Homology, Homotopy and Applications, vol.2, No.3, 2000, pp.29–40 ISSN 1512–0139

ON THE COHOMOLOGY OF SOME HOPF ALGEBROIDS AND HATTORI-STONG THEOREMS

ANDREW BAKER

(communicated by Hvedri Inassaridze)

Abstract We apply group cohomological methods to calculate the cohomology of K(n)_{∗}BP as a K(n)_{∗}K(n)-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for BP based on Johnson and Wilson’s E(n), showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral sequence associated with the natural map BP ⟶ E(n), extending calculations of Clarke for the Todd orientation MU ⟶ KU.

Introduction

In an interesting paper [4], Hovey and Sadofsky showed amongst other things that the Chromatic Spectral Sequence associated to a p-local v_{n}-periodic ring spectrum such as Johnson and Wilson’s E(n) is obtained by taking the 0 to n th columns of the Chromatic Spectral Sequence associated with BP -theory. They also identified K(n)_{∗}BP as a comodule over the Hopf algebroid K(n)_{∗}K(n) = K(n)_{∗}(E(n)). In [2], we also analysed the structure of algebras of the form

R_{∗}K(n) = R_{∗} ⊗_{BP_{∗}} BP_{∗}K(n),

where R_{∗} is a BP_{∗}-algebra such as BP_{∗} or E(n)_{∗}, and used our results to give an alternative approach to such results.

Clarke [3] discussed the change of Hopf algebroid spectral sequence associated to the Todd orientation τ: MU ⟶ KU and showed that

Ext_{KU_{∗}KU}^{1,∗}(KU_{∗}, KU_{∗}MU) ≠ 0,

despite the Hattori-Stong Theorem, which was taken as evidence that the KU-theory Adams Spectral Sequence might be trivial since

MU_{∗} ≅ Ext_{KU_{∗}KU}^{0,∗}(KU_{∗}, KU_{∗}MU).

Of course, in the two decades since the publication of [3], our understanding of the target π_{∗}(L_{KU}MU) of this spectral sequence has improved and such a result no longer appears surprising.

In this paper we use methods of Galois cohomology to rederive certain cohomology groups over the Hopf algebra K(n)_{∗}K(n) and explore some consequences of our current knowledge of the Chromatic Spectral Sequence for

Ext_{E(n)_{∗}E(n)}^{∗,∗}(E(n)_{∗}, E(n)_{∗}BP),

Received 5 January 2000, revised 15 February 2000; published on 9 March 2000. 1991 Mathematics Subject Classification: Primary 55N20 55N22 55T15 Key words and phrases: Hopf algebroid, comodule, Brown-Peterson spectrum, Johnson-Wilson spectrum, Galois cohomology \begin{matrix}c \\ ◯ \end{matrix} 2000, Andrew Baker. Permission to copy for private use granted.

and compute this completely, relating the result to a change of Hopf algebroid spectral sequence

E_{2}^{s,t,∗} = Ext_{BP_{∗}BP}^{r,∗} (BP_{∗}, Ext_{E(n)_{∗}E(n)}^{s,∗}(E(n)_{∗}, E(n)_{∗}BP))
⟹ Ext_{E(n)_{∗}E(n)}^{s+t,∗}(E(n)_{∗}, E(n)_{∗}).

This generalizes Clarke’s work and we show that there are always non-zero elements in

Ext_{E(n)_{∗}E(n)}^{n,∗}(E(n)_{∗}, E(n)_{∗})

playing an analogous rôle to the elements he identified when n = 1. We also state some generalizations of the Hattori-Stong Theorem following from our analysis.

I would like to thank Ken Brown, Francis Clarke, Steve Cohen, Mark Hovey and Hal Sadofsky for their help and inspiration, and also the referee of an earlier version whose perceptive comments were invaluable.

1. Review of some results of Hovey and Sadofsky

Throughout this paper, we follow well established (but nevertheless unfortunate) fashion in writing

K(n)_{∗}K(n) = K(n)_{∗}(E(n)) ⊊ K(n)_{∗}(K(n)).

By [4, theorem B], we have a good description of the K(n)_{∗}K(n)-comodule structure of K(n)_{∗}BP; note that this is not claimed to give a description of the K(n)_{∗}K(n)-comodule algebra structure.

Theorem 1.1. There is an isomorphism of left K(n)_{∗}K(n)-comodules

K(n)_{∗}BP ≅ K(n)_{∗}E(n)[w_{i} : i ⩾ 1],

where w_{i} ∈ K(n)_{2p^{n}(p^{i}−1)}BP is primitive with respect to the coaction.

K(n)_{∗}BP is an extended K(n)_{∗}K(n)-comodule, so for s > 0,

Ext_{K(n)_{∗}K(n)}^{s,∗}(K(n)_{∗}, K(n)_{∗}BP) = 0.

It is straightforward to deduce the following consequence.

Corollary 1.2. E(n)_{∗}BP/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞}) is an extended left E(n)_{∗}E(n)-comodule,

E(n)_{∗}BP/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})

≅ E(n)_{∗}E(n) ⊗_{E(n)_{∗}} E(n)_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})[w̃_{i} : i ⩾ 1],

where w̃_{i} has degree 2p^{n}(p^{i} − 1) and is coaction primitive. Hence for s > 0,

Ext_{E(n)_{∗}E(n)}^{s,∗}(E(n)_{∗}, E(n)_{∗}BP/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})) = 0.

In [4, theorem 5.1] the Chromatic Spectral Sequence CSS(S; E(n)) based on E(n) for the sphere spectrum S was shown to be obtained by truncating that based on BP. There is an obvious generalization to the Chromatic Spectral Sequence CSS(X; E(n)) of a connective torsion free spectrum X based on E(n) and converging to

Ext_{E(n)_{∗}E(n)}^{∗,∗}(E(n)_{∗}, E(n)_{∗}X).

Recall that the E_{1}-term of the Chromatic Spectral Sequence CSS(X; BP) based on BP and converging to

Ext_{BP_{∗}BP}^{∗,∗} (BP_{∗}, BP_{∗}X)

has the form

E_{1}^{n,t,∗}(X; BP) = Ext_{E(n)_{∗}E(n)}^{t,∗}(E(n)_{∗}, E(n)_{∗}X/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})).  (1.1)

Theorem 1.3. In the E_{1}-term of the spectral sequence CSS(X; E(n)),

E_{1}^{k,t,∗}(X; E(n)) = \begin{cases}E_{1}^{k,t,∗}(X; BP) & if k ⩽ n, \\ 0 & if k > n, \end{cases}

and the differential d_{1}^{k,t,∗} with domain E_{1}^{k,t,∗}(X; E(n)) is obtained from the corresponding differential of CSS(X; BP) whenever E_{1}^{k,t,∗}(X; E(n)) ≠ 0.

When X = BP, Corollary 1.2 implies that the only possible non-zero differential in each of the spectral sequences CSS(BP; E(n)) and CSS(BP; BP) is d_{1}.

2. The cohomology of K(n)_{∗}BP as Galois cohomology

Our aim in this section is to reprove the results of Theorem 1.1 and Corollary 1.2 using our previous work [1, 2] and making use of continuous cohomology of profinite groups.

From now on, let R_{∗} be an algebra over 𝔽_{p^{n}} ⊗ BP_{∗} annihilated by I_{n} and in which there is a unit u satisfying v_{n} = u^{p^{n}−1}. In our applications to K(n)_{∗}BP we will take

R_{∗} = 𝔽_{p^{n}} ⊗ (v_{n}^{−1}BP_{∗}/I_{n})[u]/(u^{p^{n}−1} − v_{n}),

but for completeness we give the general version.

We may form the R_{∗}-algebra

R_{∗}K(n) = R_{∗} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} K(n)_{∗}.

From [2, Theorem A.3] we have

Theorem 2.1. R_{∗}K(n) is a free R_{∗}-module. Moreover, there is an exhaustive filtration by subalgebras

R_{∗} = R_{∗}K(n)^{(1)} ⊆ R_{∗}K(n)^{(2)} ⊆ ⋯ ⊆ R_{∗}K(n)^{(k)} ⊆ ⋯ ⊆ R_{∗}K(n)

in which each extension R_{∗}K(n)^{(k)} ⟶ R_{∗}K(n)^{(k+1)} is a free R_{∗}K(n)^{(k)}-module and a Galois extension of Artin-Schreier type with Galois group 𝔽_{p^{n}}.

There is an action of the Morava stabiliser group 𝕊_{n} on R_{∗}K(n) with respect to which

R_{∗}K(n)^{(k)} = R_{∗}K(n)_{∗}^{𝕊_{n}^{[k]}},

the fixed point set of the closed subgroup

𝕊_{n}^{[k]} = {1 + ∑_{k⩽r} α_{r}S^{r} : ∀r, α_{r}^{p^{n}} = α_{r}} ⩽ 𝕊_{n}.

Here an extension A ⟶ B of commutative unital 𝔽_{q}-algebras is of Artin-Schreier type if

B = A[t]/(t^{q} − t − a)

for some a ∈ A; its Galois group is Gal(B/A) = 𝔽_{q} ⩽ Aut_{A}(B) where γ ∈ 𝔽_{q} acts by γ · f(t) = f(t + γ). Notice that B is a free A-module of rank q and B^{Gal(B/A)} = A, however in general Gal(B/A) ≠ Aut_{A}(B) (consider for example the case where a = 0).

We will apply this result to calculate the continuous cohomology of BP_{∗}K(n) with respect to with action of 𝕊_{n}, where this continuous cohomology is equivalent to the cohomology over the K(n)_{∗}-Hopf algebra K(n)_{∗}K(n). Further information the relationship of the action of 𝕊_{n} and the K(n)_{∗}K(n)-comodule structure can be found in [1].

Theorem 2.2. The continuous cohomology of R_{∗}K(n) with respect to 𝕊_{n} is given by

H_{c}^{s}(𝕊_{n}; R_{∗}K(n)) = \begin{cases}R_{∗} & if s = 0, \\ 0 & if s ≠ 0. \end{cases}

In order to apply Galois cohomological machinery we will require two technical Galois theoretic results surely known to experts but for which we know of no convenient references.

Lemma 2.3 (Normal Basis Theorem for Artin-Schrier extensions). Let A be a commutative unital 𝔽_{q}-algebra. Let B = A[t]/(f(t)) be the Artin-Schrier extension associated with the polynomial f(X) = X^{q} − X + a ∈ A[X]. Then the elements (t + γ)^{q−1} (γ ∈ 𝔽_{q}) form an A-basis for B. Hence B is a free A[Gal(B/A)]-module.

Proof. We will show that in the 𝔽_{q}-vector subspace spanned by the monomials t^{k} (0 ⩽ k ⩽ q − 1), the stated elements form a basis. Let ω ∈ 𝔽_{q}^{×} be a generator, so the non-zero elements of 𝔽_{q} are expressible as powers of ω.

By the Binomial Theorem,

(t + γ)^{q−1} = ∑_{0⩽r⩽q−1} \binom{q − 1}{r}γ^{r}t^{q−1−r},

so it suffices to show the non-singularity of the matrix

[\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 1 & \binom{q−1}{1} & \binom{q−1}{2} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω & \binom{q−1}{2}ω^{2} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω^{2} & \binom{q−1}{2}ω^{4} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω^{3} & \binom{q−1}{2}ω^{6} & ⋯ & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & \binom{q−1}{1}ω^{q−2} & \binom{q−1}{2}ω^{2(q−2)} & ⋯ & 1 \end{matrix}]

whose determinant is

|\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 1 & \binom{q−1}{1} & \binom{q−1}{2} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω & \binom{q−1}{2}ω^{2} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω^{2} & \binom{q−1}{2}ω^{4} & ⋯ & 1 \\ 1 & \binom{q−1}{1}ω^{3} & \binom{q−1}{2}ω^{6} & ⋯ & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & \binom{q−1}{1}ω^{q−2} & \binom{q−1}{2}ω^{2(q−2)} & ⋯ & 1 \end{matrix}| = |\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 1 & \binom{q−1}{1} & \binom{q−1}{2} & ⋯ & 1 \\ 0 & \binom{q−1}{1}ω & \binom{q−1}{2}ω^{2} & ⋯ & 1 \\ 0 & \binom{q−1}{1}ω^{2} & \binom{q−1}{2}ω^{4} & ⋯ & 1 \\ 0 & \binom{q−1}{1}ω^{3} & \binom{q−1}{2}ω^{6} & ⋯ & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & \binom{q−1}{1}ω^{q−2} & \binom{q−1}{2}ω^{2(q−2)} & ⋯ & 1 \end{matrix}|

= \binom{q − 1}{1}\binom{q − 1}{2} ⋯ \binom{q − 1}{q − 2} |\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & 1 & ⋯ & 1 \\ 0 & ω & ω^{2} & ⋯ & 1 \\ 0 & ω^{2} & ω^{4} & ⋯ & 1 \\ 0 & ω^{3} & ω^{6} & ⋯ & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & ω^{q−2} & ω^{2(q−2)} & ⋯ & 1 \end{matrix}|

= \binom{q − 1}{1}\binom{q − 1}{2} ⋯ \binom{q − 1}{q − 2} ∏_{0⩽r<s⩽q−2}(ω^{r} − ω^{s}).

This is non-zero because the binomial coefficients satisfy

\binom{q − 1}{r} ≡ \binom{p − 1}{r_{0}}\binom{p − 1}{r_{1}} ⋯ \binom{p − 1}{r_{d}} mod (p),

where r = r_{0} +r_{1}p+⋯+r_{m}p^{m} is the p-adic expansion of r, and the factors are never divisible by p. So the (t + γ)^{q−1} form a basis in place of the t^{k}. As this is true over 𝔽_{q}, it is also true over A.

The next result will allow an inductive calculation of cohomology in the situation of interest.

Lemma 2.4. Let A ⊆ B ⊆ C be a tower of commutative unital rings. Let G ⩽ Aut_{A}(C) be finite and N ◁ G with C^{G} = A, C^{N} = B, and C ≅ B[N] as a B[N]-module. Then

H^{∗}(G; C) ≅ H^{∗}(G/N; B).

Proof. There is a Hochschild-Serre Spectral Sequence [7, chapter II §4],

E_{2}^{r,s} = H^{r}(G/N; H^{s}(N; C)) ⟹ H^{r+s}(G; C).

Since

H^{∗}(N; C) ≅ H^{∗}(N; B[N]) = H^{0}(N; B[N]) ≅ H^{0}(N; C) = B,

the E_{2}-term is trivial except for E_{2}^{r,0} = H^{r}(G/N; B), so the result follows.

Proof of Theorem 2.2. From the proof of [2, theorem A.3], for k ⩾ 1, R_{∗}K(n)^{(k)} is a free R_{∗}-module with a basis consisting of the monomials

t_{1}^{s_{1}}t_{2}^{s_{2}} ⋯ t_{k−1}^{s_{k−1}}, (0 ⩽ s_{k} ⩽ p^{n} − 1).  (2.1)

For each of these basis elements, the stabilizer is an open subgroup of 𝕊_{n} since it contains an open subgroup of the form 𝕊_{n}^{[k]}. Hence the action of 𝕊_{n} on R_{∗}K(n) is proper in the sense that the stabilizer of each point is an open subgroup and this allows us to apply the cohomological machinery of Shatz [7] to R_{∗}K(n). In particular, [7, chapter 3 theorem 7] gives

H_{c}^{∗}(𝕊_{n}; R_{∗}K(n)) = \underrightarrow{lim}_{k} H^{∗}(𝕊_{n}/𝕊_{n}^{[k]}; R_{∗}K(n)^{(k)}).  (2.2)

To complete the proof we will show by induction on k ⩾ 1 that

H^{∗}(𝕊_{n}/𝕊_{n}^{[k]}; R_{∗}K(n)^{(k)}) = H^{0}(𝕊_{n}/𝕊_{n}^{[k]}; R_{∗}K(n)^{(k)}) = R_{∗}.

For k = 1, the result is certainly true. Suppose that it holds for some k = m ⩾ 1. The extension R_{∗}K(n)^{(m)} ⊆ R_{∗}K(n)^{(m+1)} satisfies the conditions of Lemma 2.3 and so we can also apply Lemma 2.4 with

A = R_{∗}, B = R_{∗}K(n)^{(m)}, C = R_{∗}K(n)^{(m+1)}, G = 𝕊_{n}/𝕊_{n}^{[m+1]}, N = 𝕊_{n}^{[m]}/𝕊_{n}^{[m+1]}.

This yields

H^{∗}(𝕊_{n}/𝕊_{n}^{[m+1]}; R_{∗}K(n)^{(m+1)}) = H^{∗}(𝕊_{n}/𝕊_{n}^{[m]}; R_{∗}K(n)^{(m)}) = R_{∗},

which verifies the inductive step.

Remark. Since 𝕊_{n}/𝕊_{n}^{[k]} is a p-group, the vanishing of H^{s}(𝕊_{n}/𝕊_{n}^{[k]}; R_{∗}K(n)^{(k)}) for s > 0 actually implies that R_{∗}K(n)^{(k)} is a projective 𝔽_{q}[𝕊_{n}/𝕊_{n}^{[k]}]-module and hence is free because this group algebra is local. Of course, if we knew an explicit permutation basis we could avoid the cohomological calculations.

Corollary 2.5.

Ext_{K(n)_{∗}K(n)}^{s,∗}(K(n)_{∗}, K(n)_{∗}BP) = \begin{cases}v_{n}^{−1}BP_{∗}/I_{n} & if s = 0, \\ 0 & if s ≠ 0. \end{cases}

Proof. We apply the Theorem in the case

R_{∗} = 𝔽_{p^{n}} ⊗ (v_{n}^{−1}BP_{∗}/I_{n})[u]/(u^{p^{n}−1} − v_{n}).

A complication comes from the fact that we have extended v_{n}^{−1}BP_{∗}/I_{n} to R_{∗} by tensoring with 𝔽_{p^{n}} then adjoining a (p^{n} − 1) st root of v_{n}. This is a Galois extension whose Galois group Γ_{n} is identified with 𝔽_{p^{n}}^{×} ⋊ ℤ/n. Furthermore,

Ext_{K(n)_{∗}K(n)}^{s,∗}(K(n)_{∗}, K(n)_{∗}BP) = H_{c}^{∗}(𝕊_{n}; R_{∗}K(n))^{Γ_{n}}.

All of this is straightforward given the well understood identification of 𝕊_{n} with the group of strict units in the maximal order O_{n} of a certain p-adic division algebra 𝔻_{n}. In fact there is a proper action on R_{∗}K(n) of a semi-direct product extension of 𝕊_{n}, O_{n}^{×} ⋊ ℤ/n, where the full group of units O_{n}^{×} can itself be viewed as an extension

1 → 𝕊_{n} ⟶ O_{n}^{×} ⟶ 𝔽_{p^{n}}^{×} → 1.

We will use the notation u_{k} = η_{L}(v_{k}), v_{k} = η_{R}(v_{k}), where

E(n)_{∗} ⟶^{η_{L}} E(n)_{∗}BP ⟵^{η_{R}} BP_{∗}

are the left and right units. As v_{k} is a coaction primitive, multiplication by v_{k} is a homomorphism of E(n)_{∗}E(n)-comodules E(n)_{∗}BP ⟶ E(n)_{∗}BP. Hence E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, ⋯, v_{n−1}^{r_{n−1}}) is a quotient E(n)_{∗}E(n)-comodule of E(n)_{∗}BP. As I_{n} = (p, v_{1}, …, v_{n−1})◁BP_{∗} is an invariant ideal,

K(n)_{∗}BP = E(n)_{∗}BP/(p, v_{1}, …, v_{n−1}) = E(n)_{∗}BP/(p, u_{1}, …, u_{n−1}).

Theorem 2.6. Suppose that 1 ⩽ r_{k} ⩽ ∞ for k = 0, 1, …, n − 1. Then

Ext_{E(n)_{∗}E(n)}^{s,∗}(E(n)_{∗}, E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{n−1}^{r_{n−1}}))

= \begin{cases}v_{n}^{−1}BP_{∗}/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{n−1}^{r_{n−1}}) & if s = 0, \\ 0 & if s ≠ 0. \end{cases}

Proof. Suppose that all of the exponents r_{k} are finite. We will prove the result by induction on the number

excess(r_{0}, r_{1}, …, r_{n−1}) = ∑_{k=0}^{n−1}(r_{k} − 1) ⩾ 0.

Corollary 2.5 implies that the result is true for the unique sequence (1, 1, …, 1) of excess 0.

Now suppose that it holds whenever excess(r_{0}, r_{1}, …, r_{n−1}) < m. Then for at least one value of ℓ in the range 0 ⩽ ℓ ⩽ (n − 1) we have r_{ℓ} > 1. Let (r′_{0}, r′_{1}, …, r′_{n−1}) be the exponent sequence for which

r′_{k} = \begin{cases}r_{ℓ} − 1 & if k = ℓ, \\ r_{k} & if k ≠ ℓ. \end{cases}

Then there is an exact sequence of E(n)_{∗}E(n)-comodules

0 → E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{ℓ−1}^{r_{ℓ−1}}, v_{ℓ}, v_{ℓ+1}^{r_{ℓ+1}}, …, v_{n−1}^{r_{n−1}}) ⟶

E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{n−1}^{r_{n−1}}) ⟶ E(n)_{∗}BP/(p^{r′_{0}}, v_{1}^{r′_{1}}, …, v_{n−1}^{r′_{n−1}}) → 0.

Here the left-hand monomorphism is induced from the homomorphism of comodules

E(n)_{∗}BP ⟶ E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{n−1}^{r_{n−1}}); x ⟼ v_{ℓ}^{r_{ℓ}−1}x.

The exponent sequences of the left and right hand terms each have excess less that m and so we obtain a long exact sequence in Ext yielding the result for

E(n)_{∗}BP/(p^{r_{0}}, v_{1}^{r_{1}}, …, v_{n−1}^{r_{n−1}}).

Hence the result for sequences of arbitrary excess follows by induction.

When some of the r_{k} are infinite the result is proved by using appropriate colimits over finite exponent cases under obvious inclusions maps.

3. Some generalizations of the Hattori-Stong Theorem

Recall the well-known fact that

Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}BP) = \begin{cases}BP_{∗} & if s = 0, \\ 0 & if s ≠ 0. \end{cases}

Using this we can deduce that CSS(BP; BP) collapses at E_{2} = E_{∞}. This gives

Theorem 3.1 (Hattori-Stong Theorem in E(n)-theory). For each n ⩾ 1, the E(n)- Hurewicz homomorphism BP_{∗} ⟶ E(n)_{∗}BP is a split monomorphism.

The following observation will prove useful in the proof.

Lemma 3.2. For n ⩾ 1,

Ext_{E(n)_{∗}E(n)}^{0,∗}(E(n)_{∗}, E(n)_{∗}BP) ⊆ E(n)_{∗}BP

is the kernel of the ℤ_{(p)}-homomorphism

ψ − 1 ⊗ Id: E(n)_{∗}BP ⟶ E(n)_{∗}E(n) ⊗_{E(n)_{∗}} E(n)_{∗}BP

in which the domain and codomain are torsion-free; hence it is an additive summand.

Proof of Theorem 3.1. There is an exact sequence

0 → Ext_{BP_{∗}BP}^{0,∗} (BP_{∗}, BP_{∗}BP) ⟶ Ext_{E(0)_{∗}E(0)}^{0,∗}(E(0)_{∗}, E(0)_{∗}BP)

⟶^{d_{1}} Ext_{E(1)_{∗}E(1)}^{0,∗}(E(1)_{∗}, E(1)_{∗}BP/(p^{∞})) ⟶^{d_{1}} ⋯

By the well known fact that

Ext_{E(0)_{∗}E(0)}^{0,∗}(E(0)_{∗}, E(0)_{∗}BP) = p^{−1}BP_{∗},

this gives rise to an exact sequence

0 → Ext_{BP_{∗}BP}^{0,∗} (BP_{∗}, BP_{∗}BP) ⟶ p^{−1}BP_{∗}

⟶^{d_{1}} Ext_{E(1)_{∗}E(1)}^{0,∗}(E(1)_{∗}, E(1)_{∗}BP/(p^{∞})) ⟶^{d_{1}} ⋯

From CSS(BP; E(1)) we obtain an exact sequence

0 → BP_{∗} ⟶ p^{−1}BP_{∗} ⟶^{d_{1}} Ext_{E(1)_{∗}E(1)}^{0,∗}(E(1)_{∗}, E(1)_{∗}BP/(p^{∞}))

implying that the Hurewicz homomorphism BP_{∗}/(p^{∞}) ⟶ E(1)_{∗}BP/(p^{∞}) is a monomorphism. This in turn shows that the Hurewicz homomorphism BP_{∗} ⟶ E(1)_{∗}BP is indivisible, hence is a split monomorphism by Lemma 3.2.

For n ⩾ 1, CSS(BP; E(n)) gives rise to an exact sequence

0 → BP_{∗} ⟶ p^{−1}BP_{∗} ⟶^{d_{1}} Ext_{E(1)_{∗}E(1)}^{0,∗}(E(1)_{∗}, E(1)_{∗}BP/(p^{∞})) ⟶^{d_{1}} ⋯

⋯ ⟶^{d_{1}} Ext_{E(n)_{∗}E(n)}^{0,∗}(E(n)_{∗}, E(n)_{∗}BP/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})).

From this we obtain

E_{∞}^{0,0,∗} ≅ Ext_{E(n)_{∗}E(n)}^{0,∗}(E(n)_{∗}, E(n)_{∗}BP) = BP_{∗},

and together with Lemma 3.2, this proves the theorem.

Theorem 3.3. For n ⩾ 1,

Ext_{E(n)_{∗}E(n)}^{s,∗}(E(n)_{∗}, E(n)_{∗}BP) ≅ \begin{cases}BP_{∗} & if s = 0, \\ BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞}) & if s = n, \\ 0 & if s ≠ 0, n, \end{cases}

Proof. This is deduced using the exact sequence derived from CSS(BP; E(n)). It only remains to identify Ext^{n} which we do by induction on n.

When n = 0, CSS(BP; E(0)) has trivial E_{1}-term

E_{1}^{0,t,∗}(BP; E(0)) = \begin{cases}p^{−1}BP_{∗} & if t = 0, \\ 0 & if t ≠ 0. \end{cases}

This gives rise to the exact sequence

0 → BP_{∗} ⟶ p^{−1}BP_{∗} ⟶ BP_{∗}/(p^{∞}) → 0,

where BP_{∗}/(p^{∞}) is identified with the image of the differential d_{1} in CSS(BP; E(1)). Consider the case n = 1 and CSS(BP; E(1)). Here the E_{1}-term is

0 → p^{−1}BP_{∗} ⟶^{d_{1}} v_{1}^{−1}BP_{∗}/(p^{∞}) → 0

in which d_{1} is the evident composite

p^{−1}BP_{∗} ⟶ BP_{∗}/(p^{∞}) ⟶ v_{1}^{−1}BP_{∗}/(p^{∞}).

Hence the E_{1}-term fits into an exact sequence

0 → BP_{∗} ⟶ p^{−1}BP_{∗} ⟶^{d_{1}} v_{1}^{−1}BP_{∗}/(p^{∞}) ⟶ BP_{∗}/(p^{∞}, v_{1}^{∞}) → 0

and the E_{∞}-term is

E_{∞}^{1,r,∗}(BP; E(1)) = \begin{cases}BP_{∗} & if r = 0, \\ BP_{∗}/(p^{∞}, v_{1}^{∞}) & if r = 1, \\ 0 & if r ≠ 0, 1. \end{cases}

If the stated result is correct for n < m, then for n = m the E_{1}-term is the complex

0 → p^{−1}BP_{∗} ⟶^{d_{1}} v_{1}^{−1}BP_{∗}/(p^{∞}) ⟶^{d_{1}} ⋯

⋯ ⟶^{d_{1}} v_{n−1}^{−1}BP_{∗}/(p^{∞}, ⋯, v_{n−2}^{∞}) ⟶^{d_{1}} v_{n}^{−1}BP_{∗}/(p^{∞}, …, v_{n−1}^{∞}) → 0,

in which the differentials are evident composites of the form

v_{k}^{−1}BP_{∗}/(p^{∞} …, v_{k−1}^{∞}) ⟶ BP_{∗}/(p^{∞}, …, v_{k−1}^{∞}, v_{k}^{∞}) ⟶ v_{k+1}^{−1}BP_{∗}/(p^{∞}, …, v_{k}^{∞}).

From this it is clear that

E_{∞}^{n,t,∗}(BP; E(n)) = \begin{cases}BP_{∗} & if t = 0, \\ BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞}) & if t = n, \\ 0 & if t ≠ 0, n. \end{cases}

This establishes the inductive step.

It is interesting to note that the case n = 1 gives the diagram of exact sequences

0 ⟶ BP_{∗} ⟶ p^{−1}BP_{∗} ⟶^{d_{1}} v_{1}^{−1}BP_{∗}/(p^{∞})
↓  ↓  ↓
0 ⟶ BP_{∗} ⟶ p^{−1}BP_{∗} ⟶ E(1)_{∗}BP/(p^{∞})

This shows that the composite map

BP_{∗}/(p^{∞}) ⟶ v_{1}^{−1}BP_{∗}/(p^{∞}) ⟶ E(1)_{∗}BP/(p^{∞})

is a monomorphism. As this map agrees with the E(1)-theory Hurewicz map

BP_{∗}/(p^{∞}) ⟶ E(1)_{∗}BP/(p^{∞}),

the Hurewicz map BP_{∗} ⟶ E(1)_{∗}BP is indivisible, hence is a split monomorphism. This is a reformulation of the classical Hattori-Stong Theorem.

One analogue of Hattori-Stong for E(n)_{∗}BP is that the E(n)-theory Hurewicz map BP_{∗} ⟶ E(n)_{∗}BP is indivisible; this follows from knowledge of the spectral sequence CSS(BP; E(n)). A more delicate consequence is that the homomorphism

BP_{∗}/(p^{∞}, …, v_{n−1}^{∞}) ⟶ v_{n}^{−1}BP_{∗}/(p^{∞}, …, v_{n−1}^{∞}) ⟶ E(n)_{∗}BP/(p^{∞}, …, v_{n−1}^{∞})

is a monomorphism, hence

BP_{∗}/(p^{∞}, …, v_{n−2}^{∞}) ⟶ E(n)_{∗}BP/(p^{∞}, …, v_{n−2}^{∞})

is v_{n−1}-indivisible.

4. A change of Hopf algebroid spectral sequence

We will make use of a spectral sequence concerning a morphism φ: (A, Γ) ⟶ (B, Σ) of Hopf algebroids over a commutative unital ring 𝕜.

Let M be a left Γ-comodule and N a left Σ-comodule. We give Γ ⊗_{A} B and M ⊗_{A} B their natural right Σ-comodule structures. We also have the cotensor product analogue of the adjunction formula of [5]. We say that a A-B-bimodule L is a Γ-Σ-bicomodule if it admits left Γ-comodule and right Σ-comodule structures

Γ ⊗_{A} L ⟵^{ψ_{L}} L ⟶^{ψ_{R}} L ⊗_{B} Σ

which satisfy the compatibility condition

(1 ⊗ ψ_{R})ψ_{L} = (ψ_{L} ⊗ 1)ψ_{R}.

If L is a Γ-Σ bicomodule, then by coassociativity

M □_{Γ}(L □_{Σ} N) ≅ (M □_{Γ} L) □_{Σ} N,

implying

M □_{Γ} ((Γ ⊗_{A} B) □_{Σ} N) ≅ (M □_{Γ}(Γ ⊗_{A} B)) □_{Σ} N ≅ (M ⊗_{A} B) □_{Σ} N.  (4.1)

Theorem 4.1. If M is a flat A-module, there is a spectral sequence

E_{2}^{s,t,∗}(M, N) = Cotor_{Γ}^{s,∗}(M, Cotor_{Σ}^{t,∗}(Γ ⊗_{A} B, N)) ⟹ Cotor_{Σ}^{s+t,∗}(M ⊗_{A} B, N)

with differentials d_{r}: E_{2}^{s,t,∗}(M, N) ⟶ E_{2}^{s+r,t−r+1,∗}(M, N).

Proof. This spectral sequence is constructed by Ravenel [6, theorem A1.3.11]. However, we feel it is illuminating to view it as a Grothendieck composite functor spectral sequence as discussed in [7, 8]. For completeness we outline this description.

It is a standing assumption that for a Hopf algebroid (A, Γ), Γ is flat as a left or right A-module. Given a 𝕜-algebra homomorphism A ⟶ B, Γ ⊗_{A} B is flat as a right B-module and B ⊗_{A} Γ is flat as a left B-module. Also, if M is a flat A-module the functors Cotor_{Γ}^{s}(M, ) (s ⩾ 0) is computable using resolutions of the second variable by weakly injective extended comodules.

Consider the following functors and their right derived functors:

G: N ⇝ (Γ ⊗_{A} B) □_{Σ} N, R^{t}G( ) = Cotor_{Σ}^{t,∗}(Γ ⊗_{A} B, ),

from the category of left Σ-comodules to left Γ-comodules, and

F: L ⇝ M □_{Γ} L, R^{s}F ( ) = Cotor_{Γ}^{s,∗}(M, ),

from the category of left Γ-comodules to 𝕜-modules. Then by Equation (4.1),

F G( ) ≅ (M ⊗_{A} B) □_{Σ}( ),

with right derived functors

R^{n}F G( ) = Cotor_{Σ}^{n,∗}(M ⊗_{A} B, ).

For an extended Σ-comodule Σ ⊗_{B} N_{0},

G(Σ ⊗_{B} N_{0}) = (Γ ⊗_{A} B) □_{Σ}(Σ ⊗_{B} N_{0}) ≅ (Γ ⊗_{A} B) ⊗_{B} N_{0},

which is an extended Γ-comodule. So for s > 0,

R^{s}F (G(Σ ⊗_{B} N_{0})) = 0.

This is the condition required to obtain the Grothendieck Spectral Sequence

E_{2}^{s,t}(M, N) = R^{s}F ∘ R^{t}G(N) ⟹ R^{s+t}F G(N).

In the following corollary, Ext_{Γ} is taken on the category of left Γ-comodules, while Ext_{Σ} is taken on the category of right Σ-comodules.

Corollary 4.2. There is a spectral sequence

E_{2}^{s,t,∗} = Ext_{Γ}^{s,∗}(A, Ext_{Σ}^{t,∗}(B, Γ ⊗_{A} B)) ⟹ Ext_{Σ}^{s+t,∗}(B, B)

with differentials d_{r}: E_{2}^{s,t,∗} ⟶ E_{2}^{s+r,t−r+1,∗}.

Proof. Recalling the standard equivalences of functors

Ext_{Γ}^{s,∗}(A, ) ≅ Cotor_{Γ}^{s,∗}(A, ), Ext_{Σ}^{s,∗}(B, ) ≅ Cotor_{Σ}^{s,∗}( , B),

this follows from the Theorem with M = A (with its right coaction twisted into a left action) and N = B.

As a particular case of this result, if Γ ⊗_{A} B is a relatively injective right Σ-comodule then

Ext_{Σ}^{n,∗}(B, B) ≅ Ext^{n,∗}(A, (Γ ⊗_{A} B) □_{Σ} B) ≅ Ext^{n,∗}(A, (Γ ⊗_{A} B) □_{Σ} B),

which is the Shapiro Lemma of Miller & Ravenel [5].

Clarke [3] pointed out that the morphism of Hopf algebroids

τ: (MU_{∗}, MU_{∗}MU) ⟶ (KU_{∗}, KU_{∗}KU)

induced by the Todd orientation MU ⟶ KU gives an example for which this assumption fails since Ext_{KU_{∗}KU}^{1,∗}(KU_{∗}, KU_{∗}) has infinitely many elements surviving from

Ext_{MU_{∗}MU}^{0,∗}(MU_{∗}, Ext_{KU_{∗}KU}^{1,∗}(KU_{∗}, MU_{∗}KU))

in the relevant spectral sequence. At each prime p this can also be seen by considering the morphism

τ_{1} : (BP_{∗}, BP_{∗}BP) ⟶ (E(1)_{∗}, E(1)_{∗}E(1)).

induced by the natural orientation BP ⟶ E(1). More generally, we will describe the relevant spectral sequence for each of the natural maps

τ_{n}: (BP_{∗}, BP_{∗}BP) ⟶ (E(n)_{∗}, E(n)_{∗}E(n)).

Let SS(n) denote the spectral sequence of Corollary 4.2 associated with τ_{n}.

Theorem 4.3. For n ⩾ 1, SS(n) has

E_{2}^{s,t,∗} = \begin{cases}Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}) & if t = 0, \\ Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞})) & if t = n, \\ 0 & if t ≠ 0, n. \end{cases}

For k > 0 and p ∤ k,

E_{2}^{0,1,−2k(p−1)p^{r}} = Ext_{BP_{∗}BP}^{0,−2k(p−1)p^{r}}(BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}))

contains a summand isomorphic to ℤ/p^{r+1}, having as generator the class represented in the

usual chromatic notation by \frac{1}{p^{r+1}v_{1}^{kp^{r}}}. Such classes are non-vanishing infinite cycles which

generate the cyclic groups in E_{∞} observed by Clarke. There are of course other elements in this group, for example \frac{v_{2}}{pv_{1}} . Such elements are killed by the differential d_{2} and in fact

d_{2} : Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞})) ⟶ Ext_{BP_{∗}BP}^{s+2,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}))

is the composition of chromatic coboundaries

Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞})) ⟶^{δ_{1}} Ext_{BP_{∗}BP}^{s+1,∗} (BP_{∗}, BP_{∗}/(p^{∞}))

⟶^{δ_{0}} Ext_{BP_{∗}BP}^{s+2,∗} (BP_{∗}, BP_{∗}).

Thus we have

d_{2} (\frac{v_{2}}{pv_{1}} ) = β_{1}.

Similarly, when n ⩾ 1, the differential d_{n+1} is a composite of chromatic coboundaries

Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞})) ⟶^{δ_{n}}

Ext_{BP_{∗}BP}^{s+1,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n−1}^{∞})) ⟶^{δ_{n−1}} ⋯

⋯ ⟶^{δ_{0}} Ext_{BP_{∗}BP}^{s+n+1,∗}(BP_{∗}, BP_{∗}).

The elements

\frac{1}{p^{r_{0}}v_{1}^{r_{1}} … v_{n}^{r_{n}}} ∈ Ext_{BP_{∗}BP}^{0,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞})) (r_{j} ⩾ 1),

are non-vanishing infinite cycles, corresponding to non-trivial elements of

Ext_{E(n)_{∗}E(n)}^{n,∗}(E(n)_{∗}, E(n)_{∗}).

The general picture is that there is an exact sequence of the form

0 → F^{0,s,∗} ⟶ Ext_{BP_{∗}BP}^{s,∗} (BP_{∗}, BP_{∗}/(p^{∞}, v_{1}^{∞}, …, v_{n}^{∞})) ⟶^{δ_{0}···δ_{n}}

Ext_{BP_{∗}BP}^{s+n+1,∗}(BP_{∗}, BP_{∗}) ⟶ F^{n,s+n+1,∗} → 0,

where the filtration groups F^{k,∗,∗} fit into short exact sequences

0 → F^{0,s,∗} ⟶ F^{n,s,∗} ⟶ Ext_{E(n)_{∗}E(n)}^{s,∗}(E(n)_{∗}, E(n)_{∗}) → 0.

References

[1] A. Baker, A version of the Landweber filtration theorem for v_{n}-periodic Hopf algebroids, Osaka J. Math. 32 (1995), 689–99.

[2] A. Baker, On the Adams E_{2}-term for elliptic cohomology, to appear in Proceedings of the 1999 Boulder Conference in Contemp. Math.

[3] F. Clarke, On Ext_{K_{∗}(K)}(π_{∗}(K), K_{∗}(MU)), Proc. Amer. Math. Soc. 74 (1979), 156–160.

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

[5] H. R. Miller & D. C. Ravenel, Morava stabilizer algebras and localization of Novikov’s E_{2}-term, Duke Math. J. 44 (1977), 433–47.

[6] D. C. Ravenel, Complex Cobordism and the Stable Homotopy Groups of Spheres, Academic Press (1986).

[7] S. Shatz, Profinite Groups, Arithmetic and Geometry, Annals of Mathematics Studies 67 (1972), Princeton University Press.

[8] C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press (1994).

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2000/n3/n3.(dvi,ps,dvi.gz,ps.gz)

Andrew Baker a.baker@maths.gla.ac.uk  http://www.maths.gla.ac.uk/∼ajb

Department of Mathematics University of Glasgow Glasgow G12 8QW Scotland.