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Glasgow Math. J. 54 (2012) 61–66. Glasgow Mathematical Journal Trust 2011.





Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, Switzerland e-mail: alain.jeanneret@math.unibe.ch


SBB, Br¨uckfeldstrasse 16, 3000 Bern, Switzerland e-mail: samuel.wuethrich@sbb.ch

(Received 22 September 2010; revised 4 January 2011; accepted 11 April 2011; first published online 2 August 2011)

Abstract. The aim of this note is to present a new, elementary proof of a result of Baas and Madsen on the mod p cohomology of certain quotients of the spectrum BP.

2010 Mathematics Subject Classification. 55P43; 55N10, 55S10.

In 1970s, Baas and Madsen [1] calculated the cohomology of certain quotients of MU, the Thom spectrum of the universal bundle over BU. Recall that

MU= π(MU) ∼= ⺪[x1, x2, . . .],

where xilies in degree|xi| = 2i. The spectra MUn1, . . . , nq considered by Baas and Madsen are defined for any string of integers 0< n1< · · · < nq of the form ni= 2(pji− 1), j

i> 0, where p is some fixed prime. They satisfy

π(MUn1, . . . , nq) ∼= ⺪[xn1, . . . , xnq].

Baas and Madsen determined their mod p cohomology, by relying heavily on the Atiyah–Hirzebruch spectral sequence and previous work of Cohen on the Hurewicz homomorphism on MU [4].

The aim of this note is to present an alternative proof of Baas–Madsen’s result without reference to any spectral sequence or to the work of Cohen. Our arguments use the techniques developed in [5] and thus are more transparent than the original ones. They are based on a construction of the spectra MUn1, . . . , nq which is algebraic in nature and which does not require the use of bordism with singularities any more.

The spectrum MU is a commutative S-algebra, see [5]. This leads to a well-behaved homotopy theory of MU-modules. In particular, the derived or homotopy category of MU-modules is a symmetric monoidal category for the smash product over MU. Our constructions take place in this category.

The spectra MUn1, . . . , nq can be obtained in a standard way, as regular quotients of MU [5]:


Instead of the xk, we may take any other regular sequence generating the kernel J of the projection MU→ MUn1, . . . , nq∗. So it is legitimate to write MU/J for MUn1, . . . , nq. As our interest lies in the mod p cohomology of these MU-modules, we might just as well consider their p-localisations MUn1, n2, . . . , nq(p). These spectra

admit a much more economical presentation, as quotients of BP. To see this, recall that

BP = ⺪(p)[v1, v2, . . .], |vi| = 2(pi− 1),

where the vi are Araki’s generators [6]. Recall also that BP can be realised as an

MU-algebra [2], and that the unitπ : MU → BP induces an isomorphism π∗:⺪(p)⊗ MU/(xk: k= pi− 1) ∼= BP.

From this, we deduce that

MUn1, . . . , nq(p) BP ∧MUMU/(xpj−1 : j= j1, . . . , jq).

As an additional advantage, this presentation exhibits the MU-modules MUn1, . . . , nq(p) as left BP-modules. Setting I= BP· J, we can unambiguously


MUn1, n2, . . . , nq(p) BP/I.

Following the convention x0= v0= p and extending the string of the niby n0= 0, we

obtain the other family of spectra considered by Baas and Madsen, MUpn1, . . . , nq = MUn1, . . . , nq(p)/p = BP/(I + (p)),

as quotients of BP as well.

Recall that the construction of regular quotients is natural in the following sense. If I1⊆ I2⊆ BPare two ideals such that I1 is generated by a regular sequence over BPand such that I2/I1is generated by a regular sequence over BP/I1, then there is

a canonical map of BP-modules BP/I1→ BP/I2 [5, V.1.]. Note that the Eilenberg–

MacLane spectrum Hpis the quotient of BP by the ideal generated by all thevi’s and will serve as a terminal object in our construction.

Let Ap denote the mod p Steenrod algebra and let Qi, i≥ 0, be the primitive element of degree 2pi− 1 of Milnor’s basis. Let us write (y

1, y2, . . .) for the left

ideal ofApgenerated by elements y1, y2, . . . ∈ Ap. Recalling that xpk−1≡ vkmodulo

decomposables, we have the following result.

THEOREM. Let I ⊂ BPbe the ideal generated by a regular sequencewi1, wi2, . . . with

wik≡ vik modulo decomposables and 0≤ i1< i2< · · · . Then the natural map BP/I →

Hpinduces an isomorphism ofAp-modules

H(BP/I; ⺖p) ∼= Ap/(Qi: i= i1, i2, . . .).


