THE GENERALIZED HOMOLOGY OF PRODUCTS
MARK HOVEY
Abstract.We constructQa spectral sequence that computes the generalized
homology E*( Xff) of a product of spectra. The E2-term of this spectral
sequence consists of the right derived functors of product in the catego*
*ry of
E*E-comodules, and the spectral sequence always converges when E is the
Johnson-Wilson theory E(n) and the Xffare Ln-local. We are able to prove
some results about the E2-term of this spectral sequence; in particular,*
* we
show that the E(n)-homology of a product of E(n)-module spectra Xffis ju*
*st
the comodule product of the E(n)*Xff. This spectral sequence is relevant*
* to
the chromatic splitting conjecture.
Introduction
The basic tools of computation in algebraic topology are homology theories.
Homology theories preserve coproducts, but can behave very badly on products.
There are examples of homology theories E and setsQof spectra (generalized spac*
*es)
{Xff}, for which E*Xff= 0 for all i and yet E*( ffXff) 6= 0. Indeed, we can ta*
*ke
E = HQ, rational homology, where we have (HQ)*(HZ=pk) = 0 for all k, but
Y Y
(HQ)*( HZ=pk) = ( Z=pk) Q 6= 0
k k
since, for example, the element (1, 1, 1, . .).is not torsion.
Despite this counterexample,Qin this paper we build a spectral sequence that
converges to E*( Xff) in good cases. The most important good case is when
E = E(n), the Johnson-Wilson theory of great importance in stable homotopy
theory. The E2 term of this spectral sequence is made up of the right derived
functors of product applied to {E*Xff}. Of course, the product is exact in the
category of E*-modules, so these derived functors are instead taken in the cate*
*gory
of E*E-comodules, where products remain mysterious.
The usefulness of this spectral sequence will depend on our knowledge of its
E2-term.Q At this point, the author knows very little about the derived functors
s ff
E(n)*E(n)M of product in the categoryQof E(n)*E(n)-comodules. The most
important conjecture about them is that sE(n)*E(n)Mff= 0 for all s N for
some N, so that the spectral sequence has a horizontal vanishing line at the E2
term (we show that the spectral sequence does have a horizontal vanishing line *
*at
some Er term). We expect that N is very close to n itself.
____________
Date: March 16, 2005.
1991 Mathematics Subject Classification. 55N45, 55P60, 55T99, 18G10, 16W30.
1
2 MARK HOVEY
We do prove that derived functors of product can be computed using relatively
injective resolutions, such as the cobar complex, rather that honest injective *
*reso-
lutions. It follows that
Y Y
E(n)*( Xff) ~= E(n)*Xff
E(n)*E(n)
for a family of E(n)-module spectra Xff. We also construct a spectral sequence
relating derived functors of product in the category of E(n)*E(n)-comodules to
derived functors of product in the category of BP*BP -comodules. The category
of BP*BP -comodules is easier to cope with since BP*BP is connective and free
over BP*. These results give the author hope that these derived functors will be
understood at some point, though at the moment he does not even understand
them in the simple case of E(1)*E(1)-comodules.
The reason for the author's interest in this spectral sequence is the chromat*
*ic
splitting conjecture [5] of Mike Hopkins. Recall that the simplest form of the
chromatic splitting conjecture is that K(n - 1)*LK(n)X is a direct sum of two
copies of K(n - 1)*X, for X a finite p-complete spectrum. Also recall that LK(n*
*)X
is a homotopy inverse limit holimI(LnX ^ S=I) analogous to completion at the
ideal (p, v1, . .,.vn-1). This result is due to Hopkins; a precise statement of*
* it can
be found in [8, Proposition 7.10]. Therefore, if one has a spectral sequence f*
*or
the E(n - 1)-homology of a homotopy inverse limit, one might be able to compute
E(n - 1)*(LK(n)X) and therefore K(n - 1)*(LK(n)X).
