{pC_{2}(k + 2))}_{7}
−t_{k+1}(v_{k+2}^{p^{2}} + v_{1}^{p^{2}}t_{k+1}^{p^{3}} − v_{1}^{p^{k+3}}t_{k+1}^{p^{2}} + pC_{2}(k + 2))
−(\underline{m_{k+2}}_{1} + \underline{m_{1}t_{k+1}^{p}}_{4} + t_{k+2})(v_{k+1}^{p^{k+2}} + p^{4}C_{k+3})
−(\underline{m_{k+3}}_{1} + \underline{m_{2}t_{k+1}^{p^{2}}}_{5} + \underline{m_{1}t_{k+2}^{p}}_{6} + t_{k+3})v_{1}^{p^{k+3}}
≡ v_{k+4} + v_{3}t_{k+1}^{p^{3}} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{1}C_{1}(k + 3)
−η_{R}(v_{k+2}^{p^{k+1}})t_{k+1} − v_{2}^{p^{k+2}}t_{k+2} − v_{1}^{p^{k+3}}t_{k+3} − v_{2}C_{2}(k + 2) mod (p)

q.e.d.

Lemma 3.3. For the differential d : Ω^{0}v_{2}^{−1}BP_{∗} → Ω^{1}v_{2}^{−1}BP_{∗}, there exist elements x_{0}, x_{1}, w_{4} and x′_{2} such that

d(x_{0}) = v_{1}t_{k+1}^{p} − v_{1}^{p^{k+1}}t_{k+1}
d(x_{1}) = v_{1}^{p}v_{2}^{p^{k+1}−1}t_{k+1} − v_{1}^{p+1}v_{2}^{−1}(t_{k+2}^{p} − C_{1}(k + 2)) mod (p, v_{1}^{p^{k+2}})
d(w_{4}) ≡ v_{1}^{p}(v_{2}t_{k+2}^{p^{2}} − v_{2}^{p^{k+2}}t_{k+2}) mod (p, v_{1}^{p+1})
d(x′_{2}) ≡ −v_{1}^{p^{2}+p}v_{2}^{−p}t_{k+2}^{p^{2}} − v_{1}^{p^{2}+2p}v_{2}^{p^{k+1}−p−1}v_{k+2}^{p^{2}−p}t_{k+1} + v_{1}^{p^{2}+2p+1}v_{2}^{−p−1}v_{k+2}^{p^{2}−p}t_{k+2}^{p}
mod (p, v_{1}^{p^{2}+2p+2}).

Proof. The first one follows immediately from Lemma 3.2 if we set x_{0} = v_{k+2}. Since d(v_{k+2}^{p}) ≡ v_{1}^{p}t_{k+1}^{p^{2}} mod (p, v_{1}^{p^{k+2}}) and d(v_{1}^{p}v_{2}^{−1}v_{k+3}) ≡ v_{1}^{p}v_{2}^{−1}(v_{2}t_{k+1}^{p^{2}} + v_{1}t_{k+2}^{p} − v_{1}C_{1}(k + 2) − v_{2}^{p^{k+1}}t_{k+1} − v_{1}^{p^{k+2}}t_{k+2}), we obtain the second.

Put w_{4} = v_{1}^{p}v_{k+4}−v_{1}^{p}v_{2}^{−p}v_{3}v_{k+3}^{p}−v_{1}^{p−1}v_{2}^{p^{k+2}−p}v_{3}v_{k+2}+v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}}x_{1}. Then, mod (p, v_{1}^{2p}),

