Morava Hopf algebras and spaces K (n)
equivalent to finite Postnikov systems
Michael J. Hopkins *
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
mjh@math.mit.edu
Douglas C. Ravenel y W. Stephen Wilson
University of Rochester Johns Hopkins University
Rochester, New York 14627 Baltimore, Maryland 21218
drav@troi.cc.rochester.edu wsw@math.jhu.edu
December 29, 1994
Abstract
We have three somewhat independent sets of results. Our first
results are a mixed blessing. We show that Morava Ktheories don't
see kinvariants for homotopy commutative Hspaces which are finite
Postnikov systems, i.e. for those with only a finite number of homo
topy groups. Since kinvariants are what holds the space together,
this suggests that Morava Ktheories will not be of much use around
such spaces. On the other hand, this gives us the Morava Ktheory of
a wide class of spaces which is bound to be useful. In particular, this
work allows the recent work in [RWY ] to be applied to compute the
BrownPeterson cohomology of all such spaces. Their BrownPeterson
cohomology turns out to be all in even degrees (as is their Morava K
theory) and flat as a BP *module for the category of finitely presented
BP *(BP ) modules. Thus these examples have extremely nice Brown
Peterson cohomology which is as good as a Hopf algebra.
______________________________
*Partially supported by the National Science Foundation
yPartially supported by the National Science Foundation
1
Our second set of results produces a large family of spaces which
behave as if they were finite Postnikov systems from the point of view
of Morava Ktheory even though they are not. This allows us to apply
the above results to an even wider class of spaces than finite Postnikov
systems. These examples come from spaces in omega spectra with
certain properties. There are many well known examples with these
properties. In particular, we compute the K(n) homology of all the
spaces in the spectra for P (q) and k(q) where q > n.
In order to prove our results on finite Postnikov systems we need
our third set of results; a beginning of an analysis of bicommutative
Hopf algebras over K(n)*.
Contents
1 Introduction 2
2 Proof of the main theorem 11
3 Strongly K(n)*acyclic connective spectra 23
4 Morava homology Hopf algebras 31
1 Introduction
The Morava Ktheories are a collection of generalized homology theories
which are intimately connected to complex cobordism ([JW75 ], [W"ur91 ]).
It is known that they play a central role in aspects of homotopy theory
([Hop87 ],[DHS88 ],[HS ]). This and their relative computability (due to a
K"unneth isomorphism) makes them a powerful tool. For each prime, p, and
n > 0, the coefficient ring for K(n)*() is K(n)* ' Fp[vn; v1n] where the
degree of vn is 2(pn  1).
We assume that all of our spaces are homotopy equivalent to a CW
complex.
2
Theorem 1.1 Let X be a connected plocal space with ssk(X) finitely gen
erated over Z(p), k > 1, and nonzero for only a finite number of k. Then
K(n)*(X) has a natural filtration by normal subHopf algebras:
K(n)* ' F(n+2) . . .F2 F1 F0 ' K(n)*(X)
with
Fq==F(q+1)' K(n)*(K(ssq(X); q))
as Hopf algebras.
If K(n)*(X) is commutative (e.g. if X is an Hspace), then K(n)*(X)
is isomorphic, as a Hopf algebra, to the associated graded object above:
O
K(n)*(X) ' K(n)*(K(ssm (X); m)):
0mn+1
This is natural if either all ssk(X), k > 1, are finite, or if they are all fre*
*e.
We believe that if K(n)*(X) is not commutative then we still have the
last isomorphism as coalgebras but have been unable to prove it. It may
require a dual Borel theorem for our Hopf algebras and we have only been
able to handle the bicommutative case.
We actually state and prove a theorem (Theorem 2.1) which does not
require the homotopy groups to be finitely generated over Z(p). We can have
copies of Q=Z(p)but our groups can only have a finite number of summands.
This is important both for our proofs and for some of our applications. The
last naturality statement is true in this case if all the homotopy groups are
torsion.
Note also that the n = 0 case of this theorem is a familiar result about
rational homology.
In [MM92 ], McClearyMcLaughlin show that for an EilenbergMac Lane
space X with finite homotopy group, K(n)*(X) has the same rank as
K(n  1)*(LX) , where LX denotes the free loop space of X. In view of
the theorem above, the following generalization of their result is immediate.
3
Corollary 1.2 Let X be a simply connected Hspace with finitely many non
trivial homotopy groups, each of which is finite. Then K(n)*(X) has the same
rank as K(n  1)*(LX).
The following corollary follows from the fact that the Morava Ktheory
of EilenbergMac Lane spaces is even degree (except for the circle) and from
the main results of [RWY ]. The condition on ss1 is needed to avoid having
copies of the circle in our space, giving us odd degree elements.
Corollary 1.3 If X is as in Theorem 1.1 and ss1(X) is torsion, then
K(n)*(X) is even degree and so is BPp^*(X) where BPp^ is the padic
completion of BP . If ssk(X) is finite for k > 1 then BP *(X) is even
degree. In either case, it is a flat BP *module for the category of finitely
presented BP *(BP )modules.
We see the Morava Ktheory cannot distinguish between the double loops
of such a space and a product of EilenbergMac Lane spaces with the same
homotopy groups. One cannot expect to generalize this too much to spaces
with an infinite number of nonzero kinvariants; the sphere, Sk, is a counter
example to that. Inverse limit problems rear their ugly head. Morava K
theory somehow looks at the whole space rather than how it is put together.
On the other hand, our result certainly does cover spaces with infinitely many
homotopy groups if the kinvariants are zero for all but a finite number of
stages. This is because the EilenbergMac Lane spaces split off as a product if
the kinvariant is zero. Another, more substantial direction of generalization
is to spaces in spectra with certain stable properties.
Although the abstract isomorphism is interesting from a theoretical point
of view, the practical value comes because the Morava Ktheory of Eilenberg
Mac Lane spaces is completely known, [RW80 ]. In particular, it is always even
degree (except for the circle). We also know that
K(n)*(K(ssm (X); m)) ' K(n)*
4
if m > n + 1, so that Morava Ktheory does not see the higher homotopy
groups of these spaces, and spaces with only higher homotopy groups are
acyclic, e.g. if X is n connected and ssn+1(X) is torsion. When we have
acyclicity we do not need Hopf algebras much:
Theorem 1.4 Let X an nconnected plocal space with ssk(X) a finitely gen
erated Z(p)module which is nontrivial for only a finite number of k. If
ssn+1(X) is torsion, then
K(n)*(X) ' K(n)*:
This theorem is rather easy and is proven quite quickly directly from
[RW80 ].
The proof of Theorem 1.1 is given in Section 2 modulo certain general
results about graded Hopf algebras over K(n)* which will be proven in Section
4. It mimics a proof for rational homology. The only place where Section 4
is used is in showing the final statement of Theorem 1.1 about the splitting
as Hopf algebras.
In Section 3, which is independent of the rest of the paper, we will prove
some results about spaces in the spectrum of a K(n)*acyclic spectrum
X. Recall that an spectrum X = {X_ i} has X_ i+1= X_i. A motivating
example for this study was produced by Richard Kramer's work computing
K(n)*(k(q)_*) when n < q.
Theorem 1.5 Let X = {X_ i} be a connective spectrum of finite type with
bottom cell in dimension 0 and K(n)*(X_ m) ' K(n)* for some m. Let X !
F be a map to a finite Postnikov system which is an equivalence through
dimension n + 1. Then X_q is K(n)*equivalent to F_q for all q 0.
This is Theorem 3.7. We can now apply Theorem 1.1 to such a spectrum
to get:
5
Theorem 1.6 Let X be as in Theorem 1.5, then, for all k 2 Z,
O
K(n)*(X_ k) ' K(n)*(K(ssi(X_ k); i))
n+1i0
as Hopf algebras. In particular, if k > n + 1, then K(n)*(X_ k) ' K(n)*.
Also, if ss0(X) is torsion, then X_n+1 is K(n)*acyclic. Furthermore,
whether K(n)*(X_ k) is trivial or not for k 0 depends only on whether
K(n)*(K(ss0(X); k)) is trivial or not.
We call such a spectrum with one space K(n)*acyclic, strongly K(n)*
acyclic, because it implies that almost all other spaces are also K(n)*acyclic.
We can get the spaces X_m , m > n + 1 to be K(n)*acyclic without reference
to the first section. Note that being strongly K(n)*acyclic implies that the
spectrum is K(n)*acyclic since K(n)*(X) = dir limK(n)*(X_ k) using the
