BRAVE NEW HOPF ALGEBROIDS AND EXTENSIONS OF M U-ALGEBRAS

BRAVE NEW HOPF ALGEBROIDS AND EXTENSIONS OF M U-ALGEBRAS

ANDREW BAKER and ALAIN JEANNERET

(communicated by J. D. Stasheff) Abstract

We apply recent work of A. Lazarev which develops an ob- struction theory for the existence ofR-algebra structures onR- modules, whereR is a commutativeS-algebra. We show that certainM U-modules have suchA_∞structures. Our results are often simpler to state for the related BP-modules under the currently unproved assumption that BP is a commutativeS- algebra. 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 thatBP admits anM U-algebra structure as doE(n) andE(n), in the latter case rederiving and[ strengthening older results of U. W¨urgler and the first author.

Introduction

Recent work of A. Lazarev [11] has developed an obstruction theory for the exis- tence ofR-algebra structures onR-modules, whereRis a commutativeS-algebra in the sense of [8]. In [4], ‘brave new Hopf algebroids’ were discussed and related to the Adams Spectral Sequence forR-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 cer- tainM U-modules have suchA_∞structures. In fact, our results are often simpler to state for the relatedBP-modules under the assumption thatBP 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 overM U. 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 anM U-algebra structure as doE(n) andE(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 orig- inated during the summer of 2000; subsequent preprints by P. Goerss, M. Hopkins

The first author would like to thank the Mathematisches Institut, Universit¨at 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.

c 2002, Andrew Baker and Alain Jeanneret. Permission to copy for private use granted.

and A. Lazarev have contained stronger results on the realizability ofBP and other M U-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 spectraS such as S[L] of [8].

Associated to this is the category ofS-modulesMS and its derived homotopy cate- goryDS. IfR is a commutativeS-algebra in the sense of [8], there is an associated category ofR-modulesMRand its derived category DR.

The following notions were introduced in [8]. LetAbe anR-module with aunit η:R−→Aandproduct ϕ:A∧

RA−→A. ThenA, or more precisely, (A, ϕ, η), is an R-algebraif the following diagrams commute in MR.

A∧

RA∧

RA ^ϕ∧id //

id∧ϕ



A∧

RA

ϕ



R∧

RA ^η∧id//

µ

!!D

DD DD DD DD A∧

RA

ϕ



A∧

RR

id∧η

oo

}}zzzzzzµzzz

A∧

RA

ϕ //A A

(1.1a)

Aiscommutative if the following diagram commutes in MR. A∧RA ^τ //

ϕBBBBBB  

BB A∧

RA

~~||||||ϕ||

A

(1.1b)

There are also weaker conditions for such a product. If the diagrams of (1.1a) commute inDR thenA is anR-ring spectrum, and if (1.1b) also commutes in DR

thenA is acommutativeR-ring spectrum. In that case the smash productA∧

RAis 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∧

RR−→A∧

RA←−R∧

RA←−^∼= A.

As discussed in [4], we have

Theorem 1.1. Let Abe 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 overR_∗;

(ii) for any R-moduleM,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 spectrumM U is a commutativeS-algebra. At the time of writing it seems not to be known whether the Brown-Peterson spectrum BP for a primepis a commutativeS-algebra. We will state many of our results both in terms of M U-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 commutativeS-algebra.

We will assume that a choice of polynomial generators xr ∈ M U2r (r>1) for M U_∗ has been made. We also assume that a rational prime p=x0 >0 has been chosen.

For any subset S ⊆ {xr : r > 0}, the sequence of elements of S conven- tionally ordered by increasing degree is regular. By successively killing the ho- motopy elements by forming mapping cones in the category of M U-modules we can form aM U-module spectrum M U /S. More precisely, this is a CW-cell M U- module whose cells are indexed by the monomial basis of the exterior algebra ΛM U_∗(τr : xr ∈ S) in which τr has bidegree (1,2r). The cell corresponding to τrhas dimension 2r+ 1 and an attaching mapS^2r_{M U}−→(M U /S)^(2r)in the homo- topy class ofxr∈π2r(M U /S)^(2r). We make this exterior algebra into anM U_∗-dga with differential d for which dτr =xr. Of course, ΛM U_∗(τr:xr∈S)_∗ is a Koszul complex providing a free resolution of theM U_∗-moduleπ_∗M U /S=M U_∗/(S),

ΛM U_∗(τr:xr∈S)_∗−→M U_∗/(S)→0.

