SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY
MARK HOVEY
Abstract.Morava E-theory is a much-studied theory important in under-
standing K(n)-local stable homotopy theory, but it is not a homology the*
*ory
in the usual sense. The object of this paper is to show that the usual c*
*ompu-
tational methods, spectral sequences, used to compute with homology theo*
*ries
also apply to Morava E-theory. Conceptually, we show that Morava E-theory
is n derived functors away from being a homology theory. Thus we estab-
lish spectral sequences with only n + 1 non-trivial filtrations to compu*
*te the
Morava E-theory of a coproduct or a filtered homotopy colimit.
Introduction
The usual approach to understanding stable homotopy theory is through a pro-
cess of localization. The first step is to localize the stable homotopy categor*
*y at a
fixed prime p. Once this is done, there are indecomposable field objects, known*
* as
the Morava K-theories K(n), for n a nonnegative integer. The case n = 0 is simp*
*le
and well-understood, so we will fix n > 0 throughout this paper, and drop it fr*
*om
the notation.
One can then further localize the (p-local) stable homotopy category at K. Th*
*is
is the most drastic localization one can perform, as the K-local stable homotopy
category has no nontrivial localizations. It was studied extensively in [HS99 *
*]. It
is a closed symmetric monoidal triangulated category, but to form the coproduct
and the smash product, one must apply the localization functor LK to the ordina*
*ry
coproduct and smash product. The result of this is that there are very few homo*
*logy
theories in the K-local stable homotopy category, because X_*(-) = ß*LK (X ^ -)
is usually not a homology theory as it fails to commute with coproducts.
Of course, we still have the homology theory K* itself. But the functor LK
is much more like an algebraic completion than an algebraic localization, in the
sense that it retains a lot of non-torsion information. This information is not
easily accessible from K*(-). So the theory E_*(-) becomes very important, even
though it is not a homology theory. Here one can take E to be LK E(n), as was
done in [HS99 ], but it seems better to take E to be the theory introduced by
Morava [Mor85 ], which we call Morava E-theory. This is the Landweber exact
homology theory with
E* ~=W Fpn[[u1, . .,.un-1]][u, u-1],
where W Fpn is the Witt vectors of Fpn, so an unramified extension of Zp of deg*
*ree
n, each of the ui has degree 0, and u has degree 2. Note that E0 is a complete
Noetherian regular local ring with maximal ideal m = (p, u1, . .,.un-1). The le*
*tter
E will denote this theory throughout the paper. It is also convenient to use t*
*he
symbol K for the spectrum E=m; this is a field spectrum with K* ~=Fpn[u, u-1].
____________
Date: March 1, 2004.
1
2 MARK HOVEY
As a spectrum, K is a finite wedge of copies of K(n), so this does not conflict*
* with
our previous notation LK .
One indication of the importance of Morava E-theory is the fact that X is sma*
*ll in
the K-local category if and only if E_*X is finite [HS99 , Theorem 8.5]. Furthe*
*rmore,
LK S0 is in the thick subcategory generated by E [HS99 , Theorem 8.9].
The purpose of this paper is to show that, although E_*(-) is not a homology
theory, many of the spectral sequences that one uses to compute with homology
theories still apply, and there are some new spectral sequences as well. These *
*are
the results one needs to compute with Morava E-theory, but we do not make any
computations in this paper. Conceptually, we show that Morava E-theory is n
derived functors away from being a homology theory.
The basic idea is that E_*(-) does not just take values in the category of E*-
modules; instead, it is a functor to the category cM of L-complete E*-modules.
These were discussed in [HS99 , Appendix A], and are studied in more detail in
Section 1. As an abelian category, Mc is closed symmetric monoidal and bicom-
plete with enough projectives, but it has some unfamiliar properties as well. M*
*ost
interestingly, direct sums are not exact in general, but instead have precisely*
* n - 1
left derived functors. As a result, projectives are not flat in cM when n > 1, *
*and
so the Torfunctor is not balanced; in computing TorcMs(M, N), it matters which *
*of
M and N one chooses to resolve.
The basic plan of the rest of the paper, then, is that if F is some functor on
K-local spectra, with an analogous right exact algebraic functor F on Mc, there
should be a spectral sequence converging to E_*(F X) whose E2-term is
E2s,t= [(LsF )(E_*X)]t,
where LsF denotes the sth derived functor of F .
In practice, it is more convenient to work in the categories DE and LK DE . H*
*ere
DE is the derived category of E-module spectra, and LK DE is its K-localization.
The point is that E_*X = ß*LK (E ^ X), and LK (E ^ X) is naturally an ele-
ment of LK DE . We discuss this category in Section 2, relying heavily on the
results of [EKMM97 ], and on the recently proved fact that E is a commutative
S-algebra [GH , Corollary 7.6]. In particular, we prove that ß*M 2 Mc when
M 2 LK DE in Proposition 2.5.
Now, for M in DE , LK M is a topological version of the completion of M at
the maximal ideal m. There should then be a spectral sequence from the derived
functors of completion applied to ß*M, converging to ß*LK M. Greenlees and May
built such a spectral sequence in [GM95 ], which we recall in Section 3. We then
discuss the universal coefficient and Kunneth spectral sequences in cohomology *
*in
Section 4. These again follow from previous work, in this case of [EKMM97 ], *
*but
this application of their work appears not to have been noticed before.
The universal coefficient and Kunneth spectral sequences in homology, discuss*
*ed
in Section 5, are more complicated, because of the failure of projectives to be*
* flat
in cM. We give two versions of the spectral sequence converging to ß*LK (M ^E N)
when M, N 2 LK DE . The first case, when ß*M is finitely generated, is just a
special case of the work of [EKMM97 ], but the second case, when ß*M is a pro*
*-free
E*-module, is new. W
In Section 6, we turn our attention to computing ß*LK ( Mi) for Mi2WLK DE .
Said another way, we construct a spectral sequence converging to E_*( Xi) whose
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 3
E2-term involves the derived functors of direct sum in Mc. In particular, when
n = 1, E_*(-) actually preserves coproducts as a functor to Mc. We use this in
Section 7 to show that, when n = 1, the homotopy groups of a K-local spectrum
commute with coproducts as a functor to cM.
Finally, we construct a spectral sequence to compute the Morava E-theory of a
filtered homotopy colimit in Section 8. This is much more technically difficult*
* than
the preceding spectral sequences, and involves some model category theory that *
*we
have included as an appendix.
The work in this paper grew out of conversations the author had with Neil
Strickland in the late 1990's. In particular, Strickland predicted that the hom*
*otopy
of a K-local spectrum would commute with coproducts when n = 1, as long as
the coproducts were taken in Mc. It is likely that some of the theorems in this
paper were also proved by Strickland. The author thanks him for many helpful
discussions over the years.
Let us recall the notation we use throughout this paper. The prime number p a*
*nd
the non-negative integer n are fixed throughout the paper. E denotes the Morava
E-theory corresponding to p and n, with maximal ideal m in E*. Also, K denotes
E=m, a version of Morava K-theory. The symbol LK denotes Bousfield localization
with respect to K, and we sometimes use the symbol X ^K Y for LK (X ^ Y ). The
symbol E_*X denotes ß*(E ^K X). The category cM is the category of L-complete
E*-modules, discussed below.
1. L-complete modules
For a spectrum X, E_*X is not just an E*-module; it is an L-complete E*-
module. To understand E_*(-), then, we need to first study the category Mc of
L-complete E*-modules, which we do in this section. Our basic reference for this
category is [HS99 , Appendix A], which is based on [GM92 ].
1.1. Basic structure. The basic issue is that the completion at m functor is not
left or right exact on the category of graded E*-modules. So we replace complet*
*ion
by the functor L0M, where
L0M = ExtnE*(E*=m1 , M).
Here E*=m1 is defined as usual in algebraic topology. Thus E*=p1 is the quotient
p-1E*=E*, and E*=(p1 , u11) is the quotient of u-11(E*=p1 ) by E*=p1 , and we
continue in this fashion.
There is a natural surjection fflM : L0M -!M^m, whose kernel is
lim1TorE*1(E*=mk, M),
by [HS99 , Theorem A.2]. Unlike completion, L0 is right exact, and so has left
derived functors Li. In fact, we have
LiM ~=Extn-iE*(E*=m1 , M)
by [HS99 , Theorem A.2(d)], and so, in particular, LiM = 0 for i > n.
The natural map M -!M^mfactors through a natural map jM : M -!L0M, and
a module M is called L-complete if jM is an isomorphism. The full subcategory of
graded L-complete E*-modules will be denote by cM. The module E_*X is always
L-complete [HS99 , Proposition 8.4].