suspension maps. Bousfield has a generalization of this which may have
useful applications together with the rest of our work. We'll discuss his
results in Section 3.
This does not lead to a calculation of the BrownPeterson cohomology
as in the case of a real finite Postnikov system because it is easy to have
spectra which are strongly K(n)*acyclic but have nontrivial K(n + 1) ho
mology. However, following [RWY ], it does lead to the calculation of the
E(n) cohomology and a host of others.
What we need now is a condition on X which implies that it is strongly
K(n)*acyclic. Associated with the "telescope" conjecture is a functor Lfn,
see [Rav93 ]. It supplies us with a class of examples. From Corollary 3.13 we
have:
Theorem 1.7 A connective spectrum X for which LfnX is contractible is
strongly K(n)*acyclic.
In particular, a suspension spectrum of a finite complex which is K(n)*
acyclic is strongly K(n)*acyclic. It is possible that the converse of this
6
Theorem is also true. This result is not phrased in the most familiar or
applicable of terms. When X is a BP module spectrum then this functor
coincides with a more familiar one. Let E(n) be the homology theory with
coefficient ring Z(p)[v1; v2; : :;:vn; v1n]. We have, from Theorem 3.14:
Theorem 1.8 Let X be a connective BP module spectrum which is
E(n)*()acyclic, or, equivalently, K(q)*()acyclic for 0 q n, then
X is strongly K(n)*acyclic.
What we need now are some concrete examples of interest. This result
can be reduced, Corollary 3.15, to a simpler statement which we can use for
this purpose.
Corollary 1.9 If X is a connective BP module spectrum in which each el
ement of ss*(X) is annihilated by some power of the ideal
In+1 = (p; v1; v2; : :;:vn) BP*;
then X is strongly K(n)*acyclic.
Thus we see that any connective BP module spectrum with some power
of In+1 mapping to zero is strongly K(n)*acyclic. There are a lot of familiar
examples in this collection. Recall that BP* ' Z(p)[v1; v2; : :]:where the
degree of vn is 2(pn  1). We have the spectra BP , with coefficient ring
BP * ' Z(p)[v1; v2; : :;:vn], see [Wil75 ] and [JW73 ]. We also have P (k; *
*n),
the spectrum with P (k; n)* ' BP *=Ik for 0 k n. These theories are
constructed using the usual BaasSullivan singularities [Baa73 ], and [BM71 ].
Many of these theories are already familiar. In particular, P (0; 1) = BP ,
P (n; n) = k(n), P (k; 1) = P (k), (see [JW75 ] and [W"ur77 ]), and P (0; n) =
BP . The spaces in the spectra for P (k; n) are important in studying
the spaces in the spectra for P (k) in [BW ]. In addition, the theories
E(k; n) = v1nP (k; n), play a prominent role in [RWY ].
From the above theorem, we see:
7
Corollary 1.10 For m q > n, P (q; m) is strongly K(n)*acyclic, so, for
k 2 Z,
O
K(n)*(P_(q;_m)_k) ' K(n)*(K(ssi(P_(q;_m)_k); i)):
n+1i0
We have explicitly computed the K(n) homology of all of the spaces in
the spectrum for all of these theories. Recall that this includes the more
familiar P (q) and k(q) for q > n. The simplicity of the answer is in stark
contrast with the calculation of K(n)*(P_(n)_*) in [RW ] and of K(n)*(k(n)_*)
in [Kra90 ].
The category of graded Hopf algebras over K(n)* is equivalent to that of
______
Hopf algebras over K(n) *' Fp, graded over Z=(2pn  2) (where we have set
vn = 1 in order to be working over a perfect field). Not everything about
connected graded Hopf algebras carries over to these Hopf algebras so one
must be somewhat careful. Being careful led us to initiate an investigation of
the type of Hopf algebras that arise when you take the Morava Ktheory of
connected homotopy commutative Hspaces. The results of this investigation
may well be of more interest than the applications to finite Postnikov systems
and Section 4 is dedicated to this study.
There, we study what we call commutative Morava homology Hopf al
gebras (for a given n > 0). These are bicommutative, biassociative, Hopf
algebras over Fp which are graded over Z=2(pn  1). Furthermore, the prim
itive filtration is exhaustive and it is the direct limit of finite dimensional*
* Fp
______
subcoalgebras. K(n) *(X), where X is an Hspace, is such an object. We
restrict our attention to such objects which are concentrated in even degrees.
Denote this category by EC(n). We show it is an abelian category. For
the sake of completeness, we show that for odd primes the bigger category
splits as EC(n) and OC(n), where OC(n) consists of exterior algebras on
odd degree primitive generators.
Since we deal only with evenly graded objects, our Morava homology
Hopf algebras are really graded over G = Z=(pn  1). Let H be the cyclic
group of order n. H acts on G via the pth power map. Writing G additively,
8
the map H x G ! G is given by (i; j) ! pij. Let fl denote an Horbit:
fl = {j; pj; p2j; : :}: G:
We have:
Theorem 1.11 For each evenly graded commutative Morava homology Hopf
algebra in EC(n), there is a natural splitting
O
A ' Afl
fl
where the tensor product is over all Horbits fl and the primitives of Aflall
have dimensions in 2fl.
To prove this theorem we construct idempotents in our category.
Theorem 1.12 For every A 2 EC(n) there are canonical idempotents efl
P
such that flefl= 1A and eflefiis trivial if fl 6= fi. These idempotents are
natural. The idempotent eflsends all primitives to zero which do not have
dimensions in 2fl and is the identity on all primitives which are in dimensions
in 2fl.
Theorem 1.11 follows immediately from this and the fact that tensor
products are the sum in this category. This splitting does even more for
us. Let EC(n)flbe the subcategory of EC(n) whose objects have primitives
only in dimensions in 2fl. What we really prove is the following:
Theorem 1.13 There are commuting idempotent functors, efl, on EC(n)
P
such that fleflis naturally equivalent to the identity functor and eflefiis t*
*he
trivial functor if fl 6= fi. As categories:
Y
EC(n) ' EC(n)fl:
fl
In particular, there is only the trivial map efl(A) = Afl! efi(B) = Bfiif
fl 6= fi. Furthermore, EC(n)flis an abelian category.
9
We can define a function __ffp: Z=(pn  1) ! Z by just taking the usual lift
to Z and then the usual ff, the sum of the coefficients in the padic expansion
of our number. __ffpis constant on an orbit fl. Usually several orbits will have
the same image under the map __ffp. The following fact is a consequence of the
computations done in [RW80 ], and is what we need to help prove Theorem
1.1.
______
Theorem 1.14 In the Hopf algebras K(n) *(K(T; q)) for any abelian torsion
______
group T and K(n) *(K(F; q+1)) for any torsion free abelian group with q 1,
all orbits fl (as in 1.11) with nontrivial factors satisfy __ffp(fl) = q.
This is proved below as Theorem 4.14.
Our results were discovered while pursuing the Johnson Question, see
[RW80 , Section 13]. This assertion is that if 0 6= x 2 BPn(X) where X is
a space, then x is not vn torsion. This is a very strong unstable condition.
At present, two of the authors have a good plausibility argument which they
hope to turn into a proof some day. The approach which led to the present
paper was just one of many dead ends. A cohomology theory can be defined
by:
Hom BP*(BP*(X); Q=Z(p)):
The classifying space for the nth group, M__n, contains the universal example
for the degree n Johnson Question. In an attempt to get some insight into
its BrownPeterson homology, the general phenomenon of Theorem 1.1 was
discovered. Each one of these spaces has only a finite number of nontrivial
homotopy groups. Each nontrivial group is a finite sum of copies of Q=Z(p).
Theorem 2.1 still applies to it to give K(n)*(M__n).
Morava Ktheory can sometimes be problematic when p = 2. Because we
are restricted to even degree objects this is not a problem for us, see Remark
4.4.
The authors thank Bill Dwyer, Takuji Kashiwabara, Richard Kramer,
Jim McClure, JeanPierre Meyer, Haynes Miller, and Hal Sadofsky, all of
whom helped this project along in one way or another.
10
2 Proof of the main theorem
For a group, G, let R[G] be the group ring for G over R. Let X" be the
universal cover for X. We have a sequence of fibrations, up to homotopy:
X" ! X ! ss1(X) ! X"! X:
Since X" and X" are connected, we see that
O
K(n)*(X) ' K(n)*(X") K(n)*[ss1(X)]:
From this we see that it is enough to restrict our attention to simply con
nected spaces; and they all have Postnikov decompositions.
We recall some basic facts about Postnikov towers. Let X be a simply
connected space. Then one has a diagram of the form
X = X2 u_____X3_ u________X4 u________._. .
  