This approach to the chromatic splitting conjecture is due to Mike Hopkins, a*
*nd
is based on the work of Paul Goerss [4], who constructed a spectral sequence fo*
*r the
mod p homology of a homotopy inverse limit of spaces. Hopkins suggested this id*
*ea
to Hal Sadofsky and the author after a talk by Goerss. Sadofsky has constructed*
* a
spectral sequence for the E(n)-homology of a homotopy inverse limit, as envisio*
*ned
by Hopkins, and has proved some results about it that are relevant to the chrom*
*atic
splitting conjecture. Unfortunately, Sadofsky has not yet made a preprint of h*
*is
work available.
The author decided instead to begin with the simpler case of products, though
the methods used in this paper can also be used to construct a version of Sadof-
sky's spectral sequence. To the author's knowledge, Sadofsky has not considered
products. But the author acknowledges his heavy debt to the work of Sadofsky. He
also would like to thank Mike Hopkins for his original suggestion, and Paul Goe*
*rss
for his paper [4], without which this paper would never have been written.
1.The modified Adams tower
The first step in constructing a spectral sequence is to resolve the object o*
*ne is
considering. In our case, the resolution we need is called the modified Adams t*
*ower
and is due to Devinatz and Hopkins [3]. The idea is to mimic the usual construc*
*tion
of an injective resolution using E*-injectives, where E is a well-behaved homol*
*ogy
theory. We will have to assume that E is a commutative ring spectrum such that
E*E is flat over E*; it is well-known [11, Proposition 2.2.8] that this implies*
* that
(E*, E*E) is a flat Hopf algebroid and that E*X is naturally a left E*E-comodule
for a spectrum X. It also implies that E*E-comodules form an abelian category [*
*11,
Theorem A1.1.3] with enough injectives [11, Lemma A1.2.2].
The following definition is taken from [3].
THE GENERALIZED HOMOLOGY OF PRODUCTS 3
Definition 1.1. Let E be a commutative ring spectrum such that E*E is flat
over E*. Define a functor D from injective E*E-comodules to the stable homotopy
category S as follows. Given an injective E*E-comodule I, consider the functor *
*DI
from spectra to abelian groups defined by
DI(X) = Hom E*E(E*X, I).
Then DI is a cohomology functor, so there is a unique spectrum D(I) such that
there is a natural isomorphism
DI(X) ~=[X, D(I)].
The hypotheses we have given on E are sufficient to define D(I), but appar-
ently insufficient to compute E*D(I). For this we need some form of the followi*
*ng
definition; this particular form comes from [6].
Definition 1.2. A ring spectrum E is called topologically flat if E is the mini*
*mal
weak colimit of a filtered diagram of finite spectra Xi such that E*Xi is a fin*
*itely
generated projective E*-module.
Minimal weak colimits are discussed in [7, Section 2.2]. Adams [1, Section II*
*I.13]
proves that many standard spectra such as BP are topologically flat; in additio*
*n,
any Landweber exact commutative ring spectrum over BP or MU is topologically
flat [6, Theorem 1.4.9]. Note that if E is topologically flat, then E*E is flat*
* over
E*, since it is the colimit of projective modules.
The following theorem is a translation of Theorem 1.5 of [3] to this terminol*
*ogy.
Theorem 1.3. Suppose E is a topologically flat commutative ring spectrum, and
I is an injective E*E-comodule. Then there is a natural isomorphism E*D(I) ~=I.
We can now describe the modified Adams tower. Let E be a topologically flat
commutative ring spectrum, and suppose we have a spectrum X. Let C = E*X,
and choose an injective resolution
0 -!C j-!I0 o0-!I1 o1-!. . .
of C in the category of E*E-comodules. Let js: Cs -!Is denote the kernel of os,
so that j0 = j.
As explained in [3, Section 1], we can use this resolution of C to build a to*
*wer
over X with good properties. More precisely, we have the following lemma, which
is easily proved by induction on n.