4 MARK HOVEY
The category cM is an abelian subcategory of E*-mod, closed under extensions
and inverse limits [HS99 , Theorem A.6]. Note that Mc contains all the finitely
generated E*-modules, since E* itself is L-complete and L0 is right exact. The
functor L0: E*-mod -! cM is left adjoint and left inverse to the inclusion func*
*tor
i: cM -! E*-mod, and therefore creates colimits in cM. That is, if F :I -! cM is
a functor, then the colimit of F in cM is L0(colimiF ). Thus cM is a bicomplete
abelian category.
Furthermore, cM is closed symmetric monoidal. The monoidal product is
__
M N = L0(M E* N)
as explained in [HS99 , Corollary A.7]. The closed structure is the usual one
Hom E*(M, N), because this is already L-complete. Indeed, write M as cokerf
for some map f between free E*-modules. Then Hom (M, N) is the kernel of
Hom (f, N), which is again L-complete since L-complete modules are closed un-
der products and kernels.
It is also known that Mc has enough projectives [HS99 , Corollary A.12]. We
will study these projectives in more detail below. But Mc will not have enough
injectives. Indeed, when n = 1, cM has no nonzero injectives at all. If I were *
*such
an injective, then by mapping the sequence
n
0 -!E* p-!E* -!E*=pn -!0
into I, we would find that pnI = I. Since I is L-complete, this implies that I *
*= 0
by [HS99 , Theorem A.6(d)].
The strangest feature of cM is that filtered colimits, and even direct sums, *
*need
not be exact. See Section 1.3.
1.2. Projectives in Mc . The projectives in Mc are described in [HS99 , Theo-
rem A.9]. They are all of the form L0F = Fm^for some free E*-module F , and as
such are sometimes called pro-free modules. It is proved in [HS99 , Corollary A*
*.11]
that products of projectives in cM are again projective. This might seem to be *
*the
complete story, but we show in this section that in fact filtered colimits of p*
*rojec-
tives in cM are again projective. It follows that L0F is projective in cM for a*
*ny flat
E*-module F .
We begin with an alternative characterization of projectives that was left out
of [HS99 , Theorem A.9].
Lemma 1.1. If P is in cM, then P is projective in cM if and only if
TorE*s(E*=mk, P ) = 0
for all k > 0 and all s > 0.
Proof.If this condition holds, then P is projective in cM by [HS99 , Theorem A.*
*9].
Conversely, again using [HS99 , Theorem A.9], if P is projective in cM, then
TorE*s(E*=m, P ) = 0
for all s > 0. We then prove by induction on k that
TorE*s(E*=mk, P ) = 0
for all k > 0 and all s > 0. To do so, we use the short exact sequence
0 -!mk-1=mk -!E*=mk -!E*=mk-1 -!0
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 5
and the fact that mk-1=mk is an E*=m-vector space.
Often, however, one has an E*-module M that is not L-complete, and one wants
to know that L0M is pro-free.
Proposition 1.2. Suppose M is an E*-module for which TorE*s(E*=m, M) = 0 for
all s > 0. Then L0M ~=M^mis pro-free and LsM = 0 for s > 0.
Proof.The argument of Lemma 1.1 implies that
TorE*s(E*=mk, M) = 0
for all s, k > 0. It follows from [HS99 , Theorem A.2(b)] that
L0M ~=M^mand LsM = 0 fors > 0.
If we can show that
TorE*1(E*=m, L0M) = 0
then Theorem A.9 of [HS99 ] will imply that L0M is pro-free, completing the pro*
*of.
Choose a projective resolution
. .-.!P2 -!P1 -!P0 -!E*=m
for E*=m where each Pi is a finitely generated free E*-module. We will show
that Hs(Po L0M) = 0 for all s > 0. Since each Pi is finitely generated, we can
apply [HS99 , Proposition A.4], which reduces us to showing that HsL0(Po M) = 0.
To do so, let K0 = E*=m and let Ki denote the image of Pi in Pi-1 for i > 0.
Since TorE*s(K0, M) = 0 for all s > 0, we have short exact sequences
Ki M -!Pi-1 M -!Ki-1 M
for all i > 0. We now apply L0 to get long exact sequences. However, since
LsM = 0 for s > 0 and Pi-1is projective, the long exact sequence degenerates in*
*to
isomorphisms
Ls+1(Ki-1 M) ~=Ls(Ki M)
for s > 0, and an exact sequence
0 -!L1(Ki-1 M) -!L0(Ki M)
-! L0(Pi-1 M) -!L0(Ki-1 M) -!0.
On the other hand, K0 M is already m-complete, since it is an E*=m-vector space.
Hence it is also L-complete by [HS99 , Theorem A.6], and so Ls(K0 M) = 0 for
all s > 0, again by [HS99 , Theorem A.6]. It follows that Ls(Ki M) = 0 for all
s > 0 and all i. Hence
L0(Ki M) -!L0(Pi-1 M) -!L0(Ki-1 M)
is a short exact sequence for all i, from which it follows that HiL0(Po M) = 0*
* for
all i > 0.
The following corollary is immediate.
Corollary 1.3. If F is a flat E*-module, then L0F = Fm^is pro-free and LsF = 0
for all s > 0.
It also follows easily that filtered colimits of projectives in cM are projec*
*tive.
6 MARK HOVEY
Theorem 1.4. If
F :I -!E*-mod
is a filtered diagram of pro-free E*-modules, then L0(colimF ) ~=(colimF )^mis *
*pro-
free. Thus projective objects in cM are closed under filtered colimits.
Proof.By Lemma 1.1, TorE*s(E*=mk, F (i)) = 0 for all s, k > 0 and all i. Hence
TorE*s(E*=mk, colimF ) = 0 for all s, k > 0. Now apply Proposition 1.2 to compl*
*ete
the proof.
Another useful fact about projectives in cM is the following lemma.
Lemma 1.5. If f :M -! N is a map in Mc , then f is surjective if and only
if f=m is surjective. If M and N are pro-free, then f is a split monomorphism
(isomorphism)if and only if f=m is a monomorphism (isomorphism).
Proof.If f is surjective, then certainly f=m is surjective. Conversely, suppose*
* f=m
is surjective. Consider the short exact sequence
imf -!N -!cokerf.
Both imf and cokerf are L-complete modules by [HS99 , Theorem A.6(e)]. Fur-
thermore, we have cokerf=m = 0 by assumption. But this means that cokerf = 0
by [HS99 , Theorem A.6(d)], and so f is surjective.
The second statement follows from the first and Proposition A.13 of [HS99 ].
1.3. Derived functors of colimit. We now investigate the failure of filtered co*
*l-
imits to be exact in cM. One can give a simple example of this for n = 1. Indee*
*d,
consider the system of short exact sequences below.
i
0 ----! Zp -xp---!Zp ----! Z=pi ----! 0
flfl ? ?
fl xp?y ?yxp
0 ----! Zp ----! Zp ----! Z=pi+1----! 0
xpi+1
With the usual colimit, this gives us the short exact sequence
0 -!Zp -!Qp -!Z=p1 -! 0.
However, upon applying L0 we get the sequence
0 -!Zp -!0 -!0 -!0
which is clearly not exact.
We will see that the direct sum is exact for n = 1, but it need not be exact
for n > 1. The following example of this was obtained in joint work with Neil
Strickland. Let n = 2, let Mi= E*=pi, and let fi: Mi-! Mi be multiplication by
ui1. We claim that
M M M
L0( fi): L0( Mi) -!L0( Mi)
i i i
is not injective. To see this, note from [HS99 , Theorem A.2(b)] that it suffic*
*es to
show that
M M
(1.6) lim1kTorE*1(E*=mk, Mi) -!lim1kTorE*1(E*=mk, Mi)
i i
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 7
is not injective. One can easily check that
TorE*1(E*=mk, Mi) ~=mk-i=mk,
where mk-i = E* for i k. The map induced by fi is then visibly 0 on this Tor
group. Since Torcommutes with direct sums, it follows that the map of 1.6 is al*
*so
0. Hence it suffices to show that
M M
lim1kTorE*1(E*=mk, Mi) ~=lim1k mk-i=mk 6= 0.
i i
For this, we note that for fixed i, the tower {mk-i=mk} is pro-trivial, becau*
*se
every i-fold composite is 0. Hence
limkmk-i=mk = lim1kmk-i=mk = 0.
Hence the map
Y Y
di: mk-i=mk -! mk-i=mk
k k
is an isomorphism, where di(xk) = (xk - _____xk+1) and _____xk+1is the image of*
* xk+1 in
mk-i=mk. Thus we get the commutative square
Q L k-i k d Q L k-i k
k im? =m ----! k im? =m
?y ?y
Q Q k-i k d Q Q k-i k
k im =m ----! k im =m
where the vertical arrows are injections, the bottom horizontal arrow is an iso*
*mor-
phism, and the cokernel of the top horizontal map is the group
M
lim1k mk-i=mk.
i
Q
Now, consider the element b of kimk-i=mk such that bki= 1 ifQk L i, and bki= 0
otherwise. Then (db)ki = 0 for i k + 1, and therefore db 2 k imk-i=mk.