  
  
f2 f3 f4 
  
  
u u u
K2 K3 K4
where Ks = K(sss(X); s), fs is a map inducing an isomorphism in the bottom
homotopy group, and Xs+1 is the fibre of fs. Xs+1 is the sconnected cover
of X. Theorem 1.1 is a slightly weaker plocal form of the following.
Theorem 2.1 Suppose X has only finitely many nontrivial homotopy groups.
Assume that ssq(X), q > 1, has finitely generated torsion free quotient, and
finitely generated subgroup of exponent pk for each p and k. Then for each
s 1 there is a natural Hopf algebra extension
K(n)* ! K(n)*(Xs+1) ! K(n)*(Xs) ! K(n)*(Ks) ! K(n)*:
In particular the map K(n)*(Xs) ! K(n)*(X) is always onetoone.
11
If X is an Hspace, then we have a natural isomorphism of Hopf algebras:
O
K(n)*(X) ' K(n)*(K(ssi(X); i  1)):
i2
If X is not an Hspace, the above isomorphism is still valid additively.
Although this result implies Theorem 1.1 in the Introduction, it is slightly
more general. In particular, it allows for homotopy groups with summands
like Q=Z(p)which we need. The definition of the filtration in Theorem 1.1
comes from the s  1connected cover, Xs:
Fs im K(n)*(Xs) ! K(n)*(X):
This filtration is natural. We are not really using the Postnikov construction
of a space, which is not natural, but a Postnikov decomposition which is.
The Postnikov construction starts with a point and builds up the space one
homotopy group at a time. We are starting with the space and taking it
apart one homotopy group at a time. Naturality is clear for the maps on
the EilenbergMac Lane spaces as they are determined by the maps on the
homotopy groups. We want a unique map from Xs+1 to Ys+1 if we inductively
have a unique map from Xs to Ys. The obstruction to uniqueness is in
[Xs+1; K(sss(Y ); s  1)]
but because Xs+1 is sconnected, this cohomology group is trivial.
Remark 2.2 A number of people have asked us questions about how these
results can be extended. For example, given a fibration of finite Postnikov
systems where the maps all can be delooped several times and the homotopy
groups of the fibre map splitinjective to the homotopy groups of the total
space, what can we say about the Morava Ktheory of everything? Since we
know the Morava Ktheory of all the spaces it seems to us that the results
and techniques used here should answer any question about this situation.
12
In the arguments that follow, it will be convenient to assume that ss2(X)
is torsion. In particular, the results of [RW80 ] imply that K(n)*(X) is
concentrated in even dimensions in this case. This assumption is harmless
for the following reason. In general (subject to the hypotheses of Theorem
2.1) we have a fibre sequence
f
X0 ! X ! L2
where L2 = K(ss2(X)=Torsion; 2) and ss2(f) is surjective. Then ss2(X0) is the
torsion subgroup of ss2(X), L2 is a finite product of circles, and
X ' X0x L2 (2:3)
because we can lift the maps of the circles to homotopy generators using the
Hspace structure. Hence it suffices to compute K(n)*(X0).
The rational case
The corresponding result for rational homology, K(0), is classical and we
will sketch its proof now, since it is a model for the proof of Theorem 2.1.
Since X has only finitely many nontrivial homotopy groups, Xs is contractible
for large s and we can argue by downward induction on s. Suppose Xs+1
has the prescribed rational homology, and consider the fibre sequence
js
2Ks ! Xs+1 ! Xs: (2:4)
For our inductive step we need to prove that
H*(Xs) ' H*(Xs+1) H*(Ks); (2:5)
where all homology groups have rational coefficients. When X is an Hspace,
we need the above isomorphism to be one of Hopf algebras. Otherwise it is
an isomorphism of coalgebras.
13
For our calculation we need the bar spectral sequence for a principle
fibration:
F ______wE

 (2:6)

u
B
which has
E2*;*' TorK(n)*(F)*;*(K(n)*(E); K(n)*) ) K(n)*(B): (2:7)
If the fibration we use is the loops on a principle fibration, then the bar
spectral sequence is a spectral sequence of Hopf algebras. Unfortunately, de
spite a fascination with the bar construction, e.g. [May72 ], [May75 ], [Mey8*
*4 ]
and [Mey86 ], this fact is not in the literature. It can, of course, be patched
up easily from what is there about the standard bar construction. Let our
fibration be
F ______wE




u
B
where (2.6) is a principle fibration. Then F has two products, one from F
and one from the loops. When B is constructed using the bar construction,
one product can be used in the construction and the other can be used to
get a product on the bar construction giving us this spectral sequence as
Hopf algebras (the coalgebra structure is no problem). This works for any
homology theory with a K"unneth isomorphism
In particular, one has the bar spectral sequence converging to H*(Xs)
with
2Ks)
E2 = TorH*( (H*(Xs+1); Q): (2:8)
14
Here the H*(2Ks)module structure on H*(Xs+1) is induced by the map
js of (2.4). We have
Lemma 2.9 The map js in (2.4) induces the trivial homomorphism in ra
tional homology.
Proof. The map js is an Hmap, so it must respect the Pontrjagin ring
structure in homology. We know that H*(2Ks) is generated by elements in
dimension s  2, while Xs+1 is (s  1)connected, so H*(js) is trivial. 2
It follows that (2.8) can be rewritten as
2Ks)
E2 = Tor H*( (H*(Xs+1); Q)
2Ks)
= Tor H*( (Q; Q) H*(Xs+1)) (2.10)
= H*(Ks) H*(Xs+1))
The rational homology bar spectral sequence collapses for Eilenberg
Mac Lane spaces and has no extension problems, which explains the last step
above. Now it follows formally that the spectral sequence collapses, because
differentials must lower filtration by at least 2, but H*(Xs+1)) is concen
trated in filtration 0, and H*(Ks) is generated by elements in filtration 1.
Thus one gets the desired extension of Hopf algebras
Q ! H*(Xs+1) ! H*(Xs) ! H*(Ks) ! Q: (2:11)
If X is an Hspace, then so is Xs, so H*(Xs) is bicommutative. The struc
ture of graded connected bicommutative Hopf algebras over Q is well known
(see [MM65 , Section 7]). In particular, we have the following splitting theo
rem.
15
Theorem 2.12 Let A be a graded connected bicommutative Hopf algebra
over Q. Then there is a canonical Hopf algebra isomorphism
O
A ' Ak
k>0
where Ak is generated by primitive elements in dimension k. Moreover, Ak
is a polynomial algebra for k even and an exterior algebra for k odd.
It follows that the extension (2.11) is split when X is an Hspace and the
rational case of Theorem 2.1 is proved.
The Morava Ktheory case for torsion spaces
Now we will give the proof of Theorem 2.1 under the additional assump
tion that ss*(X) is all torsion. The general setup is the same as in the ration*
*al
case. The Morava Ktheory of EilenbergMac Lane spaces was computed in
[RW80 ]. We have the bar spectral sequence as in the rational case, and we
have the following analog of Lemma 2.9.
Lemma 2.13 The map js in (2.4) induces the trivial homomorphism in
Morava Ktheory. (Here we do not require that ss*(X) be all torsion.)
______
Proof. (See the introduction for K(n) *().) We will make use of Theorems
1.11 and 1.14. We are studying the map
____
______ K(n)*(js)______
K(n) *(2Ks) ! K(n) *(Xs+1):
Recall that we are using downward induction on s so we can assume Theorem
2.1 for all t > s. Assume our map is nontrivial. Choose the largest t so that
its image is contained in K(n)*(Xt). Then the composition
_____ _____ _____ _____ _____
K(n) *(2Ks) ! K(n)*(Xt) ! K(n)*(Xt)==K(n) *(Xt+1) = K(n) *(Kt)
16
must be nontrivial. This is a Hopf algebra map, and both the source and
target are subject to the splitting theorem 1.11, with the constraints imposed
______ __
by 1.14. The factors of K(n) *(2Ks) correspond to orbits fl with ffp(fl) =
______ __
s  2 or s  3, while those of K(n) *(Kt) have ffp(fl) = t  1 or t  2. These
orbits are distinct since s < t n + 1. Since there are no nontrivial Hopf
algebra homomorphisms between factors corresponding to distinct orbits, the
result follows. 2
It follows that the analog of (2.10) holds, namely, in the bar spectral
sequence,
2Ks)
E2 = TorK(n)*( (K(n)*; K(n)*) K(n)*(Xs+1):
2Ks) 2
Now, Tor K(n)*( (K(n)*; K(n)*) is the E term of the bar spectral se
quence converging to K(n)*(Ks), and this was completely determined in
[RW80 ]. There is a map to this spectral sequence from the one we are study
ing, given by the following commutative diagram, in which each row is a fibre
sequence.
js
2Ks ______wXs+1 ______wXs
  