d(v_{1}^{p}v_{k+4}) ≡ v_{1}^{p}(\underline{v_{3}t_{k+1}^{p^{3}}}_{1} + v_{2}t_{k+2}^{p^{2}} + v_{1}t_{k+3}^{p} − v_{2}C_{2}(k + 2)
−v_{1}C_{1}(k + 3) − \underline{v_{3}^{p^{k+1}}t_{k+1}}_{3} − v_{2}^{p^{k+2}}t_{k+2})
d(−v_{1}^{p}v_{2}^{−p}v_{3}v_{k+3}^{p}) ≡ −v_{1}^{p}v_{2}^{−p}v_{3}(\underline{v_{2}^{p}t_{k+1}^{p^{3}}}_{1} − \underline{v_{2}^{p^{k+2}}t_{k+1}^{p}}_{2})
d(−v_{1}^{p−1}v_{2}^{p^{k+2}−p}v_{3}v_{k+2}) ≡ −\underline{v_{1}^{p}v_{2}^{p^{k+2}−p}v_{3}t_{k+1}^{p}}_{2}
d(v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}}x_{1}) ≡ v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}}(\underline{v_{1}^{p}v_{2}^{p^{k+1}−1}t_{k+1}}_{3} − v_{1}^{p+1}v_{2}^{−1}(t_{k+2}^{p} − C_{1}(k + 2)))

Here C_{2}(k + 2) ≡ 0 mod (p, v_{1}^{p}). This holds for k > 1. If k = 1, put w_{4} = v_{1}^{p}v_{k+4} − v_{1}^{p}v_{2}^{−p}v_{3}v_{k+3}^{p} − \frac{1}{2}v_{1}^{p−1}v_{2}^{p^{k+2}−p}v_{k+2}^{2} + v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}}x_{1}. Then we compute mod (p, v_{1}^{p+1})

d(v_{1}^{p}v_{k+4}) ≡ v_{1}^{p}(\underline{v_{3}t_{k+1}^{p^{3}}}_{1} + v_{2}t_{k+2}^{p^{2}} − v_{2}C_{2}(k + 2) − \underline{v_{3}^{p^{k+1}}t_{k+1}}_{3} − v_{2}^{p^{k+2}}t_{k+2})
d(−v_{1}^{p}v_{2}^{−p}v_{3}v_{k+3}^{p}) ≡ −v_{1}^{p}v_{2}^{−p}v_{3}(\underline{v_{2}^{p}t_{k+1}^{p^{3}}}_{1} − \underline{v_{2}^{p^{k+2}}t_{k+1}^{p}}_{2})
d(−\frac{1}{2}v_{1}^{p−1}v_{2}^{p^{k+2}−p}v_{k+2}^{2}) ≡ −\underline{v_{1}^{p}v_{2}^{p^{k+2}−p}v_{3}t_{k+1}^{p}}_{2}
d(v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}}x_{1}) ≡ v_{2}^{−p^{k+1}+1}v_{3}^{p^{k+1}} \underline{v_{1}^{p}v_{2}^{p^{k+1}−1}t_{k+1}}_{3}

Here v_{k+2} = v_{3}.

Put x′_{2} = x_{1}^{p} − v_{1}^{p^{2}−1}v_{2}^{p^{k+2}−p}v_{k+2} − v_{1}^{p^{2}+2p}v_{2}^{−p−1}v_{k+2}^{p^{2}−p}v_{k+3}. Then

d(x_{1}^{p}) ≡ v_{1}^{p^{2}}v_{2}^{p^{k+2}−p}t_{k+1}^{p} − v_{1}^{p^{2}+p}v_{2}^{−p}(t_{k+2}^{p^{2}} − C_{1}(k + 2)^{p}) mod (p, v_{1}^{p^{k+3}})
≡ v_{1}^{p^{2}}v_{2}^{p^{k+2}−p}t_{k+1}^{p} − v_{1}^{p^{2}+p}v_{2}^{−p}(t_{k+2}^{p^{2}} − v_{1}^{p}v_{k+2}^{p^{2}−p}t_{k+1}^{p^{2}}) mod (p, v_{1}^{p^{2}+3p})
d(−v_{1}^{p^{2}−1}v_{2}^{p^{k+2}−p}v_{k+2}) ≡ −v_{1}^{p^{2}}v_{2}^{p^{k+2}−p}t_{k+1}^{p}
d(v_{1}^{p^{2}+2p}v_{2}^{−p−1}v_{k+2}^{p^{2}−p}v_{k+3}) ≡ v_{1}^{p^{2}+2p}v_{2}^{−p−1}v_{k+2}^{p^{2}−p}(v_{2}t_{k+1}^{p^{2}} + v_{1}t_{k+2}^{p} − v_{2}^{p^{k+1}}t_{k+1}) mod (p, v_{1}^{p^{2}+2p+2}).