COROLLARY. There are canonical isomorphisms ofAp-modules:

H(MUn1, . . . , nq; ⺖p) ∼= Ap/(Q0, Qj1, . . . , Qjq);

H(MUpn1, . . . , nq; ⺖p) ∼= Ap/(Qj1, . . . , Qjq).

Proof. We first prove the result in the particular case of the left BP-modules P(n)= BP/In, where Inis the ideal of BP∗generated by the elementsv0, . . . , vn−1. That

is, P(0)= BP by definition and P(n)∼= ⺖p[vn, vn+1, . . .] for n ≥ 1. Let φn: P(n)→ H⺖p be the natural map and set C(n)= H(P(n);p). The cofibre sequence of left BP-modules and left BP-morphisms

· · · → P(n) vn

→ P(n) ηn

→ P(n + 1) ∂n

→ P(n) → · · · induces a long exact sequence ofAp-modules in cohomology

· · · → C(n) vn

→ C(n) ∂n

−→ C(n + 1) ηn

−→ C(n) → · · · .

The accurate reader has noticed that we have suppressed the mention of suspension coordinates. As a consequence,n∗is not a morphism of degree 0 but rather of degree 2pn− 1. Proposition B.5.15(b) in [7] implies that the image of v

nunder the Hurewicz homomorphism P(n)→ H(P(n);p) is trivial. Therefore, (vn)∗: H(P(n);p)→

H(P(n);p) is trivial, and by duality the same holds for vn: C(n)→ C(n). As a consequence, we obtain a short exact sequence ofAp-modules:

0→ C(n)


−→ C(n + 1) ηn

−→ C(n) → 0. (1)

It obviously splits in the category of ⺖p-vector spaces. We now inductively define elements qnj0+1,...,jl ∈ C(n + 1) of degreelk=0(2pjk− 1), for any l ∈ {0, . . . , n}:

ηn  qnj0+1,...,jl=  qn j0,...,jl if jl< n, 0 if jl= n. (2)

For C(0) we take q0= φ0: P(0)→ H⺖p. Assume we have chosen elements qkj0,...,jl

C(k) as indicated for k≤ n. For degree reasons, the qn

j0,...,jl admit unique lifts q


j0,...,jl to

C(n+ 1). For jl= n, we define

qnj0+1,...,jl = ∂n∗qnj0,...,jl−1. (3)

Observe that the product on BP induces a coalgebra structure on C(0). Moreover, the action of BP on P(n) gives rise to left C(0)-comodule structures on the C(n) with respect to which equation (1) is a short exact sequence of C(0)-comodules. We show by induction that there are isomorphisms of C(0)-comodules

C(n) ∼= C(0) ⊗p ⎛ ⎝  0≤j0<···<jl<npqnj0,...,jl⎠ , (4)


so that under this isomorphism,ηnmaps y⊗ qnj+1

0,...,jl to y⊗ η

n(qnj0,...,jl). This is clear for

n= 0. For the inductive step, note that comodules of this form are relatively injective [6, A1]. Since equation (1) splits overpand is a sequence of C(0)-comodules, it splits as a sequence of C(0)-comodules, which proves the claim.

From equation (4), it follows that the primitives of C(n) (with respect to the C(0)-comodule structure) are given by




We now show by induction that

C(k) ∼= Ap/(Qk, Qk+1, . . .) (5) and that φk(Qj0· · · Qjl)=  αjlq k j0,...,jl if jl< k, 0 otherwise, (6)

for some non-zeroαjl ∈ ⺖p. In fact, one can show thatαjl = 1 for all l, but it will be

enough to know equation (6) for our purposes.

It is well known that equation (5) holds for k= 0, see the original paper [3] or Theorem 4.1.12 in [6] for the mod p homology. Also, equation (6) is trivial in this case. Assume inductively that equations (5) and (6) are true for k≤ n. For the inductive step, recall that qnj+1

0,...,jl is uniquely determined by equation (2) for jl< n. Thus, equation (6)

certainly holds for k= n + 1 in this case. Now note that φn: P(n)→ H⺖p induces an isomorphism in degrees< 2(pn− 1) on homotopy groups, and so φ

n:Ap→ C(n) is an isomorphism in degrees < 2pn− 3. This implies that φ

n+1 sends Qn to some non-zero primitive element in C(n+ 1). The only primitives in C(n + 1) of degree |Qn| = 2pn− 1 are the scalar multiples of qnn+1 (because of


i=0(2pi− 1) < 2pn− 1). Therefore,φn+1(Qn)= αnqnn+1for someαn∈ ⺖p.