  
  
=   
  
  
u u u
2Ks ________wpt. ________Ksw
In our spectral sequence (the one for the top row) the extra factor of
K(n)*(Xs+1) is concentrated in the even degrees (by induction) of filtration
0. In a spectral sequence of Hopf algebras the basic differentials must go from
generators to primitives, see [Smi70 , page 78]. In the bar spectral sequence,
differentials must start in filtration greater than one. All generators here
are in even degrees and so must go to odd degree primitives, all of which
are in filtration one [RW80 , Theorem 11.5]. So, the start and finish of the
generating differentials are all in the part which maps isomorphically to the
bottom row. It follows that in our spectral sequence,
E1 = E0K(n)*(Ks) K(n)*(Xs+1);
17
where E0K(n)*(Ks) denotes the E1 term of the lower spectral sequence,
which was determined in [RW80 ]. In particular, our E1 term is concentrated
in even dimensions.
Again the edge homomorphism gives us the desired Hopf algebra exten
sion, namely
K(n)* ! K(n)*(Xs+1) ! K(n)*(Xs) ! K(n)*(Ks) ! K(n)*: (2:14)
The following result is where we need the assumption that ss*(X) is tor
sion.
Lemma 2.15 When X is an Hspace with torsion homotopy, the extension
(2.14) is split naturally.
Proof. We use Theorems 1.11 and 1.14, and assume inductively that
______
K(n) *(Xs+1) is as advertised. This means that its factors under Theorem
______
1.11 all correspond to orbits fl with __ffp(fl) s, while those of K(n) *(Ks)
with __ffp(fl) = s  1. Thus Theorem 1.13 assures us that the extension is split
naturally. 2
Notice that the proof above would fail if the torsion subgroup of sss(X)
and the torsion free quotient of sss+1(X) were both nontrivial, because in that
______ ______
case, both K(n) *(Xs+1) and K(n) *(Ks) could have factors corresponding
to the same orbit fl. In particular, there is a short exact sequence of Hopf
algebras,
_____ _____ pi_____
Fp ! K(n)*(K(Z=(pi); j))! K(n)*(K(Z(p); j + 1))!K(n)*(K(Z(p); j +!1))Fp;
which shows how maps can drop filtration (from j to (j + 1)). This
example prevents us from having a natural Hopf algebra splitting in general.
We do have an unnatural splitting for the general case however, and we
produce that now.
18
Removing the torsion condition
We have given the proof of Theorem 2.1 in the case when ss*(X) is all
torsion. We needed the torsion condition to get the desired Hopf algebra
structure when X is an Hspace. We did not need it to show that K(n)*(Xs)
is a subalgebra of K(n)*(X). Now we will describe an alternate approach
to the Hopf algebra question which does not require ss*(X) to be torsion.
We need to use the rationalization XQ of X. For an Hspace (more
generally, a nilpotent space), Y , one has a fibre sequence
T Y  ! Y  ! Y Q
where ss*(Y Q) = ss*(Y ) Q and ss*(T Y ) is all torsion. For a plocal Hspace
Y (such as X(p)), Y Q is the homotopy direct limit of
[p] [p] [p]
Y  ! Y  ! Y  ! . . .
where [p] is the Hspace pth power map. In this case Y Q and T Y are both
Hspaces.
Now for X as in Theorem 2.1, we have a fibre sequence
2XQ ! T X !i X: (2:16)
Before we can proceed, we need to bring in the generalized Atiyah
Hirzebruch spectral sequence, see [Dol62 ] and [Dye69 ]. For a fibration like
(2.6), we have the AtiyahHirzebruchSerre spectral sequence:
E2 ' H*(B; K(n)*(F )) =) K(n)*(E): (2:17)
The map of E to B maps this spectral sequence to the usual AtiyahHirzebruch
spectral sequence:
19
E2 ' H*(B; K(n)*) =) K(n)*(B): (2:18)
Lemma 2.19 If ss2(X) Q = 0, then the map i of (2.16) induces an iso
morphism in K(n)* for all n > 0.
As remarked above (2.3), this assumption on ss2(X) can be made without
loss of generality.
Proof of Lemma 2.19. We will use the AtiyahHirzebruch spectral sequence
for Morava Ktheory to compute K(n)*(T X). We have the following com
mutative diagram in which each row is a fibre sequence.
i
2XQ ______wT X _______wX
  
  
  
 i  = (2:20)
  
  
u u u
pt. _________Xw ________Xw=
The hypothesis on ss2(X) implies that 2XQ is path connected. We know
that any rational loop space is a product of rational EilenbergMac Lane
spaces. From [RW80 , Corollary 12.2] we have
dir limK(n)*(K(Z=(pi); q)) ' K(n)*(K(Z; q + 1)): (2:21)
We see that iterating [p] on the left kills everything, so the product of
rational EilenbergMac Lane spaces has the Morava Ktheory of a point.
This means that the left vertical arrow in (2.20) is a K(n)*equivalence. The
right vertical map is a homology equivalence (since it is the identity map) so
we get an isomorphism of AtiyahHirzebruchSerre spectral sequences, so i
is a K(n)*equivalence. 2
20
Now the torsion case of Theorem 2.1 tells us that
O
K(n)*(T X) ' K(n)*(K(ssi(T X); i  1));
i2
and this is K(n)*(X) by Lemma 2.19. Thus Theorem 2.1 will follow from
Lemma 2.22 For a simply connected space X with ss2(X) torsion,
O O
K(n)*(K(ssi(X); i  1)) ' K(n)*(K(ssi(T X); i  1)):
i2 i2
as Hopf algebras.
Proof. There is a split short exact sequence
0 ! ssi+1(X) Q=Z ! ssi(T X) ! Tor1(ssi(X); Q=Z) ! 0:
Note that Tor1(ssi(X); Q=Z) is the torsion subgroup of ssi(X), while ssi+1(X)
Q=Z is the tensor product of Q=Z with the torsion free quotient of ssi+1(X).
With this in mind, we also have a split short exact sequence
0 ! Tor1(ssi(X); Q=Z) ! ssi(X) ! ssi(X)=Torsion ! 0:
According to (2.21)
K(n)*(K(Z; i + 1)) ' K(n)*(K(Q=Z; i)) for i 1;
so we have
K(n)*(K(ssi+1(X) Q=Z; i  1)) ' K(n)*(K(ssi+1(X)=Torsion; i)):
It follows that
O
K(n)*(K(ssi(T X); i  1))
i2
O
= K(n)*(K(ssi+1(X) Q=Z; i  1))
i2
21
O
K(n)*(K(Tor 1(ssi(X); Q=Z); i  1))
i2
O
= K(n)*(K(ssi+1(X)=Torsion; i))
i2
O
K(n)*(K(Tor 1(ssi(X); Q=Z); i  1))
i2
O
= K(n)*(K(ssi(X)=Torsion; i  1))
i2
O
K(n)*(K(Tor 1(ssi(X); Q=Z); i  1))
i2
O
= K(n)*(K(ssi(X); i  1)):
i2
2
Remark 2.23 The above process lost us our naturality in the splitting, but
not if all of the homotopy groups are free or torsion. The problems only come
up if we try to mix them.
Proof of Theorem 1.4. First let us assume that ssn+1(X) = ssn+2(X) = 0.
Theorem 1.1 tells us that K(n)*(X) is trivial and our result follows from
the bar spectral sequence.
Remark 2.24 We do not have to use Theorem 1.1 here at all. We can do
our downward induction on the Postnikov system directly. Everything will
be trivial so there is no difficulty showing the maps are also trivial and the
spectral sequence is trivial. One can just use Lemma 3.3 over and over again.
We use this result several times in our study of spectra; the point is that tho*
*se
results are independent of Theorem 1.1.
Next we consider the case where ssn+1(X) = 0. If ssn+2(X) = 0 or is
torsion, the same proof works. It is only if ssn+2(X) has copies of Z(p)in it
that we could have a problem. We take our usual fibration with map:
22
Kn+2 ______wXn+3 ________wX
  
