The Chromatic E

New York Journal of Mathematics

New York J. Math.6(2000)21–54.

The Chromatic E

-term H

M

for p > 3

Hirofumi Nakai

Abstract. In this paper, study the module structure of Ext⁰_BP∗BP(BP∗, BP∗/(p, v₁^∞, v^∞₂ )),

which is regarded as the chromaticE1-term converging to the second line of the Adams-NovikovE2-term for the Moore spectrum. The main difficulty here is to construct elementsx(sp^r/j;k) from the Miller-Ravenel-Wilson elements (x^s_3,r/v^j₂)^pk∈H⁰M₂¹. We achieve this by developing some inductive methods of constructingx(sp^r/j;k) onk.

Contents

1. Introduction 21

2. BP_∗-homology and Bockstein spectral sequence 24

3. Definitions of some elements 28

4. Preliminary calculations 34

5. Proof of the main theorem 46

References 54

1. Introduction

LetBP be the Brown-Peterson spectrum for a fixed primep. As is well known, the pair of homotopy groupsBP_∗ and theBP_∗-homologyBP_∗BP forms a Hopf- algebroid

(BP_∗, BP_∗BP) = (Z_(p)[v₁, v₂, . . .], BP_∗[t₁, t₂, . . .] ).

The Adams-Novikov spectral sequence (ANSS) is one of the fundamental tools to compute the p-component of the stable homotopy groups π_∗^SX_(p) for a spectrum X:

E^∗,∗₂ = Ext^∗_BP∗BP(BP∗, BP∗X) =⇒ π_∗^SX_(p).

Here, for any BP_∗BP-comodule M, Ext^∗_BP∗BP(BP_∗, M) is regarded as the right derived functor of Hom_BP∗BP(BP_∗, M). We abbreviate Ext^s_BP∗BP(BP_∗, M) to H^sM as usual.

Received May 5, 1999.

Mathematics Subject Classification. 55Q45; secondary 55Q51, 55T15, 55N22.

Key words and phrases. Stable homotopy of spheres, Adams-Novikov spectral sequence, chro- matic spectral sequence.

ISSN 1076-9803/00

21

As an important example of finite spectra, we have the Smith-Toda spectrum V(n) for each primepwithBP∗V(n)∼=BP∗/(p, v1,· · · , vn), although its existence is verified only for 0 ≤ n ≤ 3 and 2n+ 1 ≤ p so far. (Recently, Lee S. Nave has shown the non-existence ofV((p+ 3)/2) forp >5.) Then the E2-term of the ANSS for V(n) is H^∗BP∗V(n). Miller, Ravenel and Wilson [2] have constructed an algebraic spectral sequence converging toH^∗BP∗V(n) as follows:

Denote theBP_∗BP-comodulesBP_∗/(p, v₁,· · · , v_n−1) byN_n⁰. Then, define N_n^m (m≥1) inductively onmby short exact sequences

0 −→ N_n^m −→ v⁻¹_m+nN_n^m −→ N_n^m+1 −→ 0.

We also defineM_n^mbyM_n^m=v⁻¹_m+nN_n^m. Indeed, they can be described directly as N_n^m = BP_∗/(p,· · · , v_n−1, v_n^∞,· · ·, v_n+m−1^∞ ),

M_n^m = v⁻¹_n+mBP_∗/(p,· · ·, v_n−1, v_n^∞,· · ·, v_n+m−1^∞ ).

Splicing the above short exact sequences, we get a long exact sequence:

0 −→ N_n⁰ −→ M_n⁰ −→ M_n¹ −→ M_n² −→ · · ·,

called the chromatic resolution of N_n⁰. Applying H^∗(−) to the above long exact sequence, we obtain a spectral sequence converging to H^∗N_n⁰ with E₁^s,t =H^tM_n^s anddr:E_r^s,t→E^s+r,t−r+1_r , calledthe chromatic spectral sequence.

The simplest example in these E1-terms is the 0-th cohomology of the n-th Morava stabilizer groupH⁰M_n⁰, which is isomorphic toZ/p[v_n^±1]. Moreover,H^tM_n⁰ (1≤t≤2) has been computed by Ravenel [5]. In general the calculation ofH^tM_n^s becomes terribly difficult ass+tincreases, except for the following case:

Theorem (Morava’s vanishing theorem). If (p−1) -n andt > n², then H^tM_n⁰= 0.