Lemma 1.4. Let E be a topologically flat commutative ring spectrum, let X be
a spectrum, and choose an injective resolution of E*X as above. Then there is a
tower g g g
X = X0 ---0- X1 ---1- X2 ---2- . . .
?? ?
yf0 ?yf1
K0 K1
over X satisfying the following properties.
(a) Ks = -sD(Is).
(b) Xs+1 is the fiber of fs.
(c) E*Xs ~= -sCs.
(d) The map fs is induced by the inclusion Cs -!Is.
4 MARK HOVEY
(e) E*gs = 0, and the boundary map Ks -! Xs+1 induces the surjection
-sIs -! -sCs+1 on E*-homology.
We call this tower the modified Adams tower for X based on E-homology. Of
course, it actually depends on the injective resolution as well. We obtain a sp*
*ectral
sequence by applying [Z, -] for any Z to get the modified Adams spectral sequen*
*ce
of Devinatz [3]; its E2-term is Ext**E*E(E*X, E*Y ), it is independent of the c*
*hoice
of resolution from the E2 page on, and in good cases it converges to [Z, LE X]*.
2.Products of comodules
In order to understand the spectral sequence for products of spectra, we need
to know a little about products of comodules. So suppose (A, ) is a flat Hopf
algebroid. As mentioned above, basic facts about the category of -comodules can
be found in [11, Appendix 1], though he does not discuss products. A more in-de*
*pth
look at the global structure of the category of -comodules, including products*
*, can
be found in [6].
The main point of interest here is that the forgetful functor to A-modules do*
*es
not preserve products. It is easiest to understand this when is freePover A. *
*In this
case, every element m in a -comodule M has a diagonal of the form fli mi,
where fli runs through a basis of as a rightQA-module, and all but finitely m*
*any
of the mi are zero. In the A-module product ffMffof comodules Mff, there may
well be elements whose diagonal would have toQbe infinitely long. In fact,Qwhen
is projective over A, the comodule product Mffis the submodule of Mff
consisting of those elements whose diagonal lands in
Y Y
A Mff ( A Mff).
To construct the product when is only assumed to be flat over A, one first
checks that Y Y
( A Nff) ~= A ( Nff)
for A-modules Nff, where A P denotes the extendedQ -comodule, in which
coacts only on the factor. One then constructs fff, where fffis an arbitra*
*ry
map of extended comodules. Finally, given arbitrary comodules Mff, we have exact
sequences of comodules
ff
0 -!Mff-_! A Mff-f-! A Nff,
where Nffis the cokernel of _, and fffis the composite
A Mff-! Nff-_! A Nff.
Q Q
It follows that Mff~=ker fff. Details can be found in [6].
This construction shows that the product of comodules is more complicated
than one would want; in particular, it is not always exact (see the example bef*
*ore
Proposition 1.2.3 of [6]). As a right adjoint, of course, the product is left *
*exact.
Since thereQare enough injective -comodules, the product will have right deriv*
*ed
functors sMfffor s 0. Almost nothing is known about these right derived
functors, but they are what will appear as the E2-term in our spectral sequence.
For the construction of our spectral sequence, we need the following proposit*
*ion.
THE GENERALIZED HOMOLOGY OF PRODUCTS 5
Proposition 2.1. Suppose E is a topologically flat commutative ring spectrum, a*
*nd
{Iff} is a family of injective E*E-comodules. Then there is a natural isomorphi*
*sm
Y Y
D( Iff) -! D(Iff).
E*E
Q
Here the notation E*Edenotes the product in the category of E*E-comodules.
Q
Proof.Note that E*EIffis again an injective comodule. The functoriality of D
guarantees the existence of this map. Now, if X is an arbitrary spectrum, we ha*
*ve
a chain of isomorphisms
Y Y Y
[X, D( Iff)] ~=Hom E*E(E*X, Iff) ~= Hom E*E(E*X, Iff)
E*E E*E
~=Y [X, D(Iff)] ~=[X, Y D(Iff)].