  
  
=   
  
  
u u u
Kn+2 _______wpt. ________Kn+2:w
Since ssn+1(Xn+3) = ssn+2(Xn+3) = 0, we have K(n)*(Xn+3) is trivial.
Thus we get an isomorphism on the E2 terms of the two bar spectral se
quences. Thus the E1 terms must be isomorphic as well. However, since we
know, from [RW80 , Theorem 12.3], that
TorK(n)*(K(Z(p);n+1))(K(n)*; K(n)*) = K(n)*;
we get our result.
All we have left to deal with is the case where ssn+1(X) is finite. The
argument is exactly the same as that just given except that the Tor is trivial
by [RW80 , Theorem 11.5]. 2
3 Strongly K(n)*acyclic connective spectra
Throughout this section n will be assumed to be positive. For a connective
spectrum X, X_ m for m 0 will denote the mth space in the associated
spectrum. Be alert to our (unusual) convention that X_m have its bottom
cell in dimension m.
Definition 3.1 A connective spectrum X is strongly K(n)*acyclic if X_m
is K(n)*acyclic for some m 2 Z.
One might guess that any K(n)*acyclic spectrum is strongly K(n)*
acyclic, but we we will see below, (3.9), that this is not the case. We will
23
show, Theorem 3.7, that for such an X each space X_m is K(n)*equivalent
to a suitable finite Postnikov system.
Remark 3.2 We take a moment to show that we need not restrict ourselves
to positive spaces in the spectrum, such as in Theorem 1.6. If m 0, then
we see from the bar spectral sequence that X_m K(n)*acyclic implies X_m+1
is K(n)*acyclic. If we have a negative number, write X_m . Let X
be the stable mconnected cover of X. Then we have a stable cofibration:
m X ! X ! F
where F is a finite Postnikov system and we have X__0 = X_m . If
this is K(n)*acyclic, then X__k is K(n)*acyclic for all k 0. We
have an unstable fibration
F_k1 ! X__m+k ! X_k:
For big k, F_k1is also K(n)*acyclic by Theorem 1.4, so X_k is K(n)*acyclic
by the next lemma.
Lemma 3.3 Let
j
F  i!E ! B
be a fibration with F K(n)*acyclic. Then the map j is a K(n)*equivalence.
Proof. There is an AtiyahHirzebruchSerre spectral sequence, (2.17), con
verging to K(n)*(E) with
E2 = H*(B; K(n)*(F )):
It maps to the usual AtiyahHirzebruch spectral sequence converging to
K(n)*(B). Since F is K(n)*acyclic, this map is an isomorphism, giving
the desired result. 2
24
Proposition 3.4 Let m 0, X be a connective spectrum with X_m K(n)*
acyclic, and let Y be any connective spectrum. Then (X_^_Y_)_mis also K(n)*
acyclic.
Recall here that our convention is that X_m and (X_^_Y_)_mhave the same
connectivity.
Proof. We will argue by skeletal induction on Y . Assume for simplicity that
the bottom cell of Y is in dimension 0, so the 0skeleton Y 0is a wedge of
spheres. Thus (X_^_Y_0)_mis a product of X_m s and is therefore K(n)*acyclic.
For k > 0 consider the cofibre sequence
1X ^ Y k=Y k1 ! X ^ Y k1 ! X ^ Y k;
which we abbreviate by A ! B ! C. Now A_m+k1 (which may be con
tractible) is K(n)*acyclic by a similar argument, and we can assume induc
tively that B_m is K(n)*acyclic. We have a fibration
j
A_m+k1 i! B_m  ! C_m
and Cm is K(n)*acyclic by 3.3. This is true for all k > 0, so (X_^_Y_)_m
is K(n)*acyclic as claimed because it is the direct limit of K(n)*acyclic
spaces. 2
We need a standard result which we include here for completeness.
Lemma 3.5 Let X be a connective spectrum and H be the integral Eilenberg
Mac Lane spectrum. Then X ^ H is the product of EilenbergMac Lane spec
tra.
Proof. We argue by skeletal induction. We have
1Xk=Xk1 ^ H ! Xk1 ^ H ! Xk ^ H:
25
By induction we have Xk1 ^H is the product of EilenbergMac Lane spectra
with homotopy H*(Xk1; Z). The first term is just a bunch of k1Hs and so
factors through K(Hk1(Xk1); k  1) in the second term so the third term
is as claimed. 2
Now we can get some control over the connectivity m.
Lemma 3.6 Let X be a connective strongly K(n)*acyclic spectrum with
bottom cell in dimension 0. Then the space X_n+3 is K(n)*acyclic.
Proof. We assume that all spectra and spaces in sight are localized at the
__
prime p. Let H denote the integer EilenbergMac Lane spectrum, and let H
denote the fibre of the map S0 ! H.
__ 0
H ! S ! H
__
The bottom cell of H is in dimension q  1, where q = 2(p  1). Thus the
cofibre sequence
__
1X ^ H ! X ^ H  ! X
gives a fibre sequence
__ j
(X_^_H)__m1 i! (X_^_H_)_m+q1 ! X_m:
(Recall our convention!) Now (X_^_H)__m1 is an (m  2)connected product
of EilenbergMac Lane spaces, so it is K(n)*acyclic for m  1 > n + 1. Thus
j is a K(n)*equivalence for m n + 3. Iterating this argument we see that
__(s)
X_n+3 is K(n)*equivalent to (X_^_H___)_n+3+s(q1)for each s > 0. The latter
is K(n)*acyclic for some s by Proposition 3.4, so X_n+3 is also K(n)*acyclic.
2
We can use this lemma to prove the following.
26
Theorem 3.7 Let X be a connective strongly K(n)*acyclic spectrum with
bottom cell in dimension 0. Let X ! F be a map to a finite Postnikov
system which is an equivalence through dimension n+1. Then X_m is K(n)*
equivalent to F_m for all m 0. In particular, X_n+2 is K(n)*acyclic, and
if ss0(X) is torsion, X_n+1 is K(n)*acyclic.
Proof. Stably we have a fibre sequence
X0 ! X ! F
where X0 is a connected cover of X having bottom cell above dimension n+1.
Consider the fibre sequence
F_m1  ! X0_m0 ! X_m
(where m0> m+n+1 depends on the connectivity of X0). From Theorem 1.4
we know that F is strongly K(n)*acyclic. Since X is also strongly K(n)*
acyclic we can use Lemma 3.3 to see that X0_m0is also strongly K(n)*acyclic.
Hence X0_n+3 is K(n)*acyclic by Lemma 3.6.
Now consider the fibre sequence
j
X0_m0 i!X_ m ! F_m:
For positive m, m0 > m + n + 1 n + 2, so X0_m0 is K(n)*acyclic. By
Lemma 3.3, j is a K(n)*equivalence as claimed. In particular, X_ n+2 is
K(n)* equivalent to F_n+2, which by Theorem 1.4 is K(n)*acyclic. Since
X was arbitrary strongly K(n)*acyclic, we also know that X0_n+2 is K(n)*
acyclic, so j is also a K(n)*equivalence for m = 0. 2
Pete Bousfield has generalized this in more than one way. His results
are not restricted to Morava Ktheories and he does not need to work with
spectra. In a short note to us he derived these results from [Boua ]. More
recently, these have been made explicit in [Boub , Section 7]. Restricting our
attention to Morava Ktheories, his result of interest to us is:
27
Theorem 3.8 (Bousfield) An (n+2)connected Hspace Y is K(n)*acyclic
if and only if Y is K(n)*acyclic.
The following example shows that not all connective K(n)*acyclic spec
tra are strongly K(n)*acyclic.
Example 3.9 Let X be the spectrum BP from [Wil75 ] and [JW73 ]
for n > 1. The third author showed in [Wil75 ] that X_ m has torsion free
homology and therefore is not K(n)*acyclic for m = 2(pn  1)=(p  1). This
would contradict Lemma 3.6 if X were strongly K(n)*acyclic, so it is not.
It is, however, K(n)*acyclic. This can be seen most easily by considering
the exterior Hopf algebra on the Milnor Bockstein, Qn, as a subHopf algebra
of the cohomology
H*(BP ; Z=(p)) ' A=A(Q0; Q1; : :;:Qn1):
Then, by MilnorMoore, 4.4, [MM65 ], H*(BP ; Z=(p)) is free over
E(Qn1) and the first differential in the AtiyahHirzebruch spectral sequence
for K(n)*(BP ) kills everything and we find BP is K(n)*acyclic.
Theorem 3.10
(i)Let X ! Y ! Z be a cofibre sequence of connective spectra. If any
two of X, Y and Z are strongly K(n)*acyclic, then so is the third.
(ii)Any retract X of a connective strongly K(n)*acyclic spectrum Y is
also strongly K(n)*acyclic.
(iii)Any connective direct limit X of connective strongly K(n)*acyclic spec