Recall from [8], the K¨unneth Spectral Sequence

E^r,s₂ = Tor^{M U}_r,s ^∗(M U_∗/(S), M U_∗/(S)) =⇒M U /S_r+s^{M U}M U /S^op. (2.1) We will need to consider situation where M U /S is an M U-ring spectrum as well as the opposite M U-ring spectrum M U /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 thatM U /S is anM U-ring spectrum. Then the K¨unneth Spectral Sequence (2.1)forM U /S^{M U}_∗ M U /S^op collapses at E2 to give

M U /S^{M U}_∗ M U /S^op= ΛM U_∗/S(τr:xr∈S), where the exterior generators satisfyτr∈M U /S_2r+1^{M U}M U /S^op.

IfM U /Sis commutative then of courseM U /S^op=M U /SasM U-ring spectra;

Theorem 2.1 then follows directly from [4, lemma 1.3]. See Strickland [16] for details on when this commutativity condition on M U /S is satisfied. In particular, when x0 =pis an odd prime, all the standard M U-module spectra of this form such as M Uhni,M Uhni, BP andBPhniare commutative M U-ring spectra, while when x0= 2, care needs to be taken so that M UhniandBPhnineed to be replaced by spectra constructed using non-standard polynomial generators forM U_∗.

As a particular case, let us assume that we have chosenS to contain a complete set of generators of M U_∗ except for xp^k−1 (0 6 k 6 n). We then have an M U- moduleM Uhni. Here we allow the possibility of x0 =p >0 corresponding to the caseM Uh−1i.

Proposition 2.2.

M Uhni^{M U}_∗ M Uhni^op= ΛM Uhni_∗(τr:r6=p^k−1, 06k6n).

Letp >0 be a prime. IfBP is a commutativeS-algebra then we can form the p-localization ofM Uhnias aBP-module, denotedBPhni, for which the following holds.

Proposition 2.3.

BPhni^BP_∗ BPhni^op= ΛBPhni_∗(τ_p^k₋₁:k>n+ 1).

Notice that for the special (commutative) caseBPh0i=HZ(p), HZ(p)BP

∗ HZ(p)= Λ_Z(p)(τp^k−1:k>1).

Similarly, in the caseBPh−1i=HF^p we have

HF^pBP_∗ HF^p= Λ_Fp(τ_p^k₋₁:k>0) and the natural map

HF^pS_∗HF^p −→HF^pBP_∗ HF^p

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

3. Extending M U-algebra structures

As a starting point we recall from [8] that for any commutative ring R, the natural homomorphismM U0 =Z−→ R lifts to a morphism of S-algebras M U −→HR, whereHR is an Eilenberg-MacLaneM U-module.HR is known to be a commutative M U-algebra. We setM Uh0i=HZsince the latter realizes the spectrum discussed above.

Theorem 3.1. Letn>0and suppose thatM Uhniadmits the structure of anM U- algebra. ThenM Uhn+ 1iadmits anM U-algebra structure so there is a morphism of M U-algebras

M Uhn+ 1i −→M Uhni

realizing the natural ring homomorphism M Uhn+ 1i_∗−→M Uhni_∗.

If BP is a commutative S-algebra and BPhni admits the structure of a BP- algebra, thenBPhn+ 1iadmits aBP-algebra structure so that there is a morphism of BP-algebras

BPhn+ 1i −→BPhni

realizing the natural ring homomorphism BPhn+ 1i_∗−→BPhni_∗.

Proof. SetM Uhn+ 1; 1i=M Uhni. We will prove by induction that the following holds for eachm>1:

• IfM Uhn+ 1;miis anM U-algebra which as anM U_∗-module satisfies π_∗M Uhn+ 1;mi=M Uhn+ 1i_∗/((xpⁿ⁺¹−1)^m),

then there is anM U-algebraM Uhn+ 1;m+ 1ifor which π_∗M Uhn+ 1;m+ 1i=M Uhn+ 1i_∗/((xpⁿ⁺¹−1)^m+1) and a morphism ofM U-algebras

M Uhn+ 1;m+ 1i −→M Uhn+ 1;mi realizing the evident homomorphism

M Uhn+ 1;m+ 1i_∗ −→M Uhn+ 1;mi_∗. There is a short exact sequence ofM U_∗-modules

0→Σ^2(pn+1−1)mM Uhni_∗−→M Uhn+ 1i_∗/((xpⁿ⁺¹−1)^m+1)

−→M Uhn+ 1i_∗/((xpⁿ⁺¹−1)^m)→0, so we need to show that this is realized by an extension ofM U-algebras, for which the fibre of the mapM Uhn+ 1;m+ 1i −→M Uhn+ 1;miis Σ^2(pn+1−1)mM Uhni.