This gives us the desired isomorphism.
3.Construction of the spectral sequence
We can now use the modified Adams towers of Lemma 1.4 to construct our
spectral sequence.
Theorem 3.1. Let E be a topologically flat commutative ring spectrum, and let
{Xff} be a family of spectra. There is a natural spectral sequence E***({Xff}) *
*with
ds,tr:Es,tr-!Es+r,t+r-1rand E2-term
Q s
Es,t2~=( E*E E*Xff)t.
This is a spectral sequence of E*E-comodules, in the sense that each ds,*ris a *
*map
of E*E comodules of degree r - 1. Furthermore, every element in E0,t2in the ima*
*ge
of the natural map M Q
E*Xff-! E*E E*Xff
is a permanent cycle.
Proof.We have modified Adams towers Xffsfor each Xff. Taking the product gives
us the tower below.
Q Q gff0Q Q gff1Q Q gff2
Xff ---- Xff1---- Xff2---- . . .
? ?
(3.2) ?yQfff0 ?yQfff1
Q Q
Kff0 Kff1
By applying E*-homology, we getQan associated exact coupleQand spectral sequenc*
*e.
That is, we let Ds,t1= Et-s( Xffs) andQEs,t1= Et-s( Kffs). We define i1: D -!D
of bidegree (-1,Q-1) by is,t1= Et-s( gffs), we define j1: D -!E of bidegree (0*
*, 0)
by js,t1= Et-s( fffs), and weQdefine k1:QE -!D of bidegree (1, 0) in bidegree *
*(s, t)
to be Et-sof the boundary map Kffs-! Xffs+1. All of these maps are maps of
comodules, and therefore the resulting spectral sequence will be a spectral seq*
*uence
of comodules, as claimed.
By combining Proposition 2.1 with Theorem 1.3, we see that
Y Y Y
Es,t1~=Et-s( -sD(Iffs)) ~=EtD( Iffs) ~=( Iffs)t.
E*E E*E
6 MARK HOVEY
Q
One can easily check that the first differential d1 is E*Eoffs, and therefore*
* that
the E2-term is as claimed.
Nautrality now follows in the usual way; a collection of maps Xff-!Y ffinduces
non-canonical maps of the injective resolutions in question, and hence the modi*
*fied
Adams towers. Taking products gives us a map of spectral sequences, which is
canonical from E2 onwards.
Finally, we can also construct a spectral sequence by taking the wedge of the
modified Adams towers ofLthe Xffand applyingLE* homology. This gives a spectral
sequence with Ds,t1= ( Cffs)tLand Es,t1= ( Iffs)t. The d1 differential is *
*the
obvious one, and so E0,*2~= E*Xffand Es,t2= 0 for s > 0. There is a map from
the spectral sequence to the spectral sequence for the product of the Xff. Anyt*
*hing
in the image of this map of spectral sequences must be a permanent cycle.
4.Convergence of the spectral sequence
We now discuss the convergence of our spectral sequence. This is a delicate
question, in general, as the example given at the beginning of the paper shows.
However, the spectral sequence always converges when E = E(n) and each Xffis
E(n)-local.
Theorem 4.1. Suppose E = E(n) and each Xffis Ln-local.Q Then the spectral
sequence of Theorem 3.1 converges strongly to E(n)*( Xff). Furthermore, it has
a horizontal vanishing line at some Er term.
Proof.First note that each Xffsis Ln-local, since Kffs= -sD(Iffs) is clearly L*
*n-
local. Each map gffs:Xffs+1-!Xffshas E(n)*(gffs) = 0. It follows from [10, Theo-
rem 5.10] that there is an N, depending on n but independentQof ff, suchQthat e*
*ach
N-fold composite Xffs+N-!Xffsis null. Hence each composite Xffs+N-! Xffsis
null, giving us our desired horizontal vanishing line. Hence
Y Y
limsE(n)*( Xffs) = lim1sE(n)*( Xffs) = 0
Q
so the spectral sequence converges conditionally to E(n)*( Xff) [2]. It is al*
*so
clear that lim1rEs,tr= 0, and so the spectral sequence converges strongly as we*
*ll [2,
Theorem 7.3].