tra is strongly K(n)*acyclic.
Proof. (i) is an easy consequence of Lemma 3.3. For (ii) note that X_m is a
retract of Y_m0 for suitable m0. For (iii) we can assume that each spectrum
in the direct system has bottom cell in dimension 0, so Y_n+2 is a direct
limit of K(n)*acyclic spaces. 2
28
We have so far produced no examples of strongly K(n)*acyclic spectra
other than finite Postnikov systems.
Proposition 3.11 Any finite K(n)*acyclic spectrum X is strongly K(n)*
acyclic.
Proof. X is a suspension spectrum of a (connected) finite complex Y , so for
some m, X_m = QY , where
QY = lim!kkY:
k
Since K(n)*(Y ) ' K(n)*(X), Y is K(n)*acyclic. All we have to do is show
that Y K(n)*acyclic implies that QY is K(n)*acyclic. There is a stable
splitting, [Sna74 ], for QY . Each piece of this stable splitting is identified
explicitly as something called e[C(j); j; Y ] from [May72 , Proposition 2.6(ii*
*),
page 14]. Since C(j) is contractible, [May72 , page 5], this is the same as
Dj Y ; for which there is a spectral sequence, [CLJ76 , Corollary 2.4, page 7]:
E2 ' H*(j; K(n)*(Y (j))) ) K(n)*(Dj Y )
where Y (j)is the jth smash product. For reduced Morava Ktheory,
K(n)*(Y (j)) = 0 because we have a K"unneth isomorphism. Thus each piece
of the stable splitting is K(n)*acyclic and we have that QY is K(n)*acyclic.
This is well known to those familiar with this but, to a novice, perhaps a bit
difficult to dig out of [Sna74 ]. 2
Corollary 3.12 Any connective spectrum which is a direct limit of finite
K(n)*acyclic spectra is strongly K(n)*acyclic.
Proof. This follows immediately from Theorem 3.10 and Proposition 3.11. 2
Now recall the localization functors Ln of [Rav84 ] and Lfnof [Rav93 ].
The former is Bousfield localization with respect to E(n), while the latter is
constructed in such a way that the fibre of the map X ! Lfnis a direct limit
of finite K(n)*acyclic spectra. Thus we get
29
Corollary 3.13 A connective spectrum X for which LfnX is contractible is
strongly K(n)*acyclic.
Proof. Since LfnX is contractible, the fibre of the map X ! Lfnis equivalent
to X. Since X is a connective spectrum which is now the direct limit of finite
K(n)*acyclic spectra, X is strongly K(n)*acyclic by Corollary 3.12. 2
There is a natural transformation n : Lfn! Ln. The telescope con
jecture, which is known to be false for n = 2, see [Rav ] and [Rav92b ], is
equivalent to the assertion that n is an equivalence. It is shown that for
n = 2 there is a spectrum X for which LnX is contractible but LfnX is not.
It is necessarily a torsion spectrum (its rational homology must vanish) and
its connective cover has the same property. However we do not know if such
a spectrum is strongly K(n)*acyclic or not.
We also know of no counterexample to the converse of Corollary 3.13,
so perhaps that is an equivalence. Alternatively, one can ask if a connective
spectrum X is strongly K(n)*acyclic if and only if E(n)*(X) = 0. In [Rav84 ,
Theorem 2.1(d)] it was shown that E(n)*(X) = 0 if and only if K(i)*(X) = 0
for 0 i n. It is also known, [JY80 ], that E(n)*(X) = 0 if and only if
v1nBP*(X) = 0.
The following is a consequence of Corollary 3.13.
Theorem 3.14 If X is a connective spectrum with E(n)*(X) = 0, then
BP ^ X is strongly K(n)*acyclic. In particular if X is also a BP module
spectrum then it is strongly K(n)*acyclic. The same holds with BP replaced
by any connective spectrum E with Bousfield class dominated by that of BP .
Proof. By the smash product theorem [Rav92a , 7.5.6], E(n)*(X) = 0 if and
only if X ^ LnS0 is contractible. We also know [Rav93 , Theorem 2.7(iii)]
that LnS0 and LfnS0 are BP*equivalent and therefore E*equivalent. Thus
we have
pt. ' E ^ X ^ LnS0 ' E ^ X ^ LfnS0 ' Lfn(E ^ X)
30
(the last equivalence is Theorem 2.7(ii) of [Rav93 ]) and E ^ X is strongly
K(n)*acyclic by Corollary 3.13. If E is a ring spectrum and X is an E
module spectrum, then it is a retract of E ^ X,
X ' S0 ^ X ! E ^ X ! X
so it is strongly K(n)*acyclic by Theorem 3.10(ii). 2
Corollary 3.15 If X is a BP module spectrum in which each element of
ss*(X) is annihilated by some power of the ideal
In+1 = (p; v1; v2; . .v.n) BP*;
then X is strongly K(n)*acyclic.
Proof. The hypothesis implies that v1nss*(X) = 0, so v1nBP*(X) = 0,
which is equivalent to E(n)*(X) = 0 as noted above. 2
Examples of spectra satisfying these hypotheses include the P (k) of
[JW75 ] (with ss*(P (k)) = BP*=Ik) for k > n and spectra obtained from
BP by killing an ideal containing some power of In+1.
4 Morava homology Hopf algebras
We want to study a category of Hopf algebras which includes the objects of
our interest: the Morava Ktheory (homology) of homotopy commutative,
connected Hspaces. We need this study to solve Hopf algebra extension
problems in the bar spectral sequence during the inductive step of the proof
of our main theorem. To do this we have a general Hopf algebra splitting
theorem which is of interest in its own right. We want to give particular
thanks to Hal Sadofsky for help with this section.
Although our Hopf algebras will be bicommutative, and so give an abelian
category, we can run into serious problems because of the cyclical grading
31
we use. In particular, we can have an element with the property x = xp. (In
______
K(n) *(K(Z=(p); n)) for example.) This wreaks havoc with all of the algebra
portion of MilnorMoore [MM65 ]. Such an algebra has no generator (i.e.,
indecomposable), and the first proposition of MilnorMoore is false for our
situation. The coalgebra portion of MilnorMoore fares much better. Before
we move on to Hopf algebras we want to indicate why our coalgebras are
still nice by making some definitions and reproving a basic result which still
holds in our setting.
Let A be a cocommutative, coassociative, coaugmented coalgebra with
counit over a ring, R. At present we are not concerned with gradings so
this could be ungraded, graded or cyclically graded. To avoid unnecessary
complications, we assume that A is flat over R. Let J(A) be the cokernel of
the coaugmentation map:
0 ! R ! A ! J(A) ! 0:
Using the iterated coproduct we define an increasing filtration, FqA, by
FqA = kerA ! J(A)q+1
for all q 0. Note that F0A ' R and F1A=F0A ' P (A), the primitives of A.
We call this the primitive filtration of A. Dualizing MilnorMoore, [MM65 ,
page 252], we could call this the coaugmentation filtration. (Milnor and
Moore reserve the name primitive filtration for their filtration on primitively
generated Hopf algebras.)
Following [Boa81 ] we say a filtration, F*A, is exhaustive (or exhausts A)
if every element of A is in some FqA. We say A is a (connected) homology
coalgebra if A is a cocommutative, coassociative, coaugmented coalgebra
over R with counit, its primitive filtration exhausts A, and it is the direct
limit of finite dimensional R subcoalgebras. The exhaustive condition on A
replaces connectivity in the graded case quite nicely. To justify the name we
have the following observation:
Lemma 4.1 Let E*() be a multiplicative homology theory and X a con
nected CW complex of finite type such that E*(X) is flat over E*, then E*(X)
32
is a homology coalgebra over E*.
Proof. Because E*(X) is flat over E*, the K"unneth isomorphism holds for
finite products of X. The diagonal, X ! X x X, induces a cocommutative,
coassociative coalgebra over E*. The map of a point into X (because X is
connected) and the map of X to a point give the coaugmentation and counit