Following [11], we need to determine the Hochschild cohomology HH_{M U}^{∗ ∗}_∗(M Uhn+ 1;mi_∗, M Uhni_∗).

We begin by calculatingM Uhn+ 1;mi^{M U}_∗ M Uhn+ 1;mi^op. There is a Koszul res- olution

ΛM U_∗(τ[n+ 1;m]r: 16r6=p^k−1, 16k6n+ 1)_∗−→M Uhn+ 1;mi_∗→0 which is anM U_∗-dga with

dτ[n+ 1;m]r=

((xpⁿ⁺¹−1)^m ifr=pⁿ⁺¹−1,

xr otherwise.

Tensoring over M U_∗ with M Uhn+ 1;mi_∗ and taking homology, we find that the K¨unneth Spectral Sequence of (2.1) collapses to give

M Uhn+ 1;mi^{M U}_∗ M Uhn+ 1;mi^op=

Λ_{M Uhn+1;mi}(τ[n+ 1;m]r: 16r6=p^k−1, 16k6n+ 1). (3.1) Similarly, using a standard divided power complex we find that

HH_{M U}^{∗ ∗}_∗(M Uhn+ 1;mi_∗, M Uhni_∗) =

M Uhni_∗[y[n+ 1;m]r: 16r6=p^k−1, 16k6n+ 1]. (3.2) Finally we require the fact that

Ext_{M U}^{∗ ∗}_∗(M Uhn+ 1;mi_∗, M Uhni_∗) =

ΛM Uhni_∗(Q[n+ 1;m]^r: 16r6=p^k−1, 16k6n+ 1), (3.3)

which is obtained using the above Koszul resolution. Here Q[n+ 1;m]^r∈Ext^1,_{M U}^∗

∗(M Uhn+ 1;mi_∗, M Uhni_∗) is the element ‘dual’ toτ[n+ 1;m]rwith respect to the M Uhni_∗-basis for

ΛM U_∗(τ[n+ 1;m]r: 16r6=p^k−1, 16k6n+ 1)1,∗

consisting of theτr’s. Then bidegQ[n+ 1;m]^r=

((1,2(pⁿ⁺¹−1)m) ifr=pⁿ⁺¹−1,

(1,2r) otherwise.

Consider the elementy[n+ 1;m]pⁿ⁺¹−1. By similar arguments to those of [11], this element gives rise to an element of

Der^2(p_{M U}^n+1−1)m+1(M Uhn+ 1;mi, M Uhni) and moreover it corresponds toQ[n+ 1;m]^pn+1−1 under the map

Der^2(p_{M U}^n+1−1)m+1(M Uhn+ 1;mi, M Uhni)−→

Ext^1,2(p_{M U∗}^n+1−1)m(M Uhn+ 1;mi_∗, M Uhni_∗) defined in that paper. Using the standard identification of elements of Ext¹ with extensions, it is easy to see that Q[n+ 1;m]^r specifies the explicit extension of M U_∗-modules

0→Σ^2(pn+1−1)mM Uhni_∗−→M Uhn+ 1;m+ 1i_∗−→M Uhn+ 1;mi_∗→0 corresponding to the fibration sequence

Σ^2(pn+1−1)mM Uhni −→M Uhn+ 1;m+ 1i −→M Uhn+ 1;mi

ofM U-modules. By Lazarev’s algebra extension machinery, there is a morphism of M U-algebrasM Uhn+ 1;m+ 1i −→M Uhn+ 1;miwhich 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

M0−→M1−→ · · · −→Mn−→ · · · is a directed system of cofibrations inMRthenhocolim

n Mnis weakly equivalent tocolim

n Mn. (b) If

N0←−N1←− · · · ←−Nn←− · · · is a directed system of fibrations inMRthenholim

n Mnis weakly equivalent to limn Mn.