Hence db must represent a nontrivial element of
M
lim1k mk-i=mk,
i
as required.
As a left adjoint, of course, direct sums and filtered colimits are right exa*
*ct.
So we should consider the left derived functors of direct sum and filtered coli*
*mit.
For this, we need to know that there are enough projectives in cM and its diagr*
*am
categories.
Lemma 1.7. If I is a small category, then the category F (I, cM) is a bicomplete
abelian category with enough projectives. Furthermore, each projective functor*
* is
pointwise projective.
Proof.It is well-known and easy to check that functor categories into abelian c*
*at-
egories are abelian. Limits and colimits are taken pointwise. Now Mc itself h*
*as
enough projectives, by [HS99 , Corollary A.12]. For an object i 2 I, the funct*
*or
8 MARK HOVEY
Evi: F (I, cM) -! cM defined by Evi(X) = Xi is exact and has a left adjoint Fi.
Here Fi is defined by
a
(FiM)j = I(i, j) x M = M,
I(i,j)
where the coproduct is of course taken in cM. The reader can check that Fi is l*
*eft
adjoint to Evi, and so preserves projectives. Given a diagram X, then, we choose
surjections`Pi -!Xi for all i 2 I. These give maps FiPi -!X of diagrams, and
the map i2IFiPi -! X is then a surjection from a projective, as desired.` In
particular, if X is itself a projective functor, then X is a retract of i2IFi*
*Pi, and
so is pointwise projective.
Recall that LiM 2 cM for all M, by [HS99 , Theorem A.6].
Theorem 1.8. Let I be either a set or a filtered category, and let colimicMdeno*
*te
the ith left derived functor of colimcM:F (I, cM) -!Mc . Then
colimicMF ~=Li(colimF ),
where the latter colimit is taken in the category of E*-modules.
Proof.Since L0: E*-mod -! cM preserves all colimits, the claim is certainly true
for i = 0. Furthermore, given a short exact sequence
0 -!F 0-!F -! F 00-!0
of functors, we get a short exact sequence
0 -!colimF 0-!colimF -! colimF 00-!0
since filtered colimits and direct sums are exact in the category of E*-modules.
This gives us a long exact sequence
. .-.!Lj+1colimF 00-!LjcolimF 0-!LjcolimF
-! LjcolimF 00-!Lj-1colimF 0-!. ...
The formal properties of derived functors will imply the theorem, then, as long*
* as
Lj(colimP ) = 0 for all j > 0 when P is a projective functor.
Since P is projective, P (i) is projective for all i 2 I. By Lemma 1.1, this *
*means
that
TorE*j(E*=mk, P (i)) = 0
for all k > 0 and all j > 0. Since Torcommutes with filtered colimits, we concl*
*ude
that
TorE*j(E*=mk, colimP ) = 0
for all j, k > 0. Proposition 1.2 now implies that Lj(colimP ) = 0 for all j > *
*0.
Theorem 1.8 implies that filtered colimits and direct sums in cM have at most*
* n
left derived functors. In fact, direct sums have at most n - 1.
L
Proposition 1.9. If {Mi} is a family of L-complete modules, then Ln( Mi) = 0.
In particular, if n = 1, then direct sums are exact.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 9
Proof.Using the embedding of the sum into the product we see that
M M
Ln( Mi) = Hom E*(E*=m1 , Mi)
is a subobject of
Y Y Y
Hom E*(E*=m1 , Mi) ~= Hom E*(E*=m1 , Mi) = LnMi.
Since each Mi is L-complete, LnMi = 0 by [HS99 , Theorem A.6], giving us the
desired result.
1.4. The Ext functor. We now investigate the Ext functor in the category cM,
proving that ExtscM(M, N) ~=ExtsE*(M, N) for all L-complete modules M and N.
We begin by noting that, although L0 is not exact, it does preserve certain pro*
*jective
resolutions.
Proposition 1.10. Suppose M 2 cM, and
. .-.!F2 -!F1 -!F0 -!M
is a resolution of M in E*-mod by flat E*-modules. Then
. .-.!L0F2 -!L0F1 -!L0F0 -!M
is a projective resolution of M in cM. In particular, M has projective dimensio*
*n at
most n in cM.
Proof of Proposition 1.10.Divide the given resolution into short exact sequences
0 -!Ki+1-! Fi-! Ki-! 0
with K0 = M. We will prove by induction on i that LjKi= 0 for all j > 0. This
is true for i = 0 since M 2 Mc, by [HS99 , Theorem A.6(b)]. Now assume that
LjKi = 0 for all j > 0, and apply L0 to the short exact sequence above. We get
isomorphisms LjKi+1~= LjFi for all j > 0. Since Fi is flat, Corollary 1.3 tells*
* us
LjFi= 0 for all j > 0.
It now follows easily that
. .-.!L0F2 -!L0F1 -!L0F0 -!M
is still a resolution of M, and Corollary 1.3 implies that it is a projective r*
*esolution
in cM.
Theorem 1.11. Suppose M, N 2 cM. Then ExtscM(M, N) ~=ExtsE*(M, N) for all
s.
Proof.Let P* -!M be a projective resolution of M, so that
ExtsE*(M, N) ~=Hs Hom E*(P*, N).
On the other hand, it follows from Proposition 1.10 that L0P* is a projective
resolution of M in cM, so
ExtscM(M, N) ~=HsMc (L0P*, N).
Now the fact that L0 is left adjoint to the inclusion functor completes the pro*
*of.
10 MARK HOVEY
1.5. The Tor functor. We_now investigate the derived functors of the symmet-
ric monoidal product M N on Mc. This is a difficult issue, because projectives
need not be flat, since direct sums are not exact. This means that TorcMs(M, N)
does not really make sense, because it depends on whether one resolves M or
N. We will define L TorcM*(M, N) to be the functors obtained by resolving M,
__
so that L TorcM*(M, N) are the left derived functors of (-) N. Similarly, we d*
*e-
fine R TorcM*(M, N) to_be the functors obtained by resolving N, which are the l*
*eft
derived functors of M (-).
Proposition 1.12. Suppose M, N 2 cM. If M is finitely generated, then
L TorcMs(M, N) ~=TorE*s(M, N).
Similarly, if N is finitely generated, then
R TorcMs(M, N) ~=TorE*s(M, N).
Proof.We just prove the first statement, as the second one is similar. Suppose
M is finitely generated. Choose a projective resolution P* -!M of M as an E*-
module such that each Ps is finitely generated. Then P* = L0P* is also a projec*
*tive
resolution of M in cM. Thus
__
L TorcMs(M, N) = Hs(P* N).
But __
P* N ~=L0(Ps E* N) ~=Ps E* N
by [HS99 , Proposition A.4]. The proposition follows.
As stated above, projectives in Mc are not flat in Mc. But they are flat as
E*-modules, by the following proposition.
Proposition 1.13. Pro-free E*-modules are flat.
Proof.It follows from [HS99 , Proposition A.13] that a pro-free module M is a
summand in a product of free modules. Since E* is Noetherian, products of free
modules are flat, and so M is flat.
2. K-local E-module spectra
We are interested in E_*X = ß*LK (E ^ X). These are the homotopy groups of a
K-local E-module spectrum. In this section, we discuss the category of K-local *
*E-
module spectra. We establish the basic structure of this stable homotopy catego*
*ry,
proving that if M is a K-local E-module spectrum, then ß*M is an L-complete
E*-module. We also construct a fundamental resolution of a K-local E-module
spectrum that we will use to build our spectral sequences in the rest of the pa*
*per.
2.1. Basic structure. The derived category DR of an S-algebra R is constructed
in [EKMM97 , Chapters III and VII]. The spectrum E is known to be a commutati*
*ve
S-algebra by [GH , Corollary 7.6]. It follows that DE is a monogenic stable hom*
*otopy
category in the sense of [HPS97 ]. This means that it is a closed symmetric mon*
*oidal
triangulated category such that the unit E is a small weak generator.
As with ordinary rings and modules, there is a forgetful functor from DE to
the ordinary stable homotopy category DS. This functor reflects isomorphisms and
has both a left and right adjoint (Proposition III.4.4 of [EKMM97 ]), so pres*
*erves
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 11
coproducts and products. The left adjoint takes X to E ^ X, or, more precisely,
an object of DE whose underlying spectrum is E ^ X. We refer to this as the free
E-module on X.
Of course, we are interested in E ^K X = LK (E ^ X). For this, we recall that*
* if
M is in DE , then the Bousfield localization LK M of the underlying spectrum of*
* M
is the underlying spectrum of LDEE^KM; in particular, the natural map M -!LK M
is (the underlying map of spectra of) a map in DE . See [EKMM97 , Chapter VII*
*I].