respectively. Again, we need connectivity to show the primitive filtration is
exhaustive. An element of E*(X) which lives on the 0cell maps trivially
to J(E*(X)). Since E*(X) is the direct limit of E*(Xq) where Xq is the q
skeleton, any element x in E*(X) comes from E*(Xq) for some q (this shows
the coalgebra is the direct limit of finite dimensional E* subcoalgebras).
Q q+1
Getting a cellular approximation to the iterated diagonal map, X ! X
we see that on some coordinate Xq maps to the 0cell and thus our element
must be in FqE*(X) and the primitive filtration is exhaustive. 2
We can also prove a standard result which, as we see, does not depend
on a grading, but just on having an exhaustive primitive filtration. This is
well known and is even somewhere in the algebraic topology literature but
we cannot remember where we have seen it. In the other literature, it could
well be that Lemma 11.0.1 on page 217 of [Swe69 ] could prove it; but it is a
lot easier to reprove it than to be sure of that.
Lemma 4.2 Let A ! B be a map of coalgebras where A is a homology
coalgebra, then the map injects if and only if the induced map on primitives
injects.
Proof. Since P (A) A, an injection on A is automatically an injection on
P (A). In the other direction our proof is by induction on the degree of the
primitive filtration of an element. We are given that P (A) ' F1A injects to
ground our induction. If we have an element, x 2 FqA but not in Fq1A,
P 0 00
then the coproduct takes x to x x in J(A) J(A) where each nonzero
x0and x00must be in a lower filtration and so they inject to J(B) J(B). 2
33
We say that A is a (connected) homology Hopf algebra if it is an asso
ciative Hopf algebra with unit such that the coalgebra structure is that of a
homology coalgebra. Clearly, if our X above is an Hspace, then E*(X) is a
homology Hopf algebra. If the algebra structure is commutative, we say A is
a commutative homology Hopf algebra . If X above is a homotopy commu
tative Hspace (e.g. any double loop space) then E*(X) is a commutative
homology Hopf algebra.
Since the exhaustive condition replaces connectivity so nicely, we can
define a Hopf algebra conjugation on homology Hopf algebras. This is done
P 0 00
inductively on the primitive filtration. Let x ! x 1 + 1 x + x x ,
then C(x) = x  x0C(x00) inductively. Following [MM65 , Proposition 8.8,
page 260], we have C O C = IA because our coalgebra is cocommutative. The
existence of C is essential to show that commutative homology Hopf algebras
are an abelian category, see [Gug66 ]. We are in a slightly generalized situati*
*on
over the usual connected bicommutative Hopf algebras of finite type so it is
worth a short discussion of our case. If A is a coalgebra and B is an algebra,
then two maps from A to B can be combined to get a third, still following
[MM65 , Section 8], by
fg
A ! A A ! B B ! B:
To get an abelian category we need Hom (A; B) to be an abelian group. In
particular, the above map must be a Hopf algebra map. For A ! A A to
be a Hopf algebra map, A must be cocommutative. For B B ! B to be a
Hopf algebra map, B must be commutative. So, we need the bicommutativ
ity of commutative homology Hopf algebras to get our composition back in
our category. The Hopf algebra conjugation above gives us our inverse and
bicommutativity shows Hom (A; B) is an abelian group. Following [MM65 ,
Sections 3 and 4], bicommutativity allows us to define, for all f : A ! B, a
kernel
A\\f ' R 2B A ' A 2B R
and a cokernel
B==f ' R A B ' B A R:
34
Tensor product is the product in our category and given maps f : A ! B
fg
and g : A ! C, we get (f; g) : A ! A A  ! B C. The rest of the
conditions to be an abelian category are easy to verify.
Theorem 4.3 The category of commutative homology Hopf algebras is abelian.
In the graded connected case over a perfect field of characteristic p, com
mutative homology Hopf algebras of finite type have been classified nicely
in [Sch70 ]. This applies to the case where E is standard mod p homology
theory. Unaware of Schoeller's work, the second author had also embarked
on such a project. His project was aborted when he learned about Schoeller's
paper and the only physical remains of this project are in [Rav75 ]. His ap
proach was quite different from hers and quite easily extends to the case at
hand, so we are now reaping the benefits of his study in this paper.
What concerns us here are what we will call commutative Morava ho
mology Hopf algebras. These are just commutative homology Hopf algebras
over Fp which are cyclically graded over Z=(2(pn  1)) for some n. These
arise naturally when you take the Morava Ktheory, K(n)*(X), where X is
a connected homotopy commutative Hspace, and set vn equal to one as we
do throughout the rest of this paper. We do only what we need with these
Hopf algebras here, but we hope to return to the problem of classifying them
in a future paper. Because of the cyclic grading there is a much richer, more
interesting collection of Hopf algebras than in the standard case. The Morava
Ktheory of EilenbergMac Lane spaces in [RW80 ] supplies lots of examples
unlike anything seen in the normal graded case.
Remark 4.4 At this stage we must point out a problem and its solution for
the prime p = 2. K(n) is not a homotopy commutative ring spectrum for
p = 2, so K(n)*(X) is not necessarily in our category. However, if K(n)*(X)
is even degree it is. For a discussion of this problem see the Appendix of
[JW82 ] where, following [W"ur77 ], [RW80 ] is shown to hold for p = 2.
35
Because of the cyclic grading, it is not unusual to find ourselves dealing
with infinite dimensional vector spaces; for example, a polynomial algebra
with one primitive generator already gets us into that situation. However,
we can use finiteness when we need it because our coalgebras are the direct
limit of finite dimensional subcoalgebras. Many Hopf algebra categories are
self dual, an extremely nice property. Unfortunately, ours is not self dual.
We can define the dual category, however.
Proposition 4.5 Let A be a commutative homology Hopf algebra over a field
R. It is the direct limit of its finite dimensional subcoalgebras Aff. Let its
dual be defined by
A* = limHom R(Aff; R):
Then A* is a compact topological bicommutative Hopf algebra under the in
verse limit topology. Moreover A is the continuous linear dual of A*.
Proof. The product and coproduct in A* are induced respectively by the
coproduct and product in A. A* is compact because it is the inverse limit
of finite dimensional vector spaces. The continuous linear dual of an inverse
limit is the direct limit of the continuous linear dual, and
Hom R(Hom R(Aff; R); R) = Aff
so the continuous linear dual of A* is A. 2
This result enables us to make the following definition.
Definition 4.6 Let A be a commutative homology Hopf algebra over Fp. The
Frobenius map F : A ! A is the Hopf algebra homomorphism that sends
each element to it pth power. The Verschiebung map V : A ! A is the
dual of the Frobenius map on A*
We are ready to look closely at the category of commutative Morava
homology Hopf algebras and say what we need to say about it. We propose
36
to split up every Hopf algebra into canonical components. Because it is
all we need for this paper, we restrict our attention to evenly graded Hopf
algebras. Thus our Morava Hopf algebras are really graded over G = Z=(pn
