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 thethe following diagrams commute in ℳ_{R}.

[figure: A∧_{R}A∧_{R}A; φ∧id; id∧φ; A∧_{R}A; φ; R∧ 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