5. Relatively injective resolutions and an application
Although we cannot prove very much about the derived functors of products, we
can at least show that one can use relatively injective comodules to compute th*
*em.
This allows us to compute the E(n)-homology of products of E(n)-module spectra.
Proposition 5.1. Let (A, ) be a flat Hopf algebroid,Qand suppose Mffis a rela-
tively injective -comodule for all ff. Then sMff= 0 for s > 0.
Proof.Since Mffis relativelyQinjective, it is a retract of A Mff. It theref*
*ore
suffices to show that s( A Mff) = 0 for all s > 0. To do so, choose an inje*
*ctive
resolution Iff*of Mffin the category of A-modules. Since is flat over A, A *
*Iff*is
a resolution of A Mffin the category of -comodules. Furthermore, each A If*
*fs
is an injective -comodule [11, Lemma A1.2.2]. Hence
Qs ff ~ s Q ff ~ s Q ff
( A M ) = H ( ( A I* )) = H ( A ( I* )).
Since products are exact on the category of A-modules, and since is flat, the*
*se
groups are 0 for s > 0.
THE GENERALIZED HOMOLOGY OF PRODUCTS 7
This yields an immediate topological corollary.
Corollary 5.2. Suppose Xffis an E(n)-module spectrum for all ff. Then
Y Y
E(n)*( Xff) ~= E(n)*(Xff).
E(n)*E(n)
In particular,
Y Y
E(n)*( E(n) ^ Xff) ~=E(n)*E(n) E(n)*( E(n)*Xff).
Proof.Since Xffis an E(n)-module spectrum, it is Ln-local. Furthermore, E(n)*Xff
is a retract of
E(n)*(E(n) ^ Xff) ~=E(n)*E(n) E(n)*E(n)*Xff,
so is relatively injective. Proposition 5.1 then implies that the E2-term of o*
*ur
spectral sequence is 0 except in bidegree (0, t). It therefore collapses, and w*
*e get
the desired isomorphism.
It also follows, using standard homological algebra, that we can use relative*
*ly
injective resolutions to compute the derived functors of product. For example, *
*we
can use the cobar resolution C*(M) described in [11, Definition A1.2.10].
Corollary 5.3. Suppose (A, ) is a flat Hopf algebroid, and {Mff} is a set of
-comodules. Let C*(Mff) denote the cobar resolution on Mff. Then
Q s ff~ s Q * ff
M = H C M .
This corollary tellsQus, for example, that if JMff= 0 for some invariant idea*
*l J
and all ff, then J sMff= 0 for all s.
6.BP*BP-comodules and E(n)*E(n)-comodules
In this section, we exploit the close relationship between BP*BP -comodules
and E(n)*E(n)-comodules studied in [9] to get some partial understanding of the
product of comodules.
We begin with BP*BP -comodules, which are easier to handle because BP*BP
is connective and projective over BP*. As mentioned inQSection 2, the product o*
*f a
family {Mff} of BP*BP -comodules is the submodule of Mffconsisting of those
elements whose diagonal has finite length.
Definition 6.1. A family of BP*BP -comodules {Mff} is uniformly bounded
below if there is a d 2 Z such that Mffn= 0 for all n < d and all ff.
The product and its derived functors are particularly simple for a uniformly
bounded below family.
Theorem 6.2. Suppose (A, ) = (BP*, BP*BP ), and {Mff} is a family of -
comodules that is uniformly bounded below. Then
Q ff~ Q ff Q s ff
M = M and M = 0 for alls > 0.