Many chromaticE1-terms are computed so far (cf. [12]), most of them are due to Miller-Ravenel-Wilson [2], Ravenel [5] and Shimomura’s works. In particular, H⁰M_n¹ is computed in [2, Theorem 5.10] and H⁰M_n² (n ≥2, p > 2) is also done in [10, Theorem 1.2]. The purpose of this paper is to prove the following theorem about thek(1)_∗-module structure of unknownH⁰M₁² forp >3.

Theorem. For each Miller-Ravenel-Wilson element x^s_3,r+k/v^jp₂ ^k ∈ H⁰M₂¹ (p-j andp-s)and1/v^mp₂ ^r ∈H⁰M₂¹ (p-m), there exists an element

x(sp^r/j;k)∈v⁻¹₃ BP_∗/(p, v^∞₂ ) which is congruent to(x^s_3,r/v₂^j)^pk mod(v₁), and

1/X_2,r^m ∈BP_∗/(p, v₂^∞) congruent to1/v₂^mpr mod (v1), so that as a k(1)∗-module

H⁰M₁²∼= k(1)_∗n

x(sp^r/j;k)/v^N₁^(s,r,j;k) k≥0, r≥0, p -s∈Z and p-j≤a_3,ro

⊕ k(1)∗

1/v₁^a2,rX_2,r^m r≥0and p-m≥1 ,

The Chromatic E1-term H M₁ for p >3 23 where the integers a2,r and a3,r are defined in 2.3.1 and 3.3.1, and the integers N(s, r, j;k)are given as follows:

[(p³+p²−p−1)p^k−3] + [p^k−3] forr= 0andp-(s−1), (1)

(p^k+1−1)/(p−1) forr= 0,p|(s−1)ands6∈N₁, (2)

[(2p²−1)p^k−2] + [p^k−2] forr= 0ands∈N1, (3)

(p^k+1−1)/(p−1)−[p^k−1] for odd r≥1,s6∈N₀ andj =a_3,r−1, (4)

p^k+p^k−1−1 ( =a2,k) for odd r≥1,s6∈N0 andj ≤a3,r−2 (5)

or evenr≥2 ands6∈N₀, p^k+1+p^k−1 ( =a2,k+1) forr= 1,s∈N0 andj=p−1, (6)

[(2p³+p²−p−1)p^k−3] + [p^k−3] forr= 1,s∈N₀ andj≤p−2, (7)

[(p²+p−1)p^k−2] + [p^k−2] for even r≥2 ands∈N₀, (8)

[(p⁴+p²+p−1)p^k−4] + [p^k−4] for odd r≥3 andj=a3,r, (9)

and for oddr≥3,s∈N₀ andj≤a_3,r−1, we have [(p³+p²−1)p^k−3] + [p^k−3] forj=a_3,r−1, (10)

same as the case(1) fora3,r−p≤j≤a3,r−2, (11)

2p^k forj=a_3,r−p−1, (12)

same as the case(7) forj=a3,r−p+ 2, (13)

orp-(j+ 1)andj≤a3,r−p−2, p^k+ (p^k+1−1)/(p−1) forp|(j+ 1)

(14)

anda_3,r−p²≤j≤a_3,r−2p, [(p²+p−1)p^k−1] + [p^k−1] forj=a3,r−p²−p,

(15)

orp|(j+ 1),p³ -(j+ 1)(j+p+ 1) andj≤a3,r−p²−3p,

[(p⁴+p³−p²+ 1)p^k−3] + [2p^k−1] forj≤a_3,r−2p²−pandp²|(j+ 1), (16)

p^k+1+p^k forj≤a_3,r−p²−2p (17)

andp²|(j+p+ 1), Here [x] is the greatest integer which does not exceedx, and

N₀ = {ap−1 | p-a}, N₁ =

(ap²−p−1)p^r+ 1 | p-a, r≥1 :odd . 2 In Section 2 we shall review BP-theory and the Bockstein spectral sequence and recall the structure of the chromaticE₁-termH⁰M₂¹computed in [2]. Then we change theFp-module basis ofH⁰M₂¹and state the method of getting the structure ofH⁰M₁²(originally due to Miller-Ravenel-Wilson). It is enough to read this section for an idea of the theoretical part. In Section 3we give the fundamental elements un,k, wn,k and X3,r, construct the new element 1/X2,r, and give the differentials on these elements (some of them are introduced in [1] and [8]). In Section 4 we