1). Denote by EC(n) the category of evenly graded commutative Morava
homology Hopf algebras (for a given n). Let H be the cyclic group of order
n. H acts on G via the pth power map. Writing G additively, the map
H x G ! G is given by (i; j) ! pij. Let fl denote an Horbit:
fl = {j; pj; p2j; : :}: G:
Recall Theorem 1.11 from the introduction, which uses these definitions.
Remark 4.7 We illustrate the result with the case p = 2 and n = 3. Then
H and G have orders 3 and 7 respectively and there are three orbits, namely
fl1 = {1; 2; 4};
fl2 = {3; 6; 5} and
fl3 = {0}:
We also know, from [RW80 ], that for i = 1; 2 or 3, K(3)*(K(T; i)) has its
primitives in dimensions in 2fli for any torsion abelian group T .
We discuss this more later, but in general, K(n)*(K(T; i)) can have fac
tors with generators in more than one orbit. For example when p = 2, n = 4
and i = 2 there are two such orbits, namely {3; 6; 9; 12} and {5; 10}. How
ever each orbit is associated with a unique value of i, namely i = __ffp(j), the
sum of the digits of the padic expansion of any j 2 fl.
Theorem 1.12 follows from Theorem 1.13 by:
Proof of Theorem 1.12. All we need to prove here, after we see how our
functors are constructed, is that there are no nontrivial maps Afl! Bfiif
fl 6= fi. Such a map, f, must commute with efiwhich is the identity on Bfi
but is the trivial map on Afland so it must be the trivial map. 2
37
Proof of Theorem 1.13. Our concern for the rest of this section is to construct
the idempotents efland prove Theorem 1.13. A Hopf algebra, A, in EC(n)
is equipped with the usual Frobenius and Verschiebung endomorphisms F
and V (see 4.6). The relation, V F = F V = p, is a simple calculation. Do
not confuse this p with multiplication by p. This is p times the identity in
the abelian group of endomorphisms of A. Thus the ring of endomorphisms
of A which ignores the grading is a module over Z(p). Because the primitive
filtration is exhaustive every element of A is annihilated by some power of V
and hence by some power of p. This means that the endomorphism ring is
also a module over the padic integers Zp.
Given an element a 2 Fp we can define an endomorphism [a] of A to be
multiplication by ai in dimension 2i. If we replace A by A Fpn, we have
an endomorphism [a] for each a 2 Fpn. Hence the endomorphism ring of
A Fpn is a module over Zp[G], the padic group ring over G = Z=(pn  1),
which is isomorphic to the multiplicative group of the field Fpn.
Remark 4.8 Although our applications in this paper only require us to study
evenly graded Hopf algebras, our study of these Hopf algebras would be in
complete without splitting the category into evenly graded Hopf algebras and
exterior Hopf algebras on odd degree generators for odd primes. Classical sign
arguments force a Hopf algebra with odd degree primitives to be an exterior
algebra. The idempotents necessary for this splitting are just ([1] [1])=2.
To check this we need only evaluate on primitives. ([1]  [1])=2 is the iden
tity on odd primitives and trivial on even primitives. The reverse is true for
([1] + [1])=2.
We want to use Zp[G] to construct idempotents of A. We need some
basic facts to do this. Let M be an R module and let H act on R and M.
Then we have a map j : R ! End (M) and H acts on f 2 End (M) by
(hf)(x) = h(f(h1(x))). Then
Lemma 4.9 If the map j is Hequivariant then the fixed points, RH act on
the fixed points MH .
38
Proof. If m 2 MH and r 2 RH then
(jr)(m) = (jh(r))(m) = [h(jr)](m) = h(jr(h1m)) = h(jr(m)):
2
We have our group H = Z=(n). H is isomorphic to the Galois group of
Fpn over Fp, and its action on the ring Zp[G] corresponds to the action of
the Galois group on the units of Fpn. We will write G multiplicatively from
now on. We want to show that if an element x 2 Zp[G] is fixed by this action
then the corresponding endomorphism leaves A = (A Fpn)H A Fpn
invariant. Hence the endomorphism ring of A is a module over the fixed
point subring Zp[G]H . By our lemma, we must just show that the map
Zp[G] ! End (A Fpn) is Hequivariant. It is enough to show this on [g].
Let a 2 A and z 2 Fpn. Then:
h
j(h([g]))(a z) = j([gp ])(a z)
ha
= a gp z
h
= h(a gaz1=p )
h
= h(j[g])(a z1=p )
= h(j[g])h1(a z):
We construct some idempotents in an even bigger ring, W (Fpn)[G], where
W (Fpn) denotes the Witt ring of Fpn, i.e., the extension of Zp obtained by
adjoining a primitive (pn1)throot of unity !. See [Haz78 , page 132, Remark
17.4.18]. We will use the notation ! for both the root of unity in W (Fpn)
and its mod p reduction in Fpn. For each j 2 G let
X !ij[!i]
ej = ________n: (4:10)
1ipn1 p  1
Note that ej is independent of the choice of the primitive root !. One sees
easily that the ej are orthogonal idempotents, i.e.,
(
ej ifj = j0
ejej0 = (4.11)
0 otherwise
39
X
and ej = 1:
j
Now let
X k
cm = !mp 2 Zp W (Fpn) (4:12)
k
where the sum is over all distinct roots of the indicated form. Thus the
number of summands is some divisor of n depending on m. This element is
in Zp because it is invariant under the action of the Galois group H, whose
generator sends each root of unity to its pth power.
Now let fl be the Horbit containing j and consider the element
X
efl = ejpk
k
X !ijpk[!i]
= _________n
i;k p  1
X cij[!i]
= _______n;
i p  1
where the first sum is as in (4.12). This element is idempotent by (4.11). It
is in Zp[G] W (Fpn)[G] since the coefficients cij all lie in Zp.
Furthermore eflis fixed by the Galois action since cijpk= cij, so it lies
in the invariant subring Zp[G]H . This is the idempotent giving the splitting
of Theorem 1.11.
In order to verify this, it suffices to show that it behaves appropriately
on primitive elements. Let x be a primitive in dimension m. Recall that the
endomorphism ring of a Hopf algebra acts additively on its primitives, i.e.,
for endomorphisms ff and fi and coefficients a; b 2 Zp, we have
(aff + bfi)(x) = aff(x) + bfi(x)
where the coefficients on the right are reduced mod p. With this in mind we
40
have
X cij[!i](x)
efl(x) = __________n
i p  1
X cij!imx
= _________n
i p  1
X !imijpkx
= _________n
i;k p  1
k!
X X !i(mjp )
= _________nx:
k i p  1
The inner sum is 1 when m = jpk and 0 otherwise, so we have
(
x ifm 2 fl
efl(x) =
0 otherwise
as desired. 2
In the introduction we defined a function __ffp: Z=(pn  1) ! Z by taking
the sum of the coefficients of the padic expansion. We observed that __ffpis
constant on orbits fl. From Theorem 1.13, we have:
Q
Corollary 4.13 Let EC(n)q = __ffp(fl)=qEC(n)flfor all 0 q n(p  1).
Then:
Y
EC(n) = EC(n)q:
0qn(p1)
This is now what we need for our Hopf algebras. We actually need very
little knowledge about K(n)*(K(Z=(pi); q)). We remind the reader that
______
K(n) *(X) K(n)*(X) K(n)*Fp
where the K(n)* module structure of Fp is given by sending vn to 1. The
following is a restatement of Theorem 1.14.
41
______
Theorem 4.14 ([RW80 ]) K(n) *(K(Z=(pi); q)) is in EC(n)q for 0 < q
______ ______
n, and is K(n) * ' Fp for q > n. K(n) *(K(Z(p); q + 1)) is the direct limit of
______
K(n) *(K(Z=(pi); q)) and is also in EC(n)q.
Proof. The theorem is read off from [RW80 , Theorem 11.1] where we see the
primitives are all in degrees 2(pi1+ pi2+ : :+:piq), 0 = i1 < i2 < . .<.iq < n,
for Z=(pi) and [RW80 , Corollary 12.2] for Z(p). In fact, all primitives come
______
from K(n) *(K(Z=(p); q)). 2
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