Hence we can think of E ^K X as an object of LDEE^KDE , the category of K-local
E-module spectra.
Of course, K = E=m is itself an E-module spectrum, as we can successively form
the cofiber sequences
E=(p, u1, , , ., ui) ui+1---!E=(p, u1, . .,.ui) -!E=(p, u1, . .,.ui+1)
in DE . Therefore we also have the Bousfield localization functor LDEKin DE .
Our first order of business is proving that these two notions of K-local E-mo*
*dule
spectra coincide. Given a commutative S-algebra R, let us denote by BousR(M)
the Bousfield class of M as an object of DR ; we can think of this as the class*
* of all
N in DR such that M ^R N = 0, ordered as usual by reverse inclusion.
Lemma 2.1. Suppose R is a commutative S-algebra and M is an object of DR .
Then BousR(R ^ M) BousR(M).
Here R ^ M is the free R-module on the underlying spectrum of M. Since
(R^M)^R N ~=M ^N as spectra, this lemma is analogous to the obvious algebraic
fact that M N = 0 implies M R N = 0. We learned the proof of this lemma
from Neil Strickland.
Proof.It suffices to show that M ^N = 0 implies that M ^R N = 0. We can assume
that M and N are cofibrant as R-modules in the model structure of [EKMM97 ,
Chapter VII]. Then M ^N is a cofibrant R^R-module, and M ^R N is the image of
M ^N under the left Quillen functor induced by the map R^R -!R of commutative
S-algebras. If M ^ N = 0 in DR , then M ^ N is contractible as a spectrum, and
so is trivially cofibrant as an R ^ R-module. Hence its image M ^R N is trivial*
*ly
cofibrant as an R-module.
This is in general all one can say. However, in our situation, equality holds.
Proposition 2.2. BousE (E ^ K) = BousE(K).
The proof of this proposition is also due to Neil Strickland.
Proof.It suffices to show that K ^E X = 0 implies K ^ X = 0. Let I =
(pi0, vi11, . .,.vin-1n-1) be an ideal such that S=I exists as a spectrum (see *
*[HS98 ],
or [HS99 , Corollary 4.14]). Then E=I = E ^ S=I, but E=I is also in the thick
subcategory of DE generated by K. Hence
S=I ^ X = E=I ^E X = 0.
Applying K* and using the Kunneth theorem, we see that K*X = 0, as required.
We can also identify the K-localization of an E-module M as being the appro-
priate version of completion at m.
12 MARK HOVEY
Theorem 2.3. Choose a sequence of regular ideals
. . .J2 J1 J0,
in E* with intersection 0, where each Jk is of the form (pi0, ui11, . .,.uin-1n*
*-1) for
some integers i0, . .,.in-1. Then the natural map M -! holimk(M ^E E=Jk) is
K-localization on DE .
Proof.Since any two such sequences are cofinal in each other, we can choose a
particular sequence for which the tower
. .-.!S=J2 -!S=J1 -!S=J0
exists, as in [HS99 , Proposition 4.22]. Then the underlying map of spectra of
M -!holimkM ^E E=Jk is the map M -!holimkM ^S=Jk, which is K-localization
by [HS99 , Proposition 7.10(e)], using the fact that M is E-local as a spectrum.
Then [EKMM97 , Proposition VIII.1.8] completes the proof.
Corollary 2.4. Suppose M 2 DE has ß*M a free E*-module. Then ß*LK M =
L0M = M^m.
Proof.The hypothesis guarantees that ß*(M ^E E=I) ~=ß*M=I for I an ideal of
the form (pi0, ui11, . .,.uin-1n-1). The result now follows from Theorem 2.3 a*
*nd the
Milnor exact sequence.
The salient feature of K-local E-module spectra for us is the following propo*
*si-
tion.
Proposition 2.5. Suppose M is a K-local E-module spectrum. Then ß*M is an
L-complete E*-module.
Proof.In view of Theorem 2.3 and the Milnor exact sequence, it suffices to show
that ß*(M ^E E=I) is an L-complete E*-module, where I = (pi0, ui11, . .,.uin-1n*
*-1).
But M ^E E=I is a module in DE over the E-ring spectrum E=I in DE , us-
ing [EKMM97 , Theorem 2.6]. It can be a little tricky to check that E=I is an
associative E-ring spectrum, but we do not need this since, even if it is not, *
*I acts
nilpotently on ß*(M ^E E=I). It follows that ß*(M ^E E=I) is already m-complete.
It is therefore L-complete by [HS99 , Theorem A.6(a)].
2.2. The fundamental resolutions. We will now construct fundamental resolu-
tions that we will use repeatedly in the paper.
Begin with an arbitrary E-module spectrum M, not necessarily K-local. Fol-
lowing [EKMM97 , Section IV.5], begin with a finite free resolution
(2.6) 0 -!Fn ffn--!.f.f.1-!F0 ff0-!ß*M -!0
of ß*M as an E*-module. This exists since E* has global dimension n and is loca*
*l,
so projectives are free. Split this resolution up into short exact sequences
0 -!Cs fis-!Fs-1 fls-1---!Cs-1 -!0
for s > 0, where Cs is the image of ffs for s 0, and so also the kernel of ff*
*s-1
when s > 0. Note that fl0 = ff0, fin = ffn, and ffs = fis O fls for 0 < s < n.
The following proposition is due to [EKMM97 ], where it is proved at the be*
*gin-
ning of Section IV.5. It can be proved straightforwardly by induction on s.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 13
Proposition 2.7. Suppose M is in DE , and choose a resolution of ß*M as in 2.6.
Then there is a tower in DE
K0 K1 Kn
? ? ?
f0?y f1?y ~=?yfn
Y0 ----!g Y1 ----! . . .----! Yn
0 g1 gn-1
with Yi= Ki= 0 for i > n, under Y0 = M, such that
(1) Ks fs-!Ys gs-!Ys+1 @s-! Ks is an exact triangle for all s.
(2) ß*Ks = sFs. In particular, Ks ~= s(E ^ Ts), where Ts is a wedge of
spheres with one for each generator of Fs.
(3) ß*Ys = sCs.
(4) ß*fs = sfls, ß*gs = 0, and ß*@s = s+1fis+1.
One useful corollary of this is the following, which also appeared as [HS99 ,
Lemma 8.11].
Corollary 2.8. Suppose M is in DE and ß*M is a finitely generated E*-module.
Then M is in the thick subcategory generated by E. In particular, if N is K-loc*
*al,
then M ^E N is also K-local.
We need an analogous resolution for a K-local E-module spectrum M. For
this, we begin with the resolution 2.6, and apply L0 to it. This gives a projec*
*tive
resolution
(2.9) 0 -!L0Fn ffn--!.f.f.2-!L0F1 ff1-!L0F0 ff0-!ß*M -!0
of ß*M in Mc, using Proposition 1.10. Split this resolution up into short exact
sequences
0 -!Cs fis-!L0Fs-1 fls-1---!Cs-1 -!0
for s > 0, where Cs is the image of ffs for s 0, and so also the kernel of ff*
*s-1
when s > 0. Note that fl0 = ff0, fin = ffn, and ffs = fis O fls for 0 < s < n.
Let us refer to a spectrum in DE of the form
` `
LK ( kiE) = E ^K Ski
i i
as pro-free. Note that M 2 LK DE is pro-free if and only if ß*M is a pro-free
E*-module. In analogy to Proposition 2.7, we have the following proposition.
Proposition 2.10. Suppose M is in LK DE , and choose a resolution of ß*M as
in 2.9. Then there is a tower in LK DE
K0 K1 Kn
? ? ?
f0?y f1?y ~=?yfn
Y0 ----!g Y1 ----! . . .----! Yn
0 g1 gn-1
with Yi= Ki= 0 for i > n, under Y0 = M, such that
(1) Ks fs-!Ys gs-!Ys+1 @s-! Ks is an exact triangle for all s.
(2) Ks is pro-free, and ß*Ks = sL0Fs.
(3) ß*Ys = sCs.
14 MARK HOVEY
(4) ß*fs = sfls, ß*gs = 0, and ß*@s = s+1fis+1.
Proof.We prove this by induction on s. Suppose we have defined the resolution
through the cofiber sequence of which gs-1 is a part, so that we have Ys with
ß*Ys = sCs. By choosing generators {xs} for sFs and looking at their images
in under sfls, we get a map from a wedge of spheres T into Ys. This gives a map
E ^ T -! Ys in DE , which extends to a map fs: Ks = E ^K T -! Ys in DE , since
Ys is K-local. Then Ks is pro-free by definition, and
ß*Ks ~=E_*T ~=L0E*T ~= sL0Fs,
using [HS99 , Proposition 8.4(c)] and the fact that L0F = Fm^ when F is a free
module. Also, ß*fs = sfls by construction. We define Ys+1 to be the cofiber
of fs in DE . It is then easy to check that ß*gs = 0, ß*Ys+1 = s+1Cs+1, and
ß*@s = s+1fis+1.