Now consider the diagram ofAp-modules

Ap −·Q n // φn  Ap φn+1  C(n) αn∂n // C(n + 1) . (7)

We have just shown that the two compositions agree on 1∈ Ap. ByAp-linearity, the diagram therefore commutes. The inductive assumption shows that equation (6) holds for jl = n in general. Extending (7) to the right, we obtain a commutative diagram of

Ap-modules 0 // Ap/(Qn, . . .) −·Q n // ¯ φn  Ap/(Qn+1, . . .) ¯ φn+1  // Ap/(Qn, . . .) // ¯ φn  0 0 // C(n) αn∂n // C(n + 1) ηn // C(n) // 0,


where the unlabelled map is the natural projection and ¯φn∗ and ¯φn+1 are induced by φ

n andφn∗+1, respectively. The reader may check that the upper sequence is exact. By

inductive assumption, ¯φn is an isomorphism, therefore ¯φn+1 is an isomorphism, too. This concludes the proof of equation (5).

We now consider the general case and determine the mod p cohomology of BP/I for any satisfying the hypotheses of the theorem. We define ideals

J0= (0) ⊂ J1= (wi1)⊂ J2 = (wi1, wi2)⊂ . . . ⊂ I

and prove by induction that

H(BP/Jk;⺖p) ∼= Ap/(Qi: i= i1, . . . , ik). (8) As BP/I is the homotopy colimit of the sequence

BP/J0 → BP/J1→ BP/J2→ . . . ,

this implies the result:

H(BP/I; ⺖p) ∼= lim H(BP/Jk;⺖p) ∼= Ap/(Qi: i= i1, i2, . . .).

For k= 0, equation (8) is just equation (5). Assume that equation (8) holds for k≤ n. By hypothesis, we have wik ≡ vik mod Iik−1. Hence Jk is contained in

Iik+1, and therefore, there are canonical maps of left BP-modules ψk: BP/Jk

P(ik+ 1). Composing them with the canonical maps P(ik+ 1) → P(ik+1) gives maps ψk: BP/Jk→ P(ik+1). Nowvik+1 andwik+1agree as endomorphisms of P(ik), because

vi: P(ik)→ P(ik) is homotopically trivial for i< ik. Hence, there is a commutative diagram of cofibre sequences

· · · // BP/Jk wik+1// ψk  BP/Jk // ψk  BP/Jk+1 (∗) // ψk+1  BP/Jk // ψk  · · · · · · // P(ik+1) vik+1

// P(ik+1) // P(ik+1+ 1) // P(ik+1) // · · · .

Square (∗) commutes by unicity of ψk+1as a lift ofψk. Taking mod p cohomology, we obtain a commutative diagram of C(0)-comodules with exact rows, of the form

0 // H(BP/Jk;⺖p) // H(BP/Jk+1;⺖p) // H(BP/Jk;⺖p) // 0 0 // C(ik+1) // ψk OO C(ik+1+ 1) // ψk+1 OO C(ik+1) ψk OO // 0. For j0, . . . , jl∈ {i1, . . . , ik+1} with j0< · · · < jl, we define

˜qjk0+1,...,jl = ψk+1qik+1+1


∈ H(BP/J


Following analogous arguments as before, we show that H(BP/Jk+1;⺖p) ∼= C(0) ⊗p ⎛ ⎜ ⎜ ⎝  j0,...,jl∈{i1,...,ik} j0<···<jlp˜qkj0+1,...,jl ⎞ ⎟ ⎟ ⎠ . From the fact that the natural map BP/Jk+1 → H⺖pfactors as

BP/Jk+1−−→ P(i ψk+1 k+1+ 1)

φik+1+1 −−−→ H⺖p,

we easily deduce equation (8) for k= n + 1, which concludes the proof.


1. N. A. Baas and I. Madsen, On the realization of certain modules over the Steenrod

algebra, Math. Scand. 31 (1972), 220–224.

2. A. Baker and A. Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras,

Homology Homotopy Appl. 4(1) (2002), 163–173 (electronic).

3. E. H. Brown, Jr. and F. P. Peterson, A spectrum whose Zpcohomology is the algebra

of reduced pthpowers, Topology 5 (1966), 149–154

4. J. M. Cohen, The Hurewicz homomorphism on MU, Invent. Math. 10 (1970), 177–186. 5. A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, Rings, modules, and algebras

in stable homotopy theory, in Mathematical surveys and monographs, Vol. 47 (American Mathematical Society, Providence, RI, 1997).

6. D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, in Pure and

Applied Mathematics, Vol. 121 (Academic Press Inc., Orlando, FL, 1986).

7. D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, in Annals of

mathematics studies, Vol. 128 (Princeton University Press, Princeton, NJ, 1992). Appendix C by Jeff Smith.


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