Let

M Uhn+ 1;∞i= lim

m M Uhn+ 1;mi, which is equivalent inDRto holim

m M Uhn+ 1;mi. We obtain anM U-algebra struc- ture onM Uhn+ 1;∞iand the Milnor exact sequence

0→lim

m

1M Uhn+ 1;mi_∗−→M Uhn+ 1;∞i_∗−→lim

m M Uhn+ 1;mi_∗→0, gives

limm M Uhn+ 1;mi_∗=M Uhn+ 1i_∗, lim

m

1M Uhn+ 1;mi_∗= 0, since each mapM Uhn+ 1;m+ 1i_∗−→M Uhn+ 1;mi_∗is surjective. Hence

M Uhn+ 1;∞i_∗=M Uhn+ 1i_∗.

An obstruction theory argument based on the fact thatM Uhn+ 1iis a cellM U- module provides an M U-module map M Uhn+ 1i −→ M Uhn+ 1;∞i which is a weak equivalence.

TheBP version follows by a parallel argument.

Theorem 3.3. There is a tower of M U-algebras

HZ=M Uh0i ←−M Uh1i ←− · · · ←−M Uhn+ 1i ←− · · ·

whose limit M Uh∞i is an M U-algebra with M Uh∞i_∗ = M U_∗/(xr : 1 6 r 6= p^k−1).

If BP is a commutativeS-algebra, there is a tower of BP-algebras HZ(p)=BPh0i ←−BPh1i ←− · · · ←−BPhni ←−BPhn+ 1i ←− · · · whose limit BPh∞iis a BP-algebra and is equivalent to BP as a BP-ring spec- trum.

Proof. The starting point is the observation of [8] that the unit M U −→ HZ = M Uh0iis an M U-algebra morphism so that realizes the augmentation M U_∗ −→

M U0 = Z. Theorem 3.1 and an induction on n shows that the tower exists as claimed.

In the Milnor exact sequence 0→lim

n

1M Uhni_∗−→M Uh∞i_∗−→lim

n M Uhni_∗→0, we have

limn M Uhni_∗=M U_∗/(xr: 16r6=p^k−1), lim

n

1M Uhni_∗= 0 since each map M Uhn+ 1i_∗−→M Uhni_∗is surjective. Hence

M Uh∞i_∗=M U_∗/(xr: 16r6=p^k−1).

In theBP case, the unit mapBP −→BPh∞iis a weak equivalence and so is an equivalence ofBP-ring spectra.

Letε:M U −→BP be the map inducing the algebraic projection (M U_∗)_(p)−→

BP_∗ due to Quillen and described in [1]. It is well known that the image of each Hazewinkel generator vn ∈ BP2(pⁿ−1) in (M U_∗)(p) actually lies in M U_∗ and we refer to this image as vn ∈ M U2(pⁿ−1). Recall also that there is an M U-module map η: M U −→BP where BP can be given the structure ofM U-ring spectrum with unitη [8, 16].

Theorem 3.4. Choose the generators xr so that xr ∈ kerε for r 6= p^k−1 and x_p^k₋₁=vk. Then there is a map of M U-ring spectraθ:BP −→M Uh∞i(p) fac- toring the unitM U −→M Uh∞i(p) as

M U −→BP −→^θ M Uh∞i(p),

where the first map is the unit for BP.θ is an equivalence of M U-ring spectra or equivalently of M U(p)-ring spectra BP −→ M Uh∞i(p). Hence the ring spectrum BP can be realized as anM U-algebra or equivalently as an M U(p)-algebra.

Proof. SinceBP is a cellM U(p)-module with cells corresponding to the generators in the Koszul resolution of (M U(p))_∗/(xr : 1 6 r 6= p^k−1), the map θ can be constructed by induction to satisfy the required properties up to homotopy.

Corollary 3.5. Each spectrum BPhni(16n)can be realized as an M U-algebra or equivalently as an M U(p)-algebra and the natural mapBPhn+ 1i −→ BPhni can be realized as a morphism of algebras.

Proof. Obstruction theory gives anM U-module mapBPhni −→M Uhni(p)which is visibly a weak equivalence.