Thus the last congruence follows. q.e.d.

Now we define x_{i}:

x_{3m+1} = x_{3m}^{p} − v_{1}^{a_{m}−2p}v_{2}^{b_{m}−p^{k+1}}v_{k+2}^{c_{m}−p^{2}+p}x′_{2}
x_{3m+2} = x_{3m+1}^{p} − v_{1}^{a_{m}p−1}v_{2}^{b_{m}p}v_{k+2}^{c_{m}p+1} + v_{1}^{a_{m}p}v_{2}^{p(b_{m}−p^{k+1})}v_{k+2}^{pc_{m}}w_{4}
x_{3m+3} = x_{3m+2}^{p},

where integers a_{m}, b_{m} and c_{m} are given by:

a_{m} = p^{3m+1} + (p^{3} + p)e(m)
b_{m} = p^{3m}(p^{k+1} − 1) + (p^{k+1} − 1)e(m)
c_{m} = (p^{2} − p)e(m)

for e(m) = (p^{3m} − 1)/(p^{3} − 1)

Lemma 3.4.

d(x_{3m+1}) ≡ v_{1}^{a_{m}}v_{2}^{b_{m}}v_{k+2}^{c_{m}}t_{k+1} mod (p, v_{1}^{a_{m}+1})
d(x_{3m+2}) ≡ −v_{1}^{pa_{m}+p}v_{2}^{pb_{m}}v_{k+2}^{pc_{m}}t_{k+2} mod (p, v_{1}^{pa_{m}+p+1}).
d(x_{3m+3}) ≡ −v_{1}^{p^{2}a_{m}+p^{2}}v_{2}^{p^{2}b_{m}}v_{k+2}^{p^{2}c_{m}}t_{k+2}^{p} mod (p, v_{1}^{p^{2}a_{m}+p^{2}+p}).

Proof. Suppose that

d(x_{3m+1}) ≡ v_{1}^{a}v_{2}^{b}v_{k+2}^{c}t_{k+1} − v_{1}^{a+1}v_{2}^{b′}v_{k+2}^{c}t_{k+2}^{p} mod (p, v_{1}^{a+2}),

where b′ = b − p^{k+1}. Then noticing that p|c, we compute mod (p, v_{1}^{ap+p+1}),

d(x_{3m+1}^{p}) ≡ v_{1}^{ap}v_{2}^{bp}v_{k+2}^{cp}t_{k+1}^{p} − v_{1}^{ap+p}v_{2}^{pb′}v_{k+2}^{pc}t_{k+2}^{p^{2}}
d(−v_{1}^{ap−1}v_{2}^{bp}v_{k+2}^{cp+1}) ≡ −v_{1}^{ap}v_{2}^{bp}v_{k+2}^{cp}t_{k+1}^{p}
d(v_{1}^{ap}v_{2}^{pb′}v_{k+2}^{pc}w_{4}) ≡ v_{1}^{ap+p}v_{2}^{pb′}v_{k+2}^{pc}(v_{2}t_{k+2}^{p^{2}} − v_{2}^{p^{k+2}}t_{k+2})

and we obtain

d(x_{3m+2}) ≡ −v_{1}^{ap+p}v_{2}^{pb}v_{k+2}^{pc}t_{k+2} mod (p, v_{1}^{ap+p+1}),

and so

d(x_{3m+3}) ≡ −v_{1}^{ap^{2}+p^{2}}v_{2}^{p^{2}b}v_{k+2}^{p^{2}c}t_{k+2}^{p} mod (p, v_{1}^{ap^{2}+p^{2}+p}).