3. Computing E_*X from E*X
In this brief section, we point out that work of Greenlees and May [GM95 ] gi*
*ves a
spectral sequence that will compute E_*X given E*X. Specialized to our situatio*
*n,
Theorem 4.2 of [GM95 ] implies the following theorem.
Theorem 3.1. For M 2 DE , there is a natural, conditionally and strongly conver-
gent, spectral sequence of E*-modules
E2s,t= (Lsß*M)t) ßs+tLK M,
where dr: Ers,t-!Ers-r,t+r-1.
This spectral sequence is easily constructed from the resolution of Proposi-
tion 2.7. Indeed, simply apply LK to that resolution and take the associated ex*
*act
couple and the resulting spectral sequence. Corollary 2.4 tells us that E1s,t=
(L0Fs)t, and then the computation of E2 follows easily.
By applying this spectral sequence to M = E ^ X, we obtain the following
corollary.
Corollary 3.2. If X is a spectrum, there is a natural, conditionally and strong*
*ly
convergent, spectral sequence of E*-modules
E2s,t= (LsE*X)t) E_s+tX,
with dr: Ers,t-!Ers-r,t+r-1.
The simplest case of this theorem is when E*X is free, or even just flat, so
E_*X = L0(E*X). The free case of this appeared as [HS99 , Proposition 8.4(c)].
Another interesting case occurs when E*X is already L-complete, in which case
Corollary 3.2 implies that E*X = E_*X.
4. The universal coefficient and Künneth spectral sequences in
cohomology
We do not need a new construction for the universal coefficient spectral sequ*
*ence
in cohomology.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 15
Theorem 4.1. Suppose M and N are in LK DE . Then there is a natural, strongly
and conditionally convergent, spectral sequence of E*-modules
Es,t2= Exts,tcM(M*, N*) ~=Exts,tE*(M*, N*) ) ßs+tFE (M, N)
where dr: Es,tr-!Es+r,t-r+1r.
Note that this spectral sequence, like all the spectral sequences we will con*
*sider,
has Es,t2= 0 for all s > n.
Proof.This is just the spectral sequence of [EKMM97 , Theorem IV.4.1], using *
*the
isomorphism of Theorem 1.11 to identify the E2-term with the Ext groups in cM.
In general, the universal coefficient spectral sequence in cohomology only conv*
*erges
conditionally, but since Es,t2= 0 for s > n, it also converges strongly by [Boa*
*99,
Theorem 7.1].
Corollary 4.2. If X is a spectrum and N 2 LK DE , there is a natural, strongly
and conditionally convergent, spectral sequence of E*-modules
Es,t2= Exts,tcM(E_-*X, N*) ) Ns+tX.
In particular, if E_*X is pro-free, then the natural map
E*X -!Hom E*(E_-*X, E*)
is an isomorphism.
A more restrictive version of the isomorphism in Corollary 4.2 was proved as
Theorem 5.1 of [Hov03 ].
Proof.Apply the spectral sequence of Theorem 4.1 with M = E ^K X. Then
FE (E ^K X, N) ~=FE (E ^ X, N) ~=F (X, N)
using the fact that N is K-local and [EKMM97 , Corollary III.6.7].
Following [EKMM97 , Section IV.4], we also get a Künneth spectral sequence
in cohomology by taking N = FE (E ^ Y, E) in the universal coefficient spectral
sequence. Note that FE (E ^ Y, E) is easily seen to be K-local since E is so,
Corollary 4.3. For any spectra X and Y , there is a natural, strongly and condi-
tionally convergent, spectral sequence of E*-modules
Es,t2= Exts,tcM(E_-*X, E*Y ) ) E*(X ^ Y ).
5. The universal coefficient and Künneth spectral sequences in
homology
The obvious thing to do to get a universal coefficient spectral sequence in h*
*omol-
ogy for ß*LK (M ^E N) is to begin with the resolution of Proposition 2.10, apply
the functor LK ((-) ^E N) to it, and take the associated spectral sequence. This
will produce a spectral sequence, but it will be difficult to identify the E1-t*
*erm
algebraically. Indeed, the E1s,t-term is ßs+tLK (Ks^E N). Since Ks is pro-free,*
* this
is a homotopy group of a coproduct in the K-local category, but we do not know
what this is, in general. To get around this, the only obvious thing to do is *
*to
assume that ß*M is a finitely generated E*-module.
16 MARK HOVEY
In this case, however, Corollary 2.8 tells us that M ^E N is already K-local,
so we can use the spectral sequence of [EKMM97 , Theorem IV.4.1] to calculate*
* it.
Recall that this spectral sequence is of the form
E2s,t= TorE*s,t(M*, N*) ) ßs+t(M ^E N),
and converges strongly and conditionally. This is the same spectral sequence we
would get by applying the method of the first paragraph.
An interesting case of this spectral sequence is when M = K, in which case the
spectral sequence is of the form
E2s,t= TorE*s,t(E*=m, N*) ) ßs+t(N=m),
for any N 2 LK DE . The edge homomorphism of this spectral sequence is
N*=m = E20,*i E10,*,! ß*(N=m).
The following theorem identifies when this edge homomorphism is an isomorphism.
Theorem 5.1. Let N be in LK DE . Then N is pro-free if and only if the reduction
map ß*N -!ß*(N=m) is surjective, which is true if and only if the natural map
(ß*N)=m -!ß*(N=m)
is an isomorphism.
Proof.If N is pro-free, then certainly (ß*N)=m ~=ß*(N=m) since (p, u1, . .,.un-*
*1)
is a regular sequence on N* by [HS99 , Theorem A.9]. Conversely, suppose ß*N -!
ß*(N=m) is surjective. For each homogeneous generator _eof ß*(N=m) = ß*(K^E N)
as a vector space over K*, choose a map e: S|e|-!N reducing to _e. This gives us
an induced map in DE `
E ^ S|e|-!N
e
which extends to a map
`
f :LK (E ^ S|e|) -!N.
e
We now apply K ^E (-) to this map. For a general E-module M, K ^ M ~=
K ^ LK M. Hence, applying Lemma 2.1, we see that K ^E M ~=K ^E LK M. We
conclude that K ^E f is the map
`
K ^ S|e|-!N=m
e
induced by the chosen generators _eof ß*(N=m). Hence K ^E f is an equivalence.
By Proposition 2.2, this implies that K ^ f is an equivalence, and hence that f*
* is
an equivalence. Hence N is pro-free.
Applying this theorem when N = E ^K X, we get the following corollary.
Corollary 5.2. Suppose X is a spectrum. Then E_*X is pro-free if and only if the
natural map E_*X -! K*X is surjective, which is true if and only if the natural
map
(E_*X)=m -!K*X
is an isomorphism.
The spectral sequence above then gives us the following universal coefficient
theorem.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 17
Theorem 5.3. Suppose X is a spectrum and N 2 LK DE . If either E_*X or N* is
a finitely generated E*-module, then there is a natural, conditionally and stro*
*ngly
convergent, spectral sequence of E*-modules
E2s,t= TorE*s,t(E_*X, N*) ) ßs+tLK (N ^ X),
with drs,t:Ers,t-!Ers-r,t+r-1.
Proof.Apply the spectral sequence of [EKMM97 , Theorem IV.4.1] with M =
LK (E ^X) and N = N in the first case, and reverse the two in the second case.
The Künneth spectral sequence then takes the following form.
Theorem 5.4. Suppose X and Y are spectra with one of E_*X and E_*Y finitely
generated over E*. Then there is a natural, conditionally and strongly converge*
*nt,
spectral sequence of E*-modules
E2s,t= TorE*s,t(E_*X, E_*Y ) ) E_s+t(X ^ Y ),
with drs,t:Ers,t-!Ers-r,t+r-1.
We then get the following corollary.
Corollary 5.5. Suppose E_*X is pro-free and E_*Y is finitely generated. Then the
natural map
__ _ _
E_*X E*Y -! E*(X ^ Y )
is an isomorphism. In particular, the natural map
__ _ _
E_*E E*X -!E*(E ^ X)
is an isomorphism for all X that are dualizable in the K(n)-local category.
Note that it is easy to give incorrect proofs of the second statement in this
corollary. One would like to say, for example, that
__ _ _
E_*E E*X -!E*(E ^ X)
is a natural transformation of exact functors, so if it is an isomorphism for t*
*he
sphere, it is an isomorphism for all dualizable spectra. There are two problems
with this argument. The first is that the left-hand side is not an exact functo*
*r of
X, since E_*E is not flat in cM. One can get around this for X with E_*X finite*
*ly
generated though. The second problem is that there are dualizable spectra that
are not in the thick subcategory of K-local spectra generated by LK S0 (see [HS*
*99 ,
p. 76]), and there seems to be no obvious way around that problem.