4. Some M U -algebras obtained by localization

From [8, 18] we know that on inverting an element u ∈ M U2d we obtain the Bousfield localization atM U[u⁻¹] as

M −→L^{M U}_{M U[u}−1]M =M U[u⁻¹] ∧

M UM for anyM U-moduleM. Furthermore, ifAis anM U-algebra then

A−→L^{M U}_{M U[u}−1]A=M U[u⁻¹] ∧

M UA

is a morphism of M U-algebras. Similar considerations would apply if BP were a commutativeS-algebra. We will use the notation

M U(n) = L^{M U}_{M U[x}−1

pn−1]M Uhni, BP(n) = L^BP_BP[v−1

n ]BPhni, where

M U(n)_∗=M Uhni_∗[x⁻_pn¹−1], BP(n)_∗=BPhni[v_n⁻¹].

Proposition 4.1. For each primepandn>1, there is a localization morphism of M U-algebras

M Uhni −→L^{M U}_{M U[x}−1

pn−1]M Uhni=M U(n)

and an equivalence of M U-ring spectra M U(n)_(p) −→E(n). Hence also E(n)ad- mits the structure of an M U-algebra. Therefore E(n) also admits the structure of anS-algebra which is a commutative S-ring spectrum.

IfBP is a commutativeS-algebra, then there is a localization morphism ofBP- algebras

BPhni −→L^BP_BP[v−1

n ]BPhni=BP(n)

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

GivenE(n) as anM U orBP-algebra, we can form the quotient moduleE(n)/I_n^k obtained by killing all the monomials v^r₀⁰v^r₁¹· · ·v_n^rn₋⁻₁¹ with Pn−1

i=0 ri = k. In [2, 5], these spectra were constructed as E(n)-modules for an appropriate[ S-algebra structure; hereE(n) is the[ In-adic completion of theS-moduleE(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)/In. By [2, 5], the tower

E(n)/In←−E(n)/I_n²←− · · · ←−E(n)/I_n^k←−E(n)/I_n^k+1←− · · · (4.1) has E(n) for its limit as an[ S-ring spectrum, however here we are working with M U,BP orE(n)-modules. This gives a new proof of the existence of anS-algebra structure on E(n). We also have[ E(n)[ ' L^S_K(n)E(n), so this result can also be proved by another application of the Bousfield localization theory ofS-algebras.

Proposition 4.2. The natural mapE(n)−→E(n)[ is a morphism ofM U-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. LetRbe a commutativeS-algebra,Abe anR-algebra andEbe anR- module andM anA-module. Then the(A∧

RE)^A-localization mapλ:M −→L^A_A∧

REM is an E^R-localization map for M. Hence there is a weak equivalence of R-modules L^R_EM −→L^A_A∧

REM.

Now takeR=S,A=M U, E=K(n) andM =E(n). Then we are done since there is an equivalence ofM U-modules

E(n)[ 'L^{M U}_{M U}∧

SK(n)E(n).

For reference we determine the Bousfield class of theM U-moduleM U∧

SK(n). Mak- ing use of ideas similar to those in the Appendix of [3] (see especially Corollary A.2) we find thatπ_∗M U∧

SK(n) =M U_∗K(n) is a freeM U_∗/In[v⁻_n¹]-module. By a stan- dard argument, there is a weak equivalence ofM U-modules

_

α

Σ^2dαM U /In[v⁻_n¹]'M U∧

SK(n),

henceM U∧

SK(n) is Bousfield equivalent toM U /In[v_n⁻¹].

Actually, Theorem 6.4 of [4] also tells us that there is an equivalence of M U- modules

L^{M U}_{M U /I}

n[v⁻n¹]E(n)'E(n);[

indeed the tower of (4.1) is constructed in [4] as a tower of M U-modules and its homotopy limit is shown to be L^{M U}_{M U /I}

n[v⁻n¹]E(n).

Having obtainedE(n) as an[ M U-algebra, ideas of [12, 15] can be used to show that there is anM U-algebraE(n)W([ F^pn) obtained by adjoining a primitive (pⁿ−1)- st root of unity toE(n)[_∗. The 2-periodic version of this spectrum is known to be a commutativeS-algebra by work of Hopkins, Miller and Goerss [9].

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Andrew Baker [email protected] http://www.maths.gla.ac.uk/∼ajb Mathematics Department,

University of Glasgow, Glasgow G12 8QW, Scotland.

Alain Jeanneret [email protected] Mathematisches Institut,

Universit¨at Bern, Bern CH 3012, Switzerland.