The p-th power shows that

d(x_{3m+3}^{p}) ≡ −v_{1}^{ap^{3}+p^{3}}v_{2}^{p^{3}b}v_{k+2}^{p^{3}c}t_{k+2}^{p^{2}} mod (p, v_{1}^{ap^{3}+p^{3}+p^{2}})

d(−v_{1}^{ap^{3}+p^{3}−p^{2}−p}v_{2}^{p^{3}b+p}v_{k+2}^{p^{3}c}x′_{2})

≡ −v_{1}^{ap^{3}+p^{3}}v_{2}^{p^{3}b+p}v_{k+2}^{p^{3}c}(−v_{2}^{−p}t_{k+2}^{p^{2}} − v_{1}^{p}v_{2}^{p^{k+1}−p−1}v_{k+2}^{p^{2}−p}t_{k+1} + v_{1}^{p+1}v_{2}^{−p−1}v_{k+2}^{p^{2}−p}t_{k+2}^{p})

mod (p, v_{1}^{ap^{3}+p^{3}+p+2}). This completes the induction. q.e.d. 4. Computation of d(r_{0}) In this section, we will prove Lemma 2.3. The coproduct Δ: Γ(k) →

Γ(k)⨂Γ(k) is given by:

Lemma 4.1. For k > 1,

Δ(t_{k+1}) = t_{k+1} ⊗ 1 + 1 ⊗ t_{k+1}
Δ(t_{k+2}) = t_{k+2} ⊗ 1 + v_{1}b_{k+1,0} + 1 ⊗ t_{k+2}
Δ(t_{k+3}) = t_{k+3} ⊗ 1 + v_{1}b_{k+2,0} + v_{2}b_{k+1,1} + 1 ⊗ t_{k+3}.

If k = 1, then

Δ(t_{k+1}) = t_{k+1} ⊗ 1 + 1 ⊗ t_{k+1}
Δ(t_{k+2}) = t_{k+2} ⊗ 1 + v_{1}b_{k+1,0} + 1 ⊗ t_{k+2}
Δ(t_{k+3}) = t_{k+3} ⊗ 1 + t_{k+1} ⊗ t_{k+1}^{p^{k+1}} + v_{1}b_{k+2,0} + v_{2}b_{k+1,1} + 1 ⊗ t_{k+3}.

Here b_{k+1,n} = − ∑_{i=1}^{p^{n}−1} \binom{p^{n}}{i}t_{k+1}^{i} ⊗ t_{k+1}^{p^{n}−i} and

−pb_{k+2,0} = Δ(t_{k+2})^{p} − ψ(t_{k+2})^{(1)} − v_{1}^{p}b_{k+1,1}.

Proof. The coproduct is given by

Δ(t_{k+n}) + ∑_{i=1}^{n−1} m_{i}Δ(t_{k+n−i})^{p^{i}} = ψ(t_{k+n}) + ∑_{i=1}^{n−1} m_{i}ψ(t_{k+n−i})^{(i)},

where ψ(t_{k+n})^{(l)} = 1 ⊗ t_{k+n}^{p^{l}} + ∑_{i=1}^{n−k−1} t_{k+i}^{p^{l}} ⊗ t_{n−i}^{p^{k+i+l}} + t_{k+n}^{p^{l}} ⊗ 1. Take n = 2, and we have

Δ(t_{k+2}) + m_{1}Δ(t_{k+1})^{p} = ψ(t_{k+2}) + m_{1}ψ(t_{k+1})^{(1)}.

Thus we have Δ(t_{k+2}).

For Δ(t_{k+3}),

Δ(t_{k+3}) + m_{1}Δ(t_{k+2})^{p} + m_{2}Δ(t_{k+1})^{p^{2}} = ψ(t_{k+3}) + m_{1}ψ(t_{k+2})^{(1)} + m_{2}ψ(t_{k+1})^{(2)}.

Note that m_{2}(ψ(t_{k+1})^{(2)} − Δ(t_{k+1})^{p^{2}}) = pm_{2}b_{k+1,1} = (v_{2} + m_{1}v_{1}^{p})b_{k+1,1}. Then

Δ(t_{k+3}) = ψ(t_{k+3}) − m_{1}(Δ(t_{k+2})^{p} − ψ(t_{k+2})^{(1)} − v_{1}^{p}b_{k+1,1}) + v_{2}b_{k+1,1}.