set up the elements X0(v^j₂, X_3,r^s ) and X(v₂^j, X_3,r^s ), each of which is congruent to X_3,r^s /v₂^jmod (v₁), and compute the differentials. We also introduce some inductive methods of constructing x(k) for large k. This is the hardest part of this paper.

Using these results, we construct the series of elements x(sp^r/j;k) and complete the proof of themain theoremin Section5.

We can deduce some applications to H²BP_∗V(0) from this result. These will appear in the forthcoming paper [3].

Acknowledgment. I wish to thank Professor Katsumi Shimomura for leading me to this field, and giving guidance and much helpful advice. I also would like to thank Professor Mikiya Masuda and Professor Zen-ichi Yosimura for their carefully reading the draft version, and for their encouragement. I could not have finished these hard calculations without stimulation from them.

2. BP

-homology and Bockstein spectral sequence

In this section we review several basic facts in BP-theory and explain how to determine the structure ofH⁰M₁².

2.1. Summary ofBP_∗-homology and related maps. For a fixed primep, there is a spectrumBP calledthe Brown-Peterson spectrum, which is characterized by

BP_∗ = Z_(p)[v₁, v₂,· · ·, v_n,· · ·], BP∗BP = BP∗[t1, t2,· · ·, tn,· · ·],

where |vn| = |tn| = 2(pⁿ −1). The pair (A,Γ) = (BP∗, BP∗BP) form a Hopf algebroid with the following structure maps:

η_L (resp.η_R) :A→Γ (left (resp. right) unit), c: Γ→Γ (conjugation), : Γ→A (augmentation), ∆ : Γ→Γ⊗AΓ (coproduct).

Given aBP∗BP-comoduleMand itsBP∗BP-comodule structureφ:M →M⊗AΓ, we use the following notation as usual:

H^∗M = Ext^∗_Γ(A, M) ∼= H^∗(Ω^∗M, d),

where the cobar complex (Ω^∗M, d) is the double gradedZ_(p)-module with ΩⁿM =M ⊗_AΓ⊗_A· · · ⊗_AΓ (nfactors of Γ), d_n(m⊗γ₁⊗ · · ·γ_n) =φ(m)⊗γ₁⊗ · · · ⊗γ_n

+X

(−1)^km⊗γ₁⊗ · · · ⊗∆(γ_k)⊗ · · · ⊗γ_n + (−1)ⁿ⁺¹m⊗γ₁⊗ · · · ⊗γ_n⊗1.

In this paper we compute only the 0-th differentiald₀=η_R−η_L:M₁²→M₁²⊗_AΓ.

By definition,d₀satisfies

d₀(xy) =d₀(x)y+η_R(x)d₀(y)

for anyx, y ∈A. This formula is frequently used for computations in this paper.

Whend0(x)≡zmod (p, J) for an elementz∈Γ and an idealJ ⊂A, we also have d0(x^p)≡z^p mod (p, J^p).

The Chromatic E1-term H M₁ for p >3 25 See also [2, Observation 5.8].

To compute d0, we summarize some known results about ηR. Ravenel [4] has shown the following congruence of formal group laws:

X

i,j≥0

FtiηR(v_j^pi)≡ X

i,j≥0

Fvit^p_jⁱ modp.

In generalη_R(v_n)≡v_n+v_n−1t^p₁ⁿ⁻¹−v^p_n−1t₁modI_n−1for the invariant prime ideal In−1= (p, v1,· · · , vn−2). Then, for instance, direct calculation shows that

ηR(v₂ⁱ)≡(v2+v1t^p₁)ⁱ−i v^p₁v₂ⁱ⁻¹t1−i(i−1)v^p+1₁ v₂ⁱ⁻²t^p+1₁ modp, v^p+2₁ . (2.1.1)

More precisely,η_R(v₃) andη_R(v₄) satisfy the following congruences (cf. [6](4.3.21)):

η_R(v₃)≡v₃+v₂t^p₁²−v^p₂t₁+v₁t^p₂−v₁²v^p−1₂ t^p₁ modp, v³₁, (2.1.2)

ηR(v4)≡v4+v3t^p₁³−v^p₃t1+v2t^p₂²−v₂^pt^p₁³⁺¹ modp, v1, v₂^p+1. (2.1.3)

2.2. Bockstein spectral sequence. In this paper will deduce thek(1)∗-module structure of H⁰M₁² from H⁰M₂¹, which has already been computed in [2]. By definitions of comodulesM₂¹ andM₁², we obtain the short exact sequence

0 −→ M₂^{1 1/v}−→¹ M₁² −→^v1 M₁² −→ 0.

ApplyingH^∗(−) to this sequence, we get the long exact sequence 0 −→ H⁰M₂^{1 1/v}−→¹ H⁰M₁² −→^v1 H⁰M₁² −→^δ H¹M₂^{1 1/v}−→ · · ·¹ .

Regarding this long sequence as an exact couple, we get a Bockstein type spectral sequence in the usual way, leading fromH^∗M₂¹toH^∗M₁². But as in [2] we compute H⁰M₁² directly by making use of the following lemma.

Lemma 2.2.1 (cf. [2], Remark 3.11). Assume that there exists ak(1)_∗-submodule B^t ofH^tM₁² for each t < N, such that the following sequence is exact:

0→H⁰M₂^11/v−→¹B⁰−→^v1 B⁰−→^δ H¹M₂^11/v−→ · · ·^{1 1/v}−→¹B^N−1−→^v1 B^N−1−→^δ H^NM₂¹, where δ : B^t → H^t+1M₂¹ is the restriction of the coboundary map δ : H^tM₁² → H^t+1M₂¹. Then the inclusion map it : B^t → H^tM₁² is an isomorphism between k(1)∗-modules for each t < N.

Sketch of the proof. BecauseH^tM₁²is av1-torsion module, we can filterB^tand H^tM₁² as

Pt(m) ={x∈B^t | v₁^mx= 0} and Qt(m) ={x∈H^tM₁² | v₁^mx= 0}.

Assume that the inclusion i_k is an isomorphism for k ≤t−1 (the t = 0 case is obvious), and consider the following commutative ladder diagram:

B^t−1 −→^δ H^tM₂^{1 1/v}−→¹ P_t(m) −→^v1 P_t(m−1) −→^δ H^t+1M₂¹

∼=↓i_t−1 k ↓i_t ↓i_t k

H^t−1M₁² −→^δ H^tM₂^{1 1/v}−→¹ Qt(m) −→^v1 Qt(m−1) −→^δ H^t+1M₂¹. Using the Five Lemma, we can show that Pt(m) ∼= Qt(m) (m≥1) by induction

onm.

We shall constructB⁰satisfying the above lemma to determine thek(1)∗-module structure ofH⁰M₁². In order to construct ak(1)∗-module basis ofB⁰, it is natural to push each element ofH⁰M₂¹ toH⁰M₁² and to divide it byv1 as many times as possible. So we need to review the module structure ofH⁰M₂¹.

2.3. H⁰M₂¹ and changing its basis. We first recall some notations defined in [2] to write down ak(2)∗-module basis ofH⁰M₂¹. Hereafter we assume thatp >2.

Definition 2.3.1([2], (5.11) and (5.13)). Define integersa_3,k by

a3,0= 1, a3,1=p, a3,2t=p a3,2t−1, a3,2t+1=p a3,2t+p−1 for t≥1 and elementsx3,k∈v₃⁻¹BP∗ by

x_3,0=v₃, x_3,1=v₃^p−v^p₂v⁻¹₃ v₄, x_3,2t=x^p_3,2t−1, x_3,2t+1=x^p_3,2t−v^a₂^3,2t+1−pv^(p−1)p₃ ^2t+1

fort≥1.

Using these notations, Miller-Ravenel-Wilson ([2, Theorem 5.10]) have shown the following:

Theorem 2.3.2. As ak(2)∗-module, H⁰M₂¹∼=k(2)_∗

x^m_3,k/v₂^a3,k | k≥0, p-m∈Z ⊕F_p

1/v₂ⁱ | i≥1 . In this paper we will consider an analogue to Miller-Ravenel-Wilson construction of the elementsx^m_3,k∈v₃⁻¹BP∗([2, Section 5]): The elementsx3,khave been defined inductively on k with x3,0 = v3, and each of them has the relation x3,k ≡ v₃^pk mod (p, v^N₂) for a small enough integer N. Motivated by this, we shall construct elements x(k) ∈ v⁻¹₃ BP_∗/(p, v^∞₂ ) inductively on k with x(0) = x^s_3,r/v₂^j so that x(k)≡(x^s_3,r/v^j₂)^pk mod (p, v₁^N) for a small enoughN. Keeping it in mind, we now change theFp-basis ofH⁰M₂¹ to “ap^k-power basis”.

Lemma 2.3.3. As ak(2)_∗-module, H⁰M₂¹∼=Fp

n(x^s_3,r/v^j₂)^pk | k≥0, r≥0, p -s∈Z, and p-j≤a3,r

o

⊕Fp

1/v₂ⁱ | i≥1 .

Proof. It is sufficient to prove that any Fp-module base x^m_3,u/v₂^l (1 ≤ l ≤ a3,u) displayed in Theorem2.3.2can be written as a linear sum

x^m_3,u/v^l₂=X

λ

a_λx_λ, (2.3.4)

wherea_λ∈F_p and eachx_λ has a form (x^s_3,r/v^j₂)^pk withp-s∈Zandp-j≤a_3,r. We shall show it by induction onmp^u. Notice that

x^m_3,u/v₂^l =v₃^mpu/v₂^l+ (elements withv3-exponents less thanmp^u).

When mp^u = 1, the only such a base is v3/v2, so it is clear. Suppose that each base x^a_3,b/v₂^c (1 ≤ap^b < mp^u) can be expressed as (2.3.4) andl =dp^e with p-d. Becausel ≤a_3,u< p^u+1, we may assume that e≤u. Definey =x^m_3,u/v₂^l−

x^m_3,u−e/v₂^d_p^e

∈ H⁰M₂¹. Since the maximalv₃-exponent in y is less thanmp^u, y has a form (2.3.4) by inductive assumption, hence so doesx^m_3,u/v^l₂.

The Chromatic E1-term H M₁ for p >3 27 2.4. The construction of the moduleB⁰. Letx(sp^r/j;k) andy(mp^r) be ele- ments ofv⁻¹₃ BP_∗/(p, v^∞₂ ) satisfying

x(sp^r/j;k)/v1= (x^s_3,r/v^j₂)^pk/v1 and y(mp^r)/v1= 1/v1v₂^mpr

in H⁰M₁², andN(s, r, j;k) (resp. N(m;r)) be the maximal integer such that each x(sp^r/j;k)/v₁ⁱ (resp. y(mp^r)/vⁱ₁) with 1≤i≤N(s, r, j;k) (resp. 1≤i≤N(m;r)) is a cycle ofH⁰M₁².

Proposition 2.4.1. As a k(1)∗-module, B⁰=k(1)_∗n

x(sp^r/j;k)/v^N₁^(s,r,j;k) k≥0, r≥0, p-s∈Z and p-j≤a_3,ro

⊕k(1)_∗n

y(mp^r)/v^N₁^(m;r) r≥0 and p-m≥1o

is isomorphic to H⁰M₁² if it satisfies the following condition for the coboundary map δ:B⁰→H¹M₂¹ in Lemma2.2.1:

nδ

x(sp^r/j;k)/v^N₁^(s,r,j;k) o

∪n δ

y(mp^r)/v^N₁^(m;r) o

isFp-linearly independent.

Proof. All exactness of the sequence 0 → H⁰M₂^{1 1/v}→¹ B⁰ →^v1 B⁰ →^δ H¹M₂¹ is obvious, but Kerδ⊂Imv₁. So we need to show only this inclusion. Separate the F_p-basis ofB⁰ into two parts

A=n

x(sp^r/j;k)/v^N₁^(s,r,j;k)o

∪n

y(mp^r)/v₁^N(m;r)o B =

x(sp^r/j;k)/v₁^l | 1≤l < N(s, r, j;k) ∪

y(mp^r)/v^l₁ | 1≤l < N(m;r) . Then it is obvious thatδ(xλ)6= 0∈H¹M₂¹forxλ∈A, and thatδ(yµ) = 0∈H¹M₂¹ foryµ∈B. Thus for any elementz=P

λaλxλ+P

µbµyµofB⁰(aλ, bµ ∈Z/p), we haveδ(z) =P

λaλδ(xλ). The condition implies that allaλare zero whenδ(z) = 0, and sov1P

µbµyµ/v1=z. This completes the proof.

We will construct the elementx(sp^r/j;k) from (x^s_3,r/v₂^j)^pk in Section5, and the element 1/X_2,r^m from 1/v₂^mpr as a candidate fory(mp^r) in Section3.

2.5. Fp-linear independence and Cokerδ. When we compute the coboundaries δ :B⁰ →H¹M₂¹ of x(sp^r/j;k)/v₁^N(s,r,j;k) andy(mp^r)/v^N₁^(m;r), we expect each of these images to have an appropriate form so that we can judge whether the set ofδ-images isF-linearly independent or not inH¹M₂¹for use in Proposition2.4.1.

Though the structure ofH¹M₂¹(p >3) has already been computed by Shimomura ([9] and [11]), we don’t need the whole structure of H¹M₂¹ for our purpose. We follow the same method as in the proof of [2, Theorem 6.1 (p. 500)].

Consider the long exact sequence

· · · −→ H⁰M₂¹ −→^δ H¹M₃^{0 1/v}−→² H¹M₂¹ −→^v2 H¹M₂¹ −→ · · ·^δ .

Since the coboundaries on elements of H⁰M₂¹ have already been computed in [2, Proposition 5.17], we can obtain generators of Ker v2|H¹M₂¹

. An easy calcula- tion shows that Coker(δ : H⁰M₂¹ →H¹M₃⁰) is isomorphic to the following direct

sum as aFp-vector space:

Fp

nv^sp₃ ^rh0 | eithers6∈N0 and evenr≥0, orp-sand oddr≥1o (2.5.1)

⊕Fp

nv^sp₃ ^rh1 | eithers6∈N0 and oddr≥1, orp-sand even r≥0o

⊕F_p{h₁} ⊕ F_pn

v^tp−1₃ h₂ | t∈Zo

⊕ K(3)_∗{ζ₃}, whereN₀=

s∈Z | p|s+ 1 and p² -s+ 1 , andh_i is the cohomology class of t^p₁ⁱ.

LetS be the set of elements

{A_k=v₃^bkh_ak/v₂^ck |0≤k≤n, 0≤a_k≤2, c₁≥ · · · ≥c_n, v₃^bkh_ak ∈Cokerδ}, and consider the following condition onS:

(ai, bi, ci)6= (aj, bj, cj) for any two elements withi6=j.

(2.5.2)

For a linear sumA=P_n

k=1λ_kA_k, assume that A= 0 inH¹M₂¹ andc₁=c₂=

· · ·=c_m for somem≤n. Because H¹M₂¹ is av₂-torsion module, we can consider the multiplication by v₂^c1−1, so thatv₂^c1−1A =P_m

k=1λk(v^b₃^khak/v2) = 0 in Kerv2

(∼= Cokerδ). By the condition (2.5.2), the set{v₃^bkhak} is linearly independent in Cokerδand thus allλk (1≤λk ≤m) should be zero. Iterating this, we conclude that all coefficientsλk are zero andS is linearly independent inH¹M₂¹.

3. Definitions of some elements

In this section we introduce the elementsu_n,k, w_n,k,X_3,r and 1/X_2,r. We will use these to define many elements in Section4.

3.1. Moreira’s elementu_n,k and Shimomura’s elementw_n,k.

Definition 3.1.1. As is done in [1,§6(4)] or [8, 2.8], we define the elementu_n,k∈ v_n⁻¹BP_∗by the following recursive formula:

u_n,0=v⁻¹_n and X

i+j=k

v_n+iu^p_n,jⁱ = 0 fork≥1.

Remark 3.1.2. In [1] un,k is defined only for k ≤ n and denoted as un+k. By definition

u_n,1=−v_n^−p−1v_n+1,

u_n,2=v^−p_n ^2−p−1v_n+1^p+1−v_n^−p2−1v_n+2,

un,3≡ −v_n^{−p3−p2−p−1}v_n+1^p2+p+1+v^−p_n ^3−p2−1(v^p_n+1² vn+2+vn+1v^p_n+2)

−v^−p_n ³⁻¹v_n+3 mod (p), and so on.

Computing the right unitη_Ronu_n,n, Moreira has shown that theK(n)_∗-module baseζ_n ofH¹M_n⁰ is homologous toζ_n^p:

Proposition 3.1.3 ([1], Theorem 6.2.1.1). ηR(un,n)−ηL(un,n)≡ζn−ζ_n^p modIn.

The Chromatic E1-term H M₁ for p >3 29 We now recall the elementswn,k introduced by Shimomura [8]. Define elements T_j(j≥0) byT₀= 1 andP_j−1

i=0t_iη_R(v_j−i^pi )≡P_j

i=1v_iT_j−i^pi−1mod (p) forj ≥1, and an elemente_n(x)∈v⁻¹_n BP_∗BP/I_n forx∈v_n⁻¹BP_∗ by the congruence η_R(x)≡e_n(x) modI_n.

Definition 3.1.4(cf. [8], 2.10). Define elements wn,k ∈ v_n⁻¹BP∗BP (n ≥ 2) in- ductively onkby

wn,0= 0 and wn,k= Xk j=1

en(u^p_n,k−j^j−1 )T_j^pn−2 fork≥1.

Remark 3.1.5. By definition, wn,1=v⁻¹_n t^p₁ⁿ⁻¹,

wn,2≡ −v^−p−1_n v_n+1t^p₁ⁿ⁻¹+v^−p_n (t^p₂ⁿ⁻¹−t^p₁^n+pn−1) +v_n⁻¹t^p₁ⁿ⁻¹⁺¹ modp, v^p₁ⁿ⁻².

In general, as an element ofM₂¹⊗AΓ w2,k= (−1)^k−1 v₃^{(pk−1−1)/(p−1)}t^p₁

v₂^{(pk−1)/(p−1)} + (elements killed byv^(pk−1)/(p−1)−1

2 ).

We note thatw_2,2 is similar toζ₂=−v^−p−1₂ v₃t^p₁+v^−p₂ (t^p₂−t^p₁^2+p) +v₂⁻¹t₂. In factw_n,n (n≥2) is related toζ_n by the following congruence (cf. [8, 4.8]):

ζn−wn,n≡ X

1≤i<j≤n

u^p_n,n−j^j−i−1



 X

i≤k≤j

tk c(t^p_j−k^k )





pⁿ⁻ⁱ⁻¹

modIn.

In particular we haveζ₂−w_2,2≡v⁻¹₂ (t₂−t^p+1₁ ) modI₂ forn= 2.

Using the notation wn,k, Shimomura [8] has proved the following proposition, which is a generalization of Moreira’s result:

Proposition 3.1.6 ([8], Proposition 2.2). Forn≥2, η_R(u_n,k)≡ X

i+j=k

u_n,it^p_jⁱ−w_n,k^p −v_n−1w_n,k+1η_R(v⁻¹_n ) modI_n−1+ (v^p_n−1).

2 3.2. Shimomura’s element X_3,r. The elements x_n,r ∈ v_n⁻¹BP_∗ (n ≥ 1 and r ≥ 0) have been defined in [2, (5.11)] to express a basis of H⁰M_n−1¹ (x_3,r was listed in2.3.1). As is shown in [2, Proposition 5.17], the differentialsd0:v⁻¹_n BP∗→ v_n⁻¹BP_∗⊗_AΓ mod (p, v₁, v₂^1+a3,r) are given by

d0(x3,0)≡v2t^p₁² and d0(x3,r)≡v₂^a3,rx^p−1_3,r−1t^p₁^δ(r) for r≥1, whereδ(r) = 0 for oddrand 1 for evenr. However, we need to calculated0(x3,r) mod (p, v1, v₂^t) with t > 1 +a3,r for our computation. So we use the following elementsX3,r defined by Shimomura [8] instead ofx3,r.