Q
Proof.Since every element of Mffmust have finite diagonal, the first statement
is clear. For the second statement, consider the cobar_resolution_C*Mffof Mff
by relatively injective comodules. We have CsMff= A s A Mff, so, since
8 MARK HOVEY
is connective, the family {CsMff} is uniformly bounded below for each s. We
therefore have Q
s ff~ s Q ff~ s Q ff
M = H C*M = H C*M = 0
for s > 0, using Corollary 5.3 and the fact that products of modules are exact.
To relate this to E(n)*E(n)-comodules, we recall from [9] and [10] the exact
functor * from BP*BP -comodules to E(n)*E(n)-comodules defined by *M =
E(n)* BP* M. The functor * has a fully faithful right adjoint *, the composite
* * is naturally isomorphic to the identity, and the composite Ln = * * is
the localization functor on the category of BP*BP -comodules with respect to the
hereditary torsion theory of vn-torsion comodules. The functor Ln is left exact*
*, but
has right derived functors Lqnfor 0 q n, studied in [10].
As a left adjoint, we do not expect * to preserve products. We do, however,
have the following result.
Theorem 6.3. Suppose {Mff} is a family of BP*BP -comodules. Then there is a
natural isomorphism
Y Y
*Mff-! *( LnMff).
E(n)*E(n) BP*BP
In fact, there is a convergent first quadrant spectral sequence Ep,qrof E(n)*E(*
*n)-
comodules with
Q p Q p+q
Ep,q2~= *( BP*BP (LqnMff)) ) E(n)*E(n) *Mff.
Proof.Since * is a right adjoint, we have
Y Y Y
*Mff~= * *( *Mff) ~= * (LnMff),
E(n)*E(n) E(n)*E(n) BP*BP
as required. The spectral sequence is the Grothendieck spectral sequence for the
derived functors of the composition, described in [12, Section 5.8]. Recall tha*
*t this
spectral sequence has Ep,q2= (RpF )(RqG)(-) and converges to Rp+q(F G)(-), un-
der the assumption that (RpFQ)(GI) = 0 for all injectives I and p > 0. In the c*
*ase
at hand, the functor F is * BP*BP (-) and the functor G is Ln (applied object-
wise to the product category). Since Ln preserves injectives [10, Corollary 2.*
*4],
the Grothendieck spectral sequence exists. Since * is exact and products of in*
*jec-
tives are injective,Qwe can use another Grothendieck spectral sequence argument*
* to
see that RpF = * pBP*BP(-). Similarly, since * is exact and preserves injec-
tives [10, Corollary 2.5], another Grothendieck spectral sequence argument shows
that Q Q
Rp+q(F G)(-) = Rp+q( E(n)*E(n) *)(-) = p+qE(n)*E(n) *(-),
completing the proof.
This proposition allows us to compute some products of E(n)*E(n)-comodules.
For example, we have
Y Y
ffE(n)* ~=E(n)* BP* ffBP*,
E(n)*E(n)
as long as the ff are bounded below. To see this, use the fact that BP* is Ln-
local [10], Theorem 6.2, and Theorem 6.3.
THE GENERALIZED HOMOLOGY OF PRODUCTS 9
In fact, we have Q
s ff
E(n)*E(n) E(n)* = 0
for 0 < s < n and
Qs ff ~ Q s-n ff 1 1 1
E(n)*E(n) E(n)* = * BP*BP BP*=(p , v1 , . .,.vn )
for s n, again under the hypothesis that the ff are bounded below. This follo*
*ws
from the spectral sequence of Theorem 6.3 and the fact [10] that LqnBP* = 0 exc*
*ept
when q = 0 and q = n, where
L0nBP* = BP* and LnnBP* = BP*=(p1 , v11, . .,.v1n).
Q
Note that we do not know whether BP*BP ffBP*=(p1 ,Qv11, . .,.v1n) is all vn-
torsion or not, and therefore we do not know whether nE(n)*E(n) ffE(n)* is ze*
*ro
or not.
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org