Note that if k = 1, then ψ(t_{k+3}) = 1 ⊗ t_{k+3} + t_{k+1} ⊗ t_{k+1}^{p^{k+1}} + t_{k+3} ⊗ 1 while ψ(t_{k+3}) = 1 ⊗ t_{k+3} + t_{k+3} ⊗ 1 otherwise. q.e.d.

Lemma 4.2. Let C_{i}(n) be the element of (3.1). Then

1. C_{i}(k + 2) ≡ v_{1}^{p^{i−1}}v_{k+2}^{p^{i}−p^{i−1}}t_{k+1}^{p^{i}} mod (p, v_{1}^{p^{k+i}}) and

d(C_{i}(k + 2)) = −v_{1}^{p^{i}}b_{k+1,i} + v_{1}^{p^{k+i+1}}b_{k+1,i−1}.

2. d(C_{i}(k+3)) ≡ −v_{2}^{p^{i}}b_{k+1,i+1}−v_{1}^{p^{i}}b_{k+2,i}+v_{2}^{p^{k+i+1}}b_{k+1,i−1}+v_{1}^{p^{k+i+2}}b_{k+2,i−1} mod (p).

Proof. The first one follows immediately since d(t_{k+1}^{p^{i+1}}) = pb_{k+1,i}. For the second, note that

pC_{i}(k+3) ≡ d(v_{k+3}^{p^{i}})−v_{2}^{p^{i}}t_{k+1}^{p^{i+2}}−v_{1}^{p^{i}}t_{k+2}^{p^{i+1}}−v_{1}^{p^{i}}C_{i+1}(k+2)+v_{2}^{p^{k+i+1}}t_{k+1}^{p^{i}}+v_{1}^{p^{k+i+2}}t_{k+2}^{p^{i}}

mod (p^{2}), and d(t_{k+2}^{p^{i}}) = v_{1}^{p^{i}}b_{k+1,i} + pb_{k+2,i−1} by the definition of b_{k+2,i−1}. Then we compute that

d(pC_{i}(k + 3)) = −pv_{2}^{p^{i}}b_{k+1,i+1} − v_{1}^{p^{i+1}+p^{i}}b_{k+1,i+1} − pv_{1}^{p^{i}}b_{k+2,i−1}
−v_{1}^{p^{i}}(−v_{1}^{p^{i+1}}b_{k+1,i+1} + v_{1}^{p^{k+i+2}}b_{k+1,i}) + pv_{2}^{p^{k+i+1}}b_{k+,1i−1}
+v_{1}^{p^{k+i+2}}(v_{1}^{p^{i}}b_{k+1,i} + pb_{k+2,i−1})

q.e.d.

Proof of Lemma 2.3. Put H_{11} = C_{1}(k + 2) + v_{1}^{p}v_{2}^{−p^{k+3}}C_{2}(k + 3) − v_{1}^{p}v_{2}^{−p^{k+3}}t_{k+3}^{p^{2}}+v_{1}^{p^{k+4}}v_{2}^{−p^{k+3}}t_{k+3}^{p}−v_{1}^{p^{k+4}−p^{2}}v_{2}^{−p^{k+3}+p}t_{k+2}^{p^{2}}+v_{1}^{p^{k+2}−1}t_{k+2}. Since H_{11} ≡ v_{1}v_{k+2}^{p−1}t_{k+1}^{p}, it suffices to show that d(H_{11}) = 0 if k > 1, which is certified by Lemma 4.2, as follows:

d(C_{1}(k + 2)) = −\underline{v_{1}^{p}b_{k+1,1}}_{1} + \underline{v_{1}^{p^{k+2}}b_{k+1,0}}_{6}
d(v_{1}^{p}v_{2}^{−p^{k+3}}C_{2}(k + 3)) = v_{1}^{p}v_{2}^{−p^{k+3}}(−\underline{v_{2}^{p^{2}}b_{k+1,3}}_{2} − \underline{v_{1}^{p^{2}}b_{k+2,2}}_{3} + \underline{v_{2}^{p^{k+3}}b_{k+1,1}}_{1}
+\underline{v_{1}^{p^{k+4}}b_{k+2,1}}_{4})
d(−v_{1}^{p}v_{2}^{−p^{k+3}}t_{k+3}^{p^{2}}) = v_{1}^{p}v_{2}^{−p^{k+3}}(\underline{v_{1}^{p^{2}}b_{k+2,2}}_{3} + \underline{v_{2}^{p^{2}}b_{k+1,3}}_{2})
d(v_{1}^{p^{k+4}}v_{2}^{−p^{k+3}}t_{k+3}^{p}) = −v_{1}^{p^{k+4}}v_{2}^{−p^{k+3}}(\underline{v_{1}^{p}b_{k+2,1}}_{4} + \underline{v_{2}^{p}b_{k+1,2}}_{5})
d(−v_{1}^{p^{k+4}−p^{2}}v_{2}^{−p^{k+3}+p}t_{k+2}^{p^{2}}) = \underline{v_{1}^{p^{k+4}−p^{2}}v_{2}^{−p^{k+3}+p}(v_{1}^{p^{2}}b_{k+1,2})}_{5}
d(v_{1}^{p^{k+2}−1}t_{k+2}) = −\underline{v_{1}^{p^{k+2}}b_{k+1,0}}_{6}

Now take r_{0} to be v_{1}^{−1}H_{11}.

For the case k = 1, add terms v_{1}^{p}v_{2}^{−p^{k+3}}t_{k+1}^{p^{2}} ⊗ t_{k+1}^{p^{k+3}} and −v_{1}^{p^{k+4}}v_{2}^{−p^{k+3}}t_{k+1}^{p} ⊗ t_{k+1}^{p^{k+2}} on the right hands of the third and fourth equation. Therefore,

d(H_{11}) = v_{1}^{p}v_{2}^{−p^{k+3}}t_{k+1}^{p^{2}} ⊗ t_{k+1}^{p^{k+3}} − v_{1}^{p^{k+4}}v_{2}^{−p^{k+3}}t_{k+1}^{p} ⊗ t_{k+1}^{p^{k+2}}.

The first term is homologous to

v_{1}^{p^{3}+2p^{2}}v_{2}^{p^{4}−2p^{2}−p}C_{1}(k + 2) ⊗ t_{k+1}^{p}

by Lemma 4.3 below, and we see that the leading term is v_{1}^{p^{3}+2p^{2}+1}v_{2}^{p^{4}−2p^{2}−p}r_{0}⊗ t_{k+2}^{p}. q.e.d.

Lemma 4.3. The element v_{1}^{p−1}t_{2}^{p^{2}}⊗t_{2}^{p^{4}} is homologous to v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}C_{1}(3)⊗ t_{3}^{p} mod (p, v_{1}^{p^{3}+2p^{2}+1}).

Proof. Let t_{31} denote an element fitting in the equation d(v_{4}) = v_{2}t_{2}^{p^{2}} − v_{2}^{p^{2}}t_{2} + v_{1}t_{31}. Then we see that d(t_{31}) ≡ 0 mod (p) and

t_{31} = t_{3}^{p} − C_{1}(3) − v_{1}^{p^{3}−1}t_{3},

and we compute

d(v_{2}^{−1}η_{R}(v_{4})t_{2}) ≡ v_{2}^{−1}t_{2} ⊗ (v_{2}t_{2}^{p^{2}} − \underline{v_{2}^{p^{2}}t_{2}}_{1} + v_{1}t_{31})
d(−\frac{1}{2}v_{2}^{p^{2}−1}t_{2}^{2}) ≡ \underline{v_{2}^{p^{2}−1}t_{2} ⊗ t_{2}}_{1},

and we have d(x) ≡ t_{2} ⊗t_{2}^{p^{2}} +v_{1}v_{2}^{−1}t_{2} ⊗t_{31} if we put x = v_{2}^{−1}η_{R}(v_{4})t_{2} − \frac{1}{2}v_{2}^{p^{2}−1}t_{2}^{2}. We further make a computation mod (p, v_{1}^{p^{3}+2p^{2}+1}),