Proof.The first statement follows from Theorem 5.4 and Proposition 1.13. The
second statement follows from the fact that E_*E is pro-free (using, for exampl*
*e,
the fact that E*E is flat and Corollary 3.2), and E_*X is finitely generated [H*
*S99 ,
Theorem 8.6].
Note that Corollary 5.5 implies that, when E_*Y is finitely generated, it is*
* a
comodule over the graded formal Hopf algebroid (E*, E_*E) described in [Hov03 ,
Section 6.2].
These theorems are not the end of the story, however. We can drop the finitely
generated hypothesis in some cases. To see this, we need the following lemma.
18 MARK HOVEY
Lemma 5.6. Suppose M and N are pro-free objects of LK DE . Then LK (M ^E N)
is pro-free, and the natural map
__
ß*M ß*N -!ß*LK (M ^E N)
is an isomorphism.
Proof.To see that LK (M ^E N) is pro-free, write M = E ^K T and N = E ^K T 0
where T and T 0are wedges of spheres. Then
LK (M ^E N) ~=LK ((E ^ T ) ^E (E ^ T 0)) ~=LK (E ^ (T ^ T 0)).
Since T ^ T 0is also a wedge of spheres, we conclude that LK (M ^E N) is pro-fr*
*ee.
Let OE denote the natural map
__
ß*M ß*N -!ß*LK (M ^E N).
The domain of OE is a pro-free E*-module, since each factor is, and the codomain
is also pro-free, as we have just seen. To see that OE is an isomorphism, then*
*, it
suffices to check that OE=m is an isomorphism. by Lemma 1.5.
Now, for a pro-free E-module, we have
ß*(K ^E M) = ß*(M=m) = ß*M=m
since (p, u1, . .,.un-1) is a regular sequence on ß*M by [HS99 , Theorem A.9]. *
*Ap-
plying [HS99 , Proposition A.4] we conclude that
__
E*=m E* ß*M ß*N ~=ß*M=m E* ß*N=m ~=ß*(K ^E M) E* ß*(K ^E N).
On the other hand, since LK (M ^E N) is pro-free,
E*=m E* ß*LK (M ^E N) ~=ß*(K ^E (M ^E N)).
Since K is a field spectrum in DE , it has a Künneth isomorphism. It follows th*
*at
OE=m is an isomorphism, as required.
By applying this lemma to LK (E ^ X) and LK (E ^ Y ), we get the following
corollary.
Corollary 5.7. If X and Y are spectra such that E_*X and E_*Y are pro-free, then
E_*(X ^ Y ) is pro-free and the natural map
__ _ _
E_*X E*Y -! E*(X ^ Y )
is an isomorphism.
We can now use Lemma 5.6 to get a version of the universal coefficient theorem
when one factor is pro-free.
Theorem 5.8. Suppose M and N are in LK DE and N is pro-free. Then there is
a natural, conditionally and strongly convergent, spectral sequence of E*-modul*
*es
E2s,t= (L TorcMs(ß*M, ß*N))t) ßs+tLK (M ^E N),
where dr: Ers,t-!Ers-r,t+r-1.
Proof.Take the resolution of Proposition 2.7, apply LK ((-) ^E N) to it, and ta*
*ke
the associated exact couple and resulting spectral sequence. That is, in the no*
*tation
of Proposition 2.7, let
D1s,t= ßs+t(Ys ^E N) and E1s,t= ßs+t(Ks ^E N).
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 19
The maps
i1: D1s,t-!D1s+1,t+1, j1: D1s,t-!E1s-1,t, and k1: E1s,t-!D1s,t
are induced by the maps in the exact triangle
LK (Ks ^E N) -!LK (Ys ^E N) -!LK (Ys+1^E N) -! LK (Ks ^E N).
The resulting spectral sequence is conditionally convergent to ß*LK (Y0 ^E N)
by [Boa99, Definition 5.10]. Since there are only finitely many differentials,*
* it
also converges strongly by [Boa99, Theorem 7.1].
To identify the E2-term, note first that
__ __
E1s,t~=(ß*Ks ß*N)s+t~=(L0Fs ß*N)t
by Lemma 5.6. One can easily check that d1 is the expected map, and so, by
definition,
E2s,t~=(L TorcMs(ß*M, ß*N))t.
The naturality of the spectral sequence then follows in the usual way. A map of
spectra induces a non-unique map of resolutions, which induces a map of spec-
tral sequences. This map is canonical from the E2-term on, since E2**depends
functorially on ß*M and ß*N.
The resulting universal coefficient theorem is the following.
Corollary 5.9. Suppose X is a spectrum and N 2 LK DE . If E_*X is pro-free,
then there is a natural, conditionally and strongly convergent, spectral sequen*
*ce of
E*-modules
E2s,t= (L TorcMs(N*, E_*X))t) ßs+tLK (N ^ X).
Similarly, if N* is pro-free, then there is a natural, conditionally and strong*
*ly con-
vergent, spectral sequence of E*-modules
E2s,t= (L TorcMs(E_*X, N*))t) ßs+tLK (N ^ X).
The Künneth theorem is similar.
Corollary 5.10. Suppose X and Y are spectra and that E_*Y is pro-free. Then
there is a natural, conditionally and strongly convergent, spectral sequence of*
* E*-
modules
E2s,t= (L TorcMs(E_*X, E_*Y ))t) E_s+t(X ^ Y ).
6. The E-theory of a coproduct
We now construct a spectral sequence to compute the Morava E-theory of a
coproduct. We begin by pointing out that homotopy groups in LK DE commutes
with direct sums in cM as long as the E-modules involved are pro-free.
Lemma 6.1. Suppose {Mi} is a family of pro-free objects in LK DE . Then the
natural map M `
L0( ß*Mi) -!ß*LK ( Mi)
i i
is an isomorphism.
20 MARK HOVEY
Proof.Write Mi= E ^K Ti for some wedge of spheres Ti. Then
` `
LK ( Mi) ~=E ^K Ti.
i i
It follows that ` M
ß*LK ( Mi) ~=L0( E*Ti).
i i
Now, applying the functor cM(-, P ), one can check using adjointness that
M M
L0( Ni) ~=L0( L0Ni)
i i
for any E*-modules Ni. Hence
` M M
ß*LK ( Mi) ~=L0( E_*Ti) ~=L0( ß*Mi),
i i i
as required.
We can now construct our spectral sequence.
Theorem 6.2. Suppose {Mi} is a family in LK DE . There is a natural, condition-
ally and strongly convergent, spectral sequence of E*-modules
M `
E2s,t= (Ls( ß*Mi))t) ßs+tLK ( Mi),
i i
where dr: Ers,t-!Ers-r,t+r-1.
Note that E2s,t= 0 in this spectral sequence for s > n - 1 by Proposition 1.9*
*. In
particular, when n = 1, homotopy actually preserves coproduct as a functor to c*
*M.
Proof.Take the K-local coproduct of the towers in Proposition 2.10. This gives *
*us
a tower in LK DE
W W W
LK ( iKi0) LK ( iKi1) LK ( iKin)
?? ? ?
y ?y ~=?y
W W W
LK ( iY0i)----! LK ( iY1i)----! . .-.---! LK ( iYni)
We now take the corresponding exact couple and the resulting spectral sequence.
That is, we let
` `
D1s,t= ßs+tLK ( Ysi) and E1s,t= ßs+tLK ( Kis).
i i
The maps
i1: D1s,t-!D1s+1,t-1, j1: D1s,t-!E1s-1,t, and k1: E1s,t-!D1s,t
are induced by the maps in the exact triangle
` ` ` `
LK ( Kis) -!LK ( Ysi) -!LK ( Ysi+1) -! LK ( Kis).
i i i i
The resulting spectral sequence is a spectral sequence of E*-modules with
drs,t:Ers,t-!Ers-r,t+r-1.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 21
It is conditionally and strongly convergent to
` `
ßs+tLK ( Ysi) ~=ßs+tLK ( Mi)
i i
by Definition 5.10 and Theorem 7.1 of [Boa99].
To compute E2s,t, note that Lemma 6.1 implies that
M McM
E1s,t~=(L0( L0Fsi))t= ( L0Fsi)t.
i i
The map d1 is then determined by its restriction to each component L0Fsiof this
coproduct. It follows that d1 is the expected map, so that E2s,tis the sth deri*
*ved
functor of the direct sum, in degree t. Theorem 1.8 then implies that
M
E2s,t~=(Ls( ß*Mi))t.
i
Naturality of the spectral sequence then follows as usual, since the E2 term is
functorial.
Applying this with Mi= E ^K Xi gives the following corollary.
Corollary 6.3. Suppose {Xi} is a family of spectra. There is a natural, conditi*
*on-
ally and strongly convergent, spectral sequence of E*-modules
M `
E2s,t= (Ls( E_*Xi))t) E_s+t( Xi),
i i
where dr: Ers,t-!Ers-r,t+r-1.
Again, this means that E_*(-) preserves coproducts as a functor to Mc when
n = 1.
7. The K(1)-local stable homotopy category
One of the difficulties in understanding the K-local stable homotopy category*
* is
that LK is not smashing, so that homotopy groups do not commute with coprod-
ucts in the K-local category. We have just seen, however, that E_*(-) preserves
coproducts when n = 1, as a functor to the L-complete category. It seems reason-
able, then, to ask whether homotopy groups also commute with coproducts in the
K-local category when n = 1, as a functor to the L-complete category. We will s*
*ee
in this section that the answer is yes.
Now, when n = 1, E* ~=Zp[u, u-1], and L0 is the 0th left derived functor of
p-completion. Of course, the homotopy of a K-local spectrum will not be an E*-
module, but it will be a Zp-module. We will therefore also use L0 to denote the*
* 0th
left derived functor of p-completion on the category of Zp-modules. We first po*
*int
out that this apparent clash of notation is in fact euphonic.
Lemma 7.1. Suppose n = 1 and M is a graded E*-module. Then
(L0M)k ~=L0Mk
where the second L0 is taken in the category of Zp-modules.
22 MARK HOVEY
Proof.We have short exact sequences
0 -!lim1TorE*(E*=pi, M) -!L0M -!M_p-! 0
and
0 -!lim1TorZp(Z=pi, Mk) -!L0Mk -!(Mk)_p-!0,
from [HS99 , Theorem A.2(b)]. One can easily check that
(lim1TorE*(E*=pi, M))k ~=lim1TorZp(Z=pi, Mk)
and
(M_p)k ~=(Mk)_p,
from which the result follows.
Now we note that ß*X is L-complete when X is K-local.
Lemma 7.2. If n = 1 and X is K-local, then ßkX is an L-complete Zp-module
for all k.
Proof.It is well known that X = holim(X=pi); see [HS99 , Proposition 7.10] for
example. Now ß*(X=pi) is bounded p-torsion, so in particular is p-complete and *
*so
L-complete. It follows from the Milnor exact sequence and [HS99 , Theorem A.5]
that ß*X is L-complete.
We can now prove the desired theorem.
Theorem 7.3. Suppose n = 1 and {Xi} is a family of K-local spectra. Then the
natural map M `
L0( ß*Xi) -!ß*LK ( Xi)
i i
is an isomorphism.
Here L0 denotes the 0th derived functor of p-completion in the category of Zp-
modules.
Proof.We will show that the collection D of all spectra F such that the natural
map M `
F :L0( F*_Xi) -!F*_( Xi)
i
is an isomorphism is a thick subcategory. Here F*_X = ß*LK (F ^ X). We know
that E is in D by the comments following Corollary 6.3 and Lemma 7.1. Since
LK S0 is in the thick subcategory generated by E [HS99 , Theorem 8.9], we see t*
*hat
LK S0 2 D, proving the theorem.
It is clear that Y 2 D if and only if Y 2 D. To see that D is closed under
retracts, simply note that if Y is a retract of Z, then Y is a retract of Z. *
*Now
suppose that
W -! Y -! Z -! W
is an exact triangle, and W, Z 2 D. We have exact sequences
. .-.!Wn_Xi-! Yn_Xi-! Z_nXi-! Wn_-1Xi-! . . .
for all i. Since direct sums are exact in cM when n = 1, we get the following e*
*xact
sequence.
M M M
(7.4) . .-.!L0( Wn_Xi) -!L0( Yn_Xi) -!L0( Z_nXi) -!. . .
i i i
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 23
On the other hand, we have an exact triangle
` ` `
LK (W ^ Xi) -!LK (Y ^ Xi) -!LK (Z ^ Xi) -! LK (W ^ _Xi).
This gives rise to the exact sequence below.
` ` `
. .-.!Wn_( Xi) -!Yn_( Xi) -!Z_n( Xi) -!. . .
There is a map from the exact sequence 7.4 to this one, which is an isomorphism
on two out of every three terms. Hence it is also an isomorphism on the third t*
*erm,
so Y 2 D, as required.
8.Filtered homotopy colimits
The object of this section is to construct a spectral sequence analogous to t*
*hat
of Theorem 6.2 for filtered homotopy colimits. The whole notion of a homotopy
colimit depends on an underlying point-set level category, and so cannot be car*
*ried
out entirely in LK DE . We thus require a fair amount of technical material on
homotopy colimits, which we have put in an appendix.
The basic idea of a homotopy colimit is as follows. We have some model catego*
*ry
M and a small category I. The colimit defines a functor MI -!M, left adjoint to
the constant diagram functor. Define a map of diagrams to be a weak equivalence*
* if
it is a weak equivalence on each level, and define hoMI to be the quotient cate*
*gory
obtained by inverting the weak equivalences (which might not be a category in
the usual sense since it may have too many morphisms). For our purposes, the
homotopy colimit is the functor
hocolim:ho MI -!ho M
left adjoint to the constant diagram functor (which obviously passes to a funct*
*or
on homotopy categories). Note that hocolimis not a colimit, because its domain
is not a diagram category. Existence of the homotopy colimit functor in general*
* is
subtle, but for our model categories M it is easy to construct it, and it in fa*
*ct exists
in complete generality. The author highly recommends [DHKS03 ] for a treatment
of homotopy colimits in full generality. As shown in Theorem A.11, the usual
sequential (weak) colimits that exist in the ordinary stable homotopy category,
DE , or LK DE are all examples of homotopy colimits.
While we are discussing diagrams, note that, for i 2 I, the evaluation functor
Evi: MI -! M preserves weak equivalences and so passes to a functor on the
homotopy category level. Now Evi, before passing to the homotopy category, has
both a left adjoint Fi and a right adjoint Ri. Here
a Y
(FiK)j = K and (RiK)j = K.
I(i,j) I(j,i)
In good cases, which includes all the cases we will discuss, these adjunctions *
*pass
to the homotopy category level as well (Proposition A.1).
Now, we begin with homotopy colimits of pro-free objects of LK DE . Let ME
denote the model category of E-modules, as in [EKMM97 , Chapter VII], and let
LK ME denote the same category ME , but with the K-local model structure,
described in [EKMM97 , Chapter VIII]. The homotopy category of ME is DE , and
the homotopy category of LK ME is LK DE .
24 MARK HOVEY
Theorem 8.1. Let X be a diagram in (LK ME )I, where I is a filtered small
category. If Xi is pro-free for all i, then the natural map
f :L0(colimiß*Xi) -!ß*hocolimXi
is an isomorphism.
Proof.The plan is to prove that hocolimXi is pro-free and that f=m is an isomor-
phism. Since the domain of f is also pro-free by Theorem 1.4, Lemma 1.5 will th*
*en
complete the proof.
In view of Theorem 5.1, to see that hocolimXi is pro-free, it suffices to show
that the map
ß*hocolimXi-! ß*(E=m ^E hocolimXi)
is surjective. Since smashing commutes with homotopy colimits by Corollary A.6,
we see that
ß*(E=m ^E hocolimXi) ~=ß*hocolim(Xi=m).
Now, to compute hocolim(Xi=m), we first compute this homotopy colimit in MIE
and then apply LK , by Corollary A.7. But, again using the fact that smashing
commutes with homotopy colimits, we see that hocolimXi, taken in ME , is a
module over the ring spectrum K. Thus it is already K-local. Since homotopy
commutes with filtered homotopy colimits in ME by Theorem A.8, we have
ß*hocolim(Xi=m) ~=colimß*(Xi=m).
This means that in the commutative diagram below
L0(colimß*Xi) ---f-! ß*hocolimXi
?? ?
y ?y
L0(colimß*Xi)=m --f=m--! (ß*hocolimXi)=m
?? ?
y ?y
colimß*(Xi=m) ----! ß*(E=m ^E hocolimXi)
the bottom horizontal arrow is an isomorphism. On the other hand, since each Xi*
*is
pro-free, the lower left-hand vertical arrow is also an isomorphism. Thus the o*
*uter
counter-clockwise composite is surjective, and so the outer clockwise composite*
* is
also surjective. Hence the map
ß*hocolimXi-! ß*(E=m ^E hocolimXi)
is also surjective. Thus hocolimXi is pro-free by Theorem 5.1, and furthermore
the lower right-hand vertical arrow in the diagram above is an isomorphism. Thus
f=m is an isomorphism, and so f is an isomorphism.
In general, we do not get an isomorphism but a spectral sequence.
Theorem 8.2. Suppose X is a diagram in (LK ME )I for a filtered small category
I. There is a natural spectral sequence of E*-modules
E2s,t= (Ls(colimiß*Xi))t) ßs+thocolimiXi,
where dr: Ers,t-!Ers-r,t+r-1. If this spectral sequence converges conditionally*
*, then
it converges strongly. If the functor
lim: Ab I-! Ab
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 25
has only finitely many right derived functors, then it converges conditionally.
It is known that lim:Ab I -!Ab has only finitely many right derived functors
when I is a directed set of cardinality @k for some k < 1 [Jen70]; in particu*
*lar,
when I is countable. In view of Theorem A.11, this includes the case of sequent*
*ial
colimits. Note that the spectral sequence may well converge in general, but we *
*are
unable to prove this.
To prove this theorem, note that, by applying ß* objectwise, we can think of
ß*X as an element of cMI. We then choose a projective resolution
(8.3) . .-.ff2!P1 ff1-!P0 ff0-!ß*X
L
of ß*X in cMI, where Ps = L0( iFiPsi) for projectives Psiin cM, using Lemma 1.*
*7.
As usual, we split this resolution into short exact sequences
0 -!Cs fis-!Ps-1 fls-1---!Cs-1 -!0
for s > 0, where Cs is the image of ffs for s 0, and also the kernel of ffs-1*
* for
s > 0. Note that fl0 = ff0 and ffs = fis O fls for s > 0.
We then have the following proposition.
Proposition 8.4. Suppose X 2 ho(LK ME )I is a diagram, and choose a resolution
of ß*X as in 8.3. Then there is a tower in ho(LK ME )I
K0 K1
? ?
f0?y f1?y
Y0 ----!g Y1 ----! . . .
0 g1
under Y0 = X, such that
(1) Ks fs-!Ys gs-!Ys+1 @s-! Ks is an exact`triangle for all s.
(2) ß*Ks = sPs, and furthermore Ks ~= i2IFiKisfor pro-free Kisin LK DE ,
where the coproduct is taken in ho(LK ME )I.
(3) ß*Ys = sCs.
(4) ß*fs = sfls, ß*gs = 0, and ß*@s = s+1fis+1.
In the statement of this proposition, we are using the functors
Fi: LK DE -! ho(LK ME )I
left adjoint to Evi, as in Proposition A.1.
Proof.We prove this by induction on s. The only step that requiresLcomment is
the construction of Ks and fs from Ys. Recall that Ps = L0( iFiPsi) for some
projectives Psiin cM. We can write Psi= L0Fsifor some free module Fsi; then by
choosing generators`for Fsiwe can find a pro-free Kis2 LK DE with ß*Kis~= sPsi.
We then let Ks = iFiKis, where the coproduct is taken in ho(LK ME )I. It then
follows from Lemma A.2 that ß*Ks ~= sPs, as required.
To construct fs, it suffices to construct maps
FiKis-!Ys,
or, by adjointness, maps
Kis-!EviYs.
26 MARK HOVEY
The same adjointness argument shows that fls: Ps -!Cs is induced by maps
Psi-!Ev iCs.
Since Kisis pro-free, this map of modules induces the desired map Kis-!EviYs.
The construction of our spectral sequence will now come as no surprise. Given
a resolution as in Proposition 8.4, we apply the homotopy colimit functor to get
a resolution of hocolimX in LK DE , take the resulting exact couple, and then t*
*he
associated spectral sequence. This spectral sequence will be a spectral sequenc*
*e of
E*-modules with
dr: Ers,t-!Ers-r,t+r-1.
Furthermore,
E1s,t~=ßs+t(hocolimKs) ~=(colimcMPs)t,
in view of Theorem 8.1. Since d1 is the expected map when restricted to each
EviPs, it is in fact the expected map. By applying Theorem 1.8, we conclude that
E2s,t~=(Ls(colimß*Xi))t.
The naturality of the spectral sequence then follows as usual, since the E2-ter*
*m is
functorial.
To complete the proof of Theorem 8.2, then, we must investigate the convergen*
*ce
of the spectral sequence. This is more subtle than it was with the other spectr*
*al
sequences of this paper, since we do not know that our resolution 8.3 stops aft*
*er
the nth stage. Certainly the fact that the E2-term vanishes above filtration n
(Theorem 1.8), implies that if the spectral sequence converges conditionally th*
*en it
converges strongly, by [Boa99, Theorem 7.1]. To see that it converges condition*
*ally,
however, we need to know that
colims(ß*hocolimiYs) = 0.
The one thing we know about Ys is that the maps
gs: Ys -!Ys+1
are zero in homotopy. This means that, for i 2 I, the map EviYs -!EviYs+1 has
positive filtration in the spectral sequence
Es,t2= Exts,tE*(ß*Ev iYs, ß*Ev iYs+1) ) DE ( s+tEv iYs, EviYs+1)
of [EKMM97 , Theorem IV.4.1]. Hence the map Ys -!Ys+n+1 of diagrams is ob-
jectwise null. Applying Proposition A.10, using the assumption that limover I-
diagrams has only finitely many derived functors, we conclude that hocolimYs -!
hocolimYs+N is null for large n. This proves that our spectral sequence converg*
*es
conditionally, completing the proof of Theorem 8.2.
We can then apply Theorem 8.2 to compute the Morava E-theory of a homotopy
colimit.
Corollary 8.5. Suppose X is a diagram in (LK MS)I for a filtered small category
I. There is a natural spectral sequence of E*-modules
E2s,t= (Ls(colimiE_*Xi))t) E_s+t(hocolimiXi),
where dr: Ers,t-!Ers-r,t+r-1. If this spectral sequence converges conditionally*
*, then
it converges strongly. If the functor
lim: Ab I-! Ab
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 27
has only finitely many right derived functors, then it converges conditionally.
In this corollary, note that the hocolimiXi is taken in the K-local category,*
* so
that we must take the ordinary homotopy colimit and relocalize it, by Corollary*
* A.7.
Proof.Apply Theorem 8.2 to the diagram LK (E ^ X). This gives a spectral se-
quence converging to ß*hocolimLK (E ^ X). Applying Corollary A.6 and Corol-
lary A.7, we see that
holimLK (E ^ X) ~=LK (E ^ hocolimX),
so our spectral sequence does converge to E_*(hocolimX).
Appendix A. Homotopy colimits
In this appendix, we give proofs of the basic facts about homotopy colimits t*
*hat
we need in Section 8. In Section A.1 we give some general results about homotopy
colimits, and in Section A.2 we prove that sequential colimits are examples of
homotopy colimits.
A.1. Structural results. Suppose we have a model category M and a small cate-
gory I. Our main examples are the model category ME of E-modules [EKMM97 ,
Chapter VII], and the model category LK ME of E-modules given the K-local
model structure [EKMM97 , Chapter VIII]. Both these model categories are poin*
*ted,
cofibrantly generated, simplicial (in fact topological), closed symmetric monoi*
*dal
in the sense of [Hov99 , Chapter 4], and stable, in the sense that the suspensi*
*on is
an equivalence on the homotopy category level.
We will prove some basic results about homotopy colimits in this situation. We
will not give the best possible results here, since we are primarily concerned *
*with
such nice model categories.
We first recall some general results about diagram categories.
Proposition A.1. Suppose M is a cofibrantly generated model category and I is
a small category.
(1) The category of diagrams MI admits a model structure in which a map f
is a fibration or weak equivalence if and only if f(i) is so for all i 2*
* I. A
cofibration in this model structure is in particular an objectwise cofib*
*ration.
(2) The colimit functor MI -! M is a left Quillen functor and so induces a
functor
hocolim: hoMI -!ho M,
left adjoint to the (derived) constant diagram functor
c: M -!MI.
(3) For i 2 I, the evaluation functor
Evi: MI -!M
is both a left and right`Quillen functor. The left adjoint to Eviis the *
*functor
Fi defined by Fi(K)j = I(i,j)K.
The first two parts of this proposition can be found in [Hir03, Section 11.6]*
*; for
the third, note that it is obvious that Evi preserves fibrations, cofibrations,*
* and
28 MARK HOVEY
weak equivalences. ItsQleft adjoint Fi is defined above, and its right adjoint *
*Ri is
defined by (RiK)j = I(j,i)K. We will abuse notation by using
Ev i: hoMI -!ho M and Fi: hoM -!ho MI
for the derived functors of Evi and Fi. Since Evi preserves weak equivalences, *
*it
passes directly to a functor between homotopy categories anyway, but for Fi we
must take a cofibrant replacement first.
Before proceeding further with the general situation, we take a minute to look
at FiM when M = LK DE and M is pro-free. Note that if X 2 ho(LK ME )I, then
we can define ß*X to be the functor that takes i 2 I to ß*Ev iX. Then ß*X is in
cMI. In analogy to Lemma 6.1, we have the following lemma.
Lemma A.2. If {Xj} is a family of objects in ho(LK ME )I such that EviXj is
pro-free for all i 2 I, then the natural map
a a
ß*Xj -!ß*