Homology, Homotopy and Applications, vol.4(1), 2002, pp.163–173

BRAVE NEW HOPF ALGEBROIDS AND EXTENSIONS OF MU-ALGEBRAS

ANDREW BAKER and ALAIN JEANNERET

(communicated by J. D. Stasheff)

Abstract We apply recent work of A. Lazarev which develops an obstruction theory for the existence of R-algebra structures on Rmodules, where R is a commutative S-algebra. We show that certain MU-modules have such A_{∞} structures. Our results are often simpler to state for the related BP -modules under the currently unproved assumption that BP is a commutative Salgebra. Part of our motivation is to clarify the algebra involved in Lazarev’s work and to generalize it to other important cases. We also make explicit the fact that BP admits an MU-algebra structure as do E(n) and { \atop E(n)}, in the latter case rederiving and strengthening older results of U. Würgler and the first author.

Introduction

Recent work of A. Lazarev [11] has developed an obstruction theory for the existence of R-algebra structures on R-modules, where R is a commutative S-algebra in the sense of [8]. In [4], ‘brave new Hopf algebroids’ were discussed and related to the Adams Spectral Sequence for R-modules, thus generalizing the classical homotopy theoretic version described in [14]. In the present work we again consider some of the main examples of that paper and apply Lazarev’s techniques to show that certain MU-modules have such A_{∞} structures. In fact, our results are often simpler to state for the related BP -modules under the assumption that BP is a commutative S-algebra. However this currently seems to remain unproved, a preprint by I. Kriz showing this apparently has so far unfilled gaps. We often state BP analogues but normally work over MU. Part of our motivation is to clarify the algebra involved in [11] and to show how it generalizes to some other important cases. We also make explicit the fact that BP admits an MU-algebra structure as do E(n) and { \atop E(n)}, in the latter case rederiving and strengthening results of [2, 5].

As a matter of history we remark that most of the material described here originated during the summer of 2000; subsequent preprints by P. Goerss, M. Hopkins

The first author would like to thank the Mathematisches Institut, Universität Bern for providing a stimulating environment in which much of this work was carried out. We would also like to thank Christian Ausoni for helpful comments. Received April 23, 2002, revised July 23, 2002; published on November 25, 2002. 2000 Mathematics Subject Classification: 55N20, 55P43, 55N45, 55S35, 55T25. Key words and phrases: R-algebra, Hopf algebroid, obstruction theory. \begin{matrix}c \\ ◯ \end{matrix} 2002, Andrew Baker and Alain Jeanneret. Permission to copy for private use granted.

and A. Lazarev have contained stronger results on the realizability of BP and other MU-algebras using more developed machinery. However, we feel that the present approach provides a good illustration of the power of the obstruction theory implicit in [11] applied to some examples of fundamental importance to homotopy theorists.

1. Brave new Hopf algebroids

Throughout, we will work in a good category of spectra 𝒮 such as 𝒮 [𝕃] of [8]. Associated to this is the category of S-modules ℳ_{S} and its derived homotopy category 𝒟_{S}. If R is a commutative S-algebra in the sense of [8], there is an associated category of R-modules ℳ_{R} and its derived category 𝒟_{R}.

The following notions were introduced in [8]. Let A be an R-module with a unit η: R ⟶ A and product φ: A∧_{R}A ⟶ A. Then A, or more precisely, (A, φ, η), is an R-algebra if the following diagrams commute in ℳ_{R}.

\xymatrix{A∧_{R}A∧_{R}A \ar[r]^{φ∧id} \ar[d]_{id∧φ} & A∧_{R}A \ar[d]^{φ} & R∧_{R}A \ar[r]^{η∧id} \ar[dr]_{μ} & A∧_{R}A \ar[d]^{φ} & A∧_{R}R \ar[l]_{id∧η} \ar[dl]^{μ} \\ A∧_{R}A \ar[r]_{φ} & A &  & A}  (1.1a)

A is commutative if the following diagram commutes in ℳ_{R}.

\xymatrix{A∧_{R}A \ar[rr]^{τ} \ar[dr]_{φ} &  & A∧_{R}A \ar[dl]^{φ} \\  & A}  (1.1b)

There are also weaker conditions for such a product. If the diagrams of (1.1a) commute in 𝒟_{R} then A is an R-ring spectrum, and if (1.1b) also commutes in 𝒟_{R} then A is a commutative R-ring spectrum. In that case the smash product A∧_{R}A is also a commutative R-ring spectrum and is also naturally an A-algebra spectrum in two different ways induced from the left and right units

A ⟶^{≅} A∧_{R}R ⟶ A∧_{R}A ⟵ R∧_{R}A ⟵^{≅} A.

As discussed in [4], we have

Theorem 1.1. Let A be a commutative R-ring spectrum. If A_{∗}^{R}A is flat as a left, or equivalently as a right, A_{∗}-module, then

(i) (A_{∗}, A_{∗}^{R}A) is a Hopf algebroid over R_{∗};

(ii) for any R-module M, A_{∗}^{R}M is a left A_{∗}^{R}A-comodule.

Such Hopf algebroids are commonly encountered and we will meet important examples in the rest of this paper.

2. Some examples

Recall that the Thom spectrum MU is a commutative S-algebra. At the time of writing it seems not to be known whether the Brown-Peterson spectrum BP for a prime p is a commutative S-algebra. We will state many of our results both in terms of MU-algebras, and also in parallel in terms of BP -algebras on the assumption that these will eventually prove to be of interest if BP is indeed shown to be a commutative S-algebra.

We will assume that a choice of polynomial generators x_{r} ∈ MU_{2r} (r ⩾ 1) for MU_{∗} has been made. We also assume that a rational prime p = x_{0} > 0 has been chosen.

For any subset S ⊆ {x_{r} : r ⩾ 0}, the sequence of elements of S conventionally ordered by increasing degree is regular. By successively killing the homotopy elements by forming mapping cones in the category of MU-modules we can form a MU-module spectrum MU/S. More precisely, this is a CW-cell MUmodule whose cells are indexed by the monomial basis of the exterior algebra Λ_{MU_{∗}}(τ_{r} : x_{r} ∈ S) in which τ_{r} has bidegree (1, 2r). The cell corresponding to τ_{r} has dimension 2r + 1 and an attaching map S_{MU}^{2r} ⟶ (MU/S)^{(2r)} in the homotopy class of x_{r} ∈ π_{2r}(MU/S)^{(2r)}. We make this exterior algebra into an MU_{∗}-dga with differential d for which d τ_{r} = x_{r}. Of course, Λ_{MU_{∗}}(τ_{r} : x_{r} ∈ S)_{∗} is a Koszul complex providing a free resolution of the MU_{∗}-module π_{∗}MU/S = MU_{∗}/(S),

Λ_{MU_{∗}}(τ_{r} : x_{r} ∈ S)_{∗} ⟶ MU_{∗}/(S) → 0.

Recall from [8], the Künneth Spectral Sequence

E_{2}^{r,s} = Tor_{r,s}^{MU_{∗}}(MU_{∗}/(S), MU_{∗}/(S)) ⟹ MU/S_{r+s}^{MU}MU/S^{op}.  (2.1)

We will need to consider situation where MU/S is an MU-ring spectrum as well as the opposite MU-ring spectrum MU/S^{op}. By [4, lemma 1.3], (2.1) is then multiplicative. The following result is proved in [12, lemma 2.6].

Theorem 2.1. Suppose that MU/S is an MU-ring spectrum. Then the Künneth Spectral Sequence (2.1) for MU/S_{∗}^{MU}MU/S^{op} collapses at E_{2} to give

MU/S_{∗}^{MU}MU/S^{op} = Λ_{MU_{∗}/S}(τ_{r} : x_{r} ∈ S),

where the exterior generators satisfy τ_{r} ∈ MU/S_{2r+1}^{MU}MU/S^{op}.

If MU/S is commutative then of course MU/S^{op} = MU/S as MU-ring spectra; Theorem 2.1 then follows directly from [4, lemma 1.3]. See Strickland [16] for details on when this commutativity condition on MU/S is satisfied. In particular, when x_{0} = p is an odd prime, all the standard MU-module spectra of this form such as MU ⟨n⟩, MU ⟨n⟩, BP and BP ⟨n⟩ are commutative MU-ring spectra, while when x_{0} = 2, care needs to be taken so that MU ⟨n⟩ and BP ⟨n⟩ need to be replaced by spectra constructed using non-standard polynomial generators for MU_{∗}.

As a particular case, let us assume that we have chosen S to contain a complete set of generators of MU_{∗} except for x_{p^{k}−1} (0 ⩽ k ⩽ n). We then have an MUmodule MU ⟨n⟩. Here we allow the possibility of x_{0} = p > 0 corresponding to the case MU ⟨−1⟩.

Proposition 2.2.

MU ⟨n⟩_{∗}^{MU} MU ⟨n⟩^{op} = Λ_{MU⟨n⟩_{∗}}(τ_{r} : r ≠ p^{k} − 1, 0 ⩽ k ⩽ n).

Let p > 0 be a prime. If BP is a commutative S-algebra then we can form the p-localization of MU ⟨n⟩ as a BP -module, denoted BP ⟨n⟩, for which the following holds.

Proposition 2.3.

BP ⟨n⟩_{∗}^{BP} BP ⟨n⟩^{op} = Λ_{BP ⟨n⟩_{∗}}(τ_{p^{k}−1} : k ⩾ n + 1).

Notice that for the special (commutative) case BP ⟨0⟩ = Hℤ_{(p)},

Hℤ_{(p)}_{∗}^{BP} Hℤ_{(p)} = Λ_{ℤ_{(p)}}(τ_{p^{k}−1} : k ⩾ 1).

Similarly, in the case BP ⟨−1⟩ = H𝔽_{p} we have

H𝔽_{p}_{∗}^{BP} H𝔽_{p} = Λ_{𝔽_{p}}(τ_{p^{k}−1} : k ⩾ 0)

and the natural map

H𝔽_{p}_{∗}^{S}H𝔽_{p} ⟶ H𝔽_{p}_{∗}^{BP} H𝔽_{p}

corresponds to the quotient of the dual Steenrod algebra by the Hopf ideal generated by the Milnor elements ζ_{i} (i ⩾ 1); this quotient is well known to be a primitively generated exterior algebra.

3. Extending MU-algebra structures

As a starting point we recall from [8] that for any commutative ring R, the natural homomorphism MU_{0} = ℤ ⟶ R lifts to a morphism of S-algebras MU ⟶ HR, where HR is an Eilenberg-MacLane MU-module. HR is known to be a commutative MU-algebra. We set MU ⟨0⟩ = Hℤ since the latter realizes the spectrum discussed above.

Theorem 3.1. Let n ⩾ 0 and suppose that MU ⟨n⟩ admits the structure of an MUalgebra. Then MU ⟨n + 1⟩ admits an MU-algebra structure so there is a morphism of MU-algebras

MU ⟨n + 1⟩ ⟶ MU ⟨n⟩

realizing the natural ring homomorphism MU ⟨n + 1⟩_{∗} ⟶ MU ⟨n⟩_{∗}.

If BP is a commutative S-algebra and BP ⟨n⟩ admits the structure of a BP - algebra, then BP ⟨n + 1⟩ admits a BP -algebra structure so that there is a morphism of BP -algebras

BP ⟨n + 1⟩ ⟶ BP ⟨n⟩

realizing the natural ring homomorphism BP ⟨n + 1⟩_{∗} ⟶ BP ⟨n⟩_{∗}.

Proof. Set MU ⟨n + 1; 1⟩ = MU ⟨n⟩. We will prove by induction that the following holds for each m ⩾ 1:

• If MU ⟨n + 1; m⟩ is an MU-algebra which as an MU_{∗}-module satisfies

π_{∗}MU ⟨n + 1; m⟩ = MU ⟨n + 1⟩_{∗} /((x_{p^{n+1}−1})^{m}),

then there is an MU-algebra MU ⟨n + 1; m + 1⟩ for which

π_{∗}MU ⟨n + 1; m + 1⟩ = MU ⟨n + 1⟩_{∗} /((x_{p^{n+1}−1})^{m+1})

and a morphism of MU-algebras

MU ⟨n + 1; m + 1⟩ ⟶ MU ⟨n + 1; m⟩

realizing the evident homomorphism

MU ⟨n + 1; m + 1⟩_{∗} ⟶ MU ⟨n + 1; m⟩_{∗} .

There is a short exact sequence of MU_{∗}-modules

0 → Σ^{2(p^{n+1}−1)m}MU ⟨n⟩_{∗} ⟶ MU ⟨n + 1⟩_{∗} /((x_{p^{n+1}−1})^{m+1}) ⟶ MU ⟨n + 1⟩_{∗} /((x_{p^{n+1}−1})^{m}) → 0,

so we need to show that this is realized by an extension of MU-algebras, for which the fibre of the map MU ⟨n + 1; m + 1⟩ ⟶ MU ⟨n + 1; m⟩ is Σ^{2(p^{n+1}−1)m}MU ⟨n⟩.

Following [11], we need to determine the Hochschild cohomology

HH_{MU_{∗}}^{∗ ∗}(MU ⟨n + 1; m⟩_{∗} , MU ⟨n⟩_{∗}).

We begin by calculating MU ⟨n + 1; m⟩_{∗}^{MU} MU ⟨n + 1; m⟩^{op}. There is a Koszul resolution

Λ_{MU_{∗}}(τ[n + 1; m]_{r} : 1 ⩽ r ≠ p^{k} − 1, 1 ⩽ k ⩽ n + 1)_{∗} ⟶ MU ⟨n + 1; m⟩_{∗} → 0

which is an MU_{∗}-dga with

d τ[n + 1; m]_{r} = \begin{cases}(x_{p^{n+1}−1})^{m} & if r = p^{n+1} − 1, \\ x_{r} & otherwise. \end{cases}

Tensoring over MU_{∗} with MU ⟨n + 1; m⟩_{∗} and taking homology, we find that the Künneth Spectral Sequence of (2.1) collapses to give

MU ⟨n + 1; m⟩_{∗}^{MU} MU ⟨n + 1; m⟩^{op} =

Λ_{MU⟨n+1;m⟩}(τ[n + 1; m]_{r} : 1 ⩽ r ≠ p^{k} − 1, 1 ⩽ k ⩽ n + 1). (3.1)

Similarly, using a standard divided power complex we find that

HH_{MU_{∗}}^{∗ ∗}(MU ⟨n + 1; m⟩_{∗} , MU ⟨n⟩_{∗}) =

MU ⟨n⟩_{∗} [y[n + 1; m]_{r} : 1 ⩽ r ≠ p^{k} − 1, 1 ⩽ k ⩽ n + 1]. (3.2)

Finally we require the fact that

Ext_{MU_{∗}}^{∗ ∗}(MU ⟨n + 1; m⟩_{∗} , MU ⟨n⟩_{∗}) =

Λ_{MU⟨n⟩_{∗}}(Q[n + 1; m]^{r} : 1 ⩽ r ≠ p^{k} − 1, 1 ⩽ k ⩽ n + 1), (3.3) which is obtained using the above Koszul resolution. Here

Q[n + 1; m]^{r} ∈ Ext_{MU_{∗}}^{1,∗}(MU ⟨n + 1; m⟩_{∗} , MU ⟨n⟩_{∗})

is the element ‘dual’ to τ[n + 1; m]_{r} with respect to the MU ⟨n⟩_{∗}-basis for

Λ_{MU_{∗}}(τ[n + 1; m]_{r} : 1 ⩽ r ≠ p^{k} − 1, 1 ⩽ k ⩽ n + 1)_{1,∗}

consisting of the τ_{r}’s. Then

bideg Q[n + 1; m]^{r} = \begin{cases}(1, 2(p^{n+1} − 1)m) & if r = p^{n+1} − 1, \\ (1, 2r) & otherwise. \end{cases}

Consider the element y[n + 1; m]_{p^{n+1}−1}. By similar arguments to those of [11], this element gives rise to an element of

Der_{MU}^{2(p^{n+1}−1)m+1}(MU ⟨n + 1; m⟩ , MU ⟨n⟩)

and moreover it corresponds to Q[n + 1; m]^{p^{n+1}−1} under the map

Der_{MU}^{2(p^{n+1}−1)m+1}(MU ⟨n + 1; m⟩ , MU ⟨n⟩) ⟶

Ext_{MU_{∗}}^{1,2(p^{n+1}−1)m}(MU ⟨n + 1; m⟩_{∗} , MU ⟨n⟩_{∗})

defined in that paper. Using the standard identification of elements of Ext^{1} with extensions, it is easy to see that Q[n + 1; m]^{r} specifies the explicit extension of MU_{∗}-modules

0 → Σ^{2(p^{n+1}−1)m}MU ⟨n⟩_{∗} ⟶ MU ⟨n + 1; m + 1⟩_{∗} ⟶ MU ⟨n + 1; m⟩_{∗} → 0

corresponding to the fibration sequence

Σ^{2(p^{n+1}−1)m}MU ⟨n⟩ ⟶ MU ⟨n + 1; m + 1⟩ ⟶ MU ⟨n + 1; m⟩

of MU-modules. By Lazarev’s algebra extension machinery, there is a morphism of MU-algebras MU ⟨n + 1; m + 1⟩ ⟶ MU ⟨n + 1; m⟩ which realizes the natural map in homotopy. We require the following lemma which is essentially a generalization of a result to be found in [7], see [6, 10, 17]. To apply this we may need to replace maps by homotopic maps which are fibrations.

Lemma 3.2. Let R be an S-algebra.

(a) If

M_{0} ⟶ M_{1} ⟶ ⋯ ⟶ M_{n} ⟶ ⋯

is a directed system of cofibrations in ℳ_{R} then hocolim_{n} M_{n} is weakly equivalent

to colim_{n} M_{n}.

(b) If

N_{0} ⟵ N_{1} ⟵ ⋯ ⟵ N_{n} ⟵ ⋯

is a directed system of fibrations in ℳ_{R} then holim_{n} M_{n} is weakly equivalent to

lim_{n} M_{n}.

Let

MU ⟨n + 1; ∞⟩ = lim_{m} MU ⟨n + 1; m⟩ ,

which is equivalent in 𝒟_{R} to holim_{m} MU ⟨n + 1; m⟩. We obtain an MU-algebra structure on MU ⟨n + 1; ∞⟩ and the Milnor exact sequence

0 → lim_{m}^{1}MU ⟨n + 1; m⟩_{∗} ⟶ MU ⟨n + 1; ∞⟩_{∗} ⟶ lim_{m} MU ⟨n + 1; m⟩_{∗} → 0,

gives

lim_{m} MU ⟨n + 1; m⟩_{∗} = MU ⟨n + 1⟩_{∗} , lim_{m}^{1}MU ⟨n + 1; m⟩_{∗} = 0,

since each map MU ⟨n + 1; m + 1⟩_{∗} ⟶ MU ⟨n + 1; m⟩_{∗} is surjective. Hence

MU ⟨n + 1; ∞⟩_{∗} = MU ⟨n + 1⟩_{∗} .

An obstruction theory argument based on the fact that MU ⟨n + 1⟩ is a cell MUmodule provides an MU-module map MU ⟨n + 1⟩ ⟶ MU ⟨n + 1; ∞⟩ which is a weak equivalence.

The BP version follows by a parallel argument.

Theorem 3.3. There is a tower of MU-algebras

Hℤ = MU ⟨0⟩ ⟵ MU ⟨1⟩ ⟵ ⋯ ⟵ MU ⟨n + 1⟩ ⟵ ⋯

whose limit MU ⟨∞⟩ is an MU-algebra with MU ⟨∞⟩_{∗} = MU_{∗}/(x_{r} : 1 ⩽ r ≠ p^{k} − 1).

If BP is a commutative S-algebra, there is a tower of BP -algebras

Hℤ_{(p)} = BP ⟨0⟩ ⟵ BP ⟨1⟩ ⟵ ⋯ ⟵ BP ⟨n⟩ ⟵ BP ⟨n + 1⟩ ⟵ ⋯

whose limit BP ⟨∞⟩ is a BP -algebra and is equivalent to BP as a BP -ring spectrum.

Proof. The starting point is the observation of [8] that the unit MU ⟶ Hℤ = MU ⟨0⟩ is an MU-algebra morphism so that realizes the augmentation MU_{∗} ⟶ MU_{0} = ℤ. Theorem 3.1 and an induction on n shows that the tower exists as claimed.

In the Milnor exact sequence

0 → lim_{n}^{1}MU ⟨n⟩_{∗} ⟶ MU ⟨∞⟩_{∗} ⟶ lim_{n} MU ⟨n⟩_{∗} → 0,

we have

lim_{n} MU ⟨n⟩_{∗} = MU_{∗}/(x_{r} : 1 ⩽ r ≠ p^{k} − 1), lim_{n}^{1}MU ⟨n⟩_{∗} = 0

since each map MU ⟨n + 1⟩_{∗} ⟶ MU ⟨n⟩_{∗} is surjective. Hence

MU ⟨∞⟩_{∗} = MU_{∗}/(x_{r} : 1 ⩽ r ≠ p^{k} − 1).

In the BP case, the unit map BP ⟶ BP ⟨∞⟩ is a weak equivalence and so is an equivalence of BP -ring spectra.

Let ε: MU ⟶ BP be the map inducing the algebraic projection (MU_{∗})_{(p)} ⟶ BP_{∗} due to Quillen and described in [1]. It is well known that the image of each Hazewinkel generator v_{n} ∈ BP_{2(p^{n}−1)} in (MU_{∗})_{(p)} actually lies in MU_{∗} and we refer to this image as v_{n} ∈ MU_{2(p^{n}−1)}. Recall also that there is an MU-module map η: MU ⟶ BP where BP can be given the structure of MU-ring spectrum with unit η [8, 16].

Theorem 3.4. Choose the generators x_{r} so that x_{r} ∈ ker ε for r ≠ p^{k} − 1 and x_{p^{k}−1} = v_{k}. Then there is a map of MU-ring spectra θ: BP ⟶ MU ⟨∞⟩_{(p)} factoring the unit MU ⟶ MU ⟨∞⟩_{(p)} as

MU ⟶ BP ⟶^{θ} MU ⟨∞⟩_{(p)} ,

where the first map is the unit for BP. θ is an equivalence of MU-ring spectra or equivalently of MU_{(p)}-ring spectra BP ⟶ MU ⟨∞⟩_{(p)}. Hence the ring spectrum BP can be realized as an MU-algebra or equivalently as an MU_{(p)}-algebra.

Proof. Since BP is a cell MU_{(p)}-module with cells corresponding to the generators in the Koszul resolution of (MU_{(p)})_{∗}/(x_{r} : 1 ⩽ r ≠ p^{k} − 1), the map θ can be constructed by induction to satisfy the required properties up to homotopy.

Corollary 3.5. Each spectrum BP ⟨n⟩ (1 ⩽ n) can be realized as an MU-algebra or equivalently as an MU_{(p)}-algebra and the natural map BP ⟨n + 1⟩ ⟶ BP ⟨n⟩ can be realized as a morphism of algebras.

Proof. Obstruction theory gives an MU-module map BP ⟨n⟩ ⟶ MU ⟨n⟩_{(p)} which is visibly a weak equivalence.

4. Some MU-algebras obtained by localization

From [8, 18] we know that on inverting an element u ∈ MU_{2d} we obtain the Bousfield localization at MU[u^{−1}] as

M ⟶ L_{MU[u^{−1}]}^{MU} M = MU[u^{−1}] ∧_{MU}M

for any MU-module M. Furthermore, if A is an MU-algebra then

A ⟶ L_{MU[u^{−1}]}^{MU} A = MU[u^{−1}] ∧_{MU}A

is a morphism of MU-algebras. Similar considerations would apply if BP were a commutative S-algebra. We will use the notation

MU(n) = L_{MU[x_{p^{n}−1}^{−1}]}^{MU} MU ⟨n⟩ , BP (n) = L_{BP [v_{n}^{−1}]}^{BP} BP ⟨n⟩ ,

where

MU(n)_{∗} = MU ⟨n⟩_{∗} [x_{p^{n}−1}^{−1}], BP (n)_{∗} = BP ⟨n⟩ [v_{n}^{−1}].

Proposition 4.1. For each prime p and n ⩾ 1, there is a localization morphism of MU-algebras

MU ⟨n⟩ ⟶ L_{MU[x_{p^{n}−1}^{−1}]}^{MU} MU ⟨n⟩ = MU(n)

and an equivalence of MU-ring spectra MU(n)_{(p)} ⟶ E(n). Hence also E(n) admits the structure of an MU-algebra. Therefore E(n) also admits the structure of an S-algebra which is a commutative S-ring spectrum.

If BP is a commutative S-algebra, then there is a localization morphism of BP - algebras

BP ⟨n⟩ ⟶ L_{BP [v_{n}^{−1}]}^{BP} BP ⟨n⟩ = BP (n)

and an equivalence of BP -ring spectra BP (n) ⟶ E(n). Hence E(n) admits the structure of a BP -algebra.

Given E(n) as an MU or BP -algebra, we can form the quotient module E(n)/I_{n}^{k} obtained by killing all the monomials v_{0}^{r_{0}}v_{1}^{r_{1}} ⋯ v_{n−1}^{r_{n−1}} with ∑_{i=0}^{n−1} r_{i} = k. In [2,

5], these spectra were constructed as { \atop E(n)}-modules for an appropriate S-algebra structure; here { \atop E(n)} is the I_{n}-adic completion of the S-module E(n). Our present approach shows that each of the natural maps E(n)/I_{n}^{k+1} ⟶ E(n)/I_{n}^{k} can be realized as a map of E(n)-modules. We obtain K(n) as E(n)/I_{n}. By [2, 5], the tower

E(n)/I_{n} ⟵ E(n)/I_{n}^{2} ⟵ ⋯ ⟵ E(n)/I_{n}^{k} ⟵ E(n)/I_{n}^{k+1} ⟵ ⋯  (4.1)

has { \atop E(n)} for its limit as an S-ring spectrum, however here we are working with MU, BP or E(n)-modules. This gives a new proof of the existence of an S-algebra structure on { \atop E(n)}. We also have { \atop E(n)} ≃ L_{K(n)}^{S} E(n), so this result can also be proved by another application of the Bousfield localization theory of S-algebras.

Proposition 4.2. The natural map E(n) ⟶ { \atop E(n)} is a morphism of MU-algebras.

Proof. We need to make use of [8, XIII.1.8]; actually, in the statement of this result it is assumed that A is a commutative R-algebra but this is unnecessary and the following correct formulation occurs as [13, XXIII.6.5].

Lemma 4.3. Let R be a commutative S-algebra, A be an R-algebra and E be an Rmodule and M an A-module. Then the (A∧_{R}E)^{A}-localization map λ: M ⟶ L_{A∧_{R}E}^{A}M

is an E^{R}-localization map for M. Hence there is a weak equivalence of R-modules

L_{E}^{R}M ⟶ L_{A∧_{R}E}^{A}M.

Now take R = S, A = MU, E = K(n) and M = E(n). Then we are done since there is an equivalence of MU-modules

{ \atop E(n)} ≃ L_{MU∧_{S}K(n)}^{MU} E(n).

For reference we determine the Bousfield class of the MU-module MU∧_{S}K(n). Mak-

ing use of ideas similar to those in the Appendix of [3] (see especially Corollary A.2) we find that π_{∗}MU∧_{S}K(n) = MU_{∗}K(n) is a free MU_{∗}/I_{n}[v_{n}^{−1}]-module. By a standard argument, there is a weak equivalence of MU-modules

⋁_{α} Σ^{2d_{α}}MU/I_{n}[v_{n}^{−1}] ≃ MU∧_{S}K(n),

hence MU∧_{S}K(n) is Bousfield equivalent to MU/I_{n}[v_{n}^{−1}].

Actually, Theorem 6.4 of [4] also tells us that there is an equivalence of MUmodules

L_{MU/I_{n}[v_{n}^{−1}]}^{MU} E(n) ≃ { \atop E(n)};

indeed the tower of (4.1) is constructed in [4] as a tower of MU-modules and its homotopy limit is shown to be L_{MU/I_{n}[v_{n}^{−1}]}^{MU} E(n).

Having obtained { \atop E(n)} as an MU-algebra, ideas of [12, 15] can be used to show that there is an MU-algebra { \atop E(n)}W(𝔽_{p^{n}}) obtained by adjoining a primitive (p^{n}−1)- st root of unity to { \atop E(n)}_{∗}. The 2-periodic version of this spectrum is known to be a commutative S-algebra by work of Hopkins, Miller and Goerss [9].

References

[1] J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press (1974).

[2] A. Baker, A_{∞} stuctures on some spectra related to Morava K-theory, Quart. J. Math. Oxf. 42 (1991), 403–419.

[3] A. Baker, On the Adams E_{2}-term for elliptic cohomology, Contemp. Math. 271 (2001), 1–15.

[4] A. Baker & A. Lazarev, On the Adams Spectral Sequence for R-modules, Algebraic & Geometric Topology 1 (2001), 173-99.

[5] A. Baker & U. Würgler, Bockstein operations in Morava K-theory, Forum Math. 3 (1991), 543–60.

[6] J. M. Boardman, Conditionally convergent spectral sequences, Contemp. Math. 239 (1999) 49–84.

[7] A. K. Bousfield & D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 (1972).

[8] A. Elmendorf, I. Kriz, M. Mandell & J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47 (1999).

[9] P. G. Goerss & M. J. Hopkins, Realizing commutative ring spectra as E_{∞} ring spectra, preprint.

[10] M. Hovey, Model Categories, Mathematical Surveys and Monographs 63 (1999).

[11] A. Lazarev, Homotopy theory of A_{∞} ring spectra and applications to MUmodules, K-theory 24 (2001), 243–81.

[12] A. Lazarev, Towers of MU-algebras and the generalized Hopkins-Miller the-

orem, preprint math.AT/0209386.

[13] J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics 91 (1996).

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

[15] R. Schwänzl, R. M. Vogt & F. Waldhausen, Adjoining roots of unity to E_{∞} ring spectra in good cases– a remark, Contemp. Math. 239 (1999), 245–249,

[16] N. P. Strickland, Products on MU-modules, Trans. Amer. Math. Soc. 351 (1999), 2569–2606.

[17] R. M. Vogt, Homotopy limits and colimits, Math. Zeit. 134 (1973), 11–52.

[18] J. J. Wolbert, Classifying modules over K-theory spectra, J. Pure Appl. Algebra 124 (1998), 289–323.

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/2002/n1a9/v4n1a9.(dvi,ps,pdf)

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

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

Alain Jeanneret alain.jeanneret@math-stat.unibe.ch

Mathematisches Institut, Universität Bern, Bern CH 3012, Switzerland.