d(v_{1}^{p−1}x^{p^{2}}) ≡ v_{1}^{p−1}(t_{2}^{p^{2}} ⊗ t_{2}^{p^{4}} + \underline{v_{1}^{p^{2}}v_{2}^{−p^{2}}t_{2}^{p^{2}} ⊗ t_{31}^{p^{2}}}_{1})
d(−v_{1}^{p^{2}−1}v_{2}^{−p^{2}}v_{3}^{p}t_{31}^{p^{2}}) ≡ −v_{1}^{p^{2}−1}v_{2}^{−p^{2}}(\underline{v_{1}^{p}t_{2}^{p^{2}}}_{1} − \underline{v_{1}^{p^{3}}t_{2}^{p}}_{2}) ⊗ t_{31}^{p^{2}}
d(−v_{1}^{p^{3}+p^{2}−2}v_{2}^{−p^{2}}v_{3}t_{31}^{p^{2}}) ≡ −v_{1}^{p^{3}+p^{2}−2}v_{2}^{−p^{2}}(\underline{v_{1}t_{2}^{p}}_{2} − \underline{v_{1}^{p^{2}}t_{2}}_{3}) ⊗ t_{31}^{p^{2}}
d(−v_{1}^{p^{3}+2p^{2}−p−2}v_{2}^{1−2p^{2}}x_{1}t_{31}^{p^{2}}) ≡ −v_{1}^{p^{3}+2p^{2}−p−2}v_{2}^{1−2p^{2}}(\underline{v_{1}^{p}v_{2}^{p^{2}−1}t_{2}}_{3} − \underline{v_{1}^{p+1}v_{2}^{−1}t_{31}}_{4}) ⊗ t_{31}^{p^{2}}
d(−v_{1}^{p^{3}+2p^{2}−1}v_{2}^{−2p^{2}−p}t_{31}η_{R}(v_{5}^{p})) ≡ v_{1}^{p^{3}+2p^{2}−1}v_{2}^{−2p^{2}−p}t_{31} ⊗ (\underline{v_{3}^{p}t_{2}^{p^{4}}}_{5} + \underline{v_{2}^{p}t_{3}^{p^{3}}}_{4} − v_{2}^{p^{4}}t_{3}^{p} − \underline{v_{3}^{p^{3}}t_{2}^{p}}_{7})
d(−v_{1}^{p^{3}+2p^{2}−1}v_{2}^{−3p^{2}−p}t_{31}η_{R}(v_{3}^{p}v_{4}^{p^{2}})) ≡ −v_{1}^{p^{3}+2p^{2}−1}v_{2}^{−3p^{2}−p}t_{31} ⊗ v_{3}^{p}(\underline{v_{2}^{p^{2}}t_{2}^{p^{4}}}_{5} − \underline{v_{2}^{p^{4}}t_{2}^{p^{2}}}_{6})
d(\frac{1}{2}v_{1}^{p^{3}+2p^{2}−p−1}v_{2}^{p^{4}−3p^{2}−p}t_{31}η_{R}(v_{3}^{2p})) ≡ \underline{−v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−3p^{2}−p}t_{31} ⊗ v_{3}^{p}t_{2}^{p^{2}}}_{6}
d(v_{1}^{p^{3}+2p^{2}−2}v_{2}^{−2p^{2}−p}t_{31}η_{R}(v_{3}^{p^{3}+1})) ≡ \underline{v_{1}^{p^{3}+2p^{2}−1}v_{2}^{−2p^{2}−p}t_{31} ⊗ v_{3}^{p^{3}}t_{2}^{p}}_{7}

Thus we have an element y such that d(y) ≡ v_{1}^{p−1}t_{2}^{p^{2}}⊗t_{2}^{p^{4}}−v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}t_{31}⊗ t_{3}^{p}. Since d(\frac{1}{2}v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}t_{3}^{2p}) ≡ v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}t_{3}^{p} ⊗ t_{3}^{p}, the element −v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}t_{31}⊗t_{3}^{p} is homologous to v_{1}^{p^{3}+2p^{2}−1}v_{2}^{p^{4}−2p^{2}−p}C_{1}(3)⊗t_{3}^{p}. q.e.d.

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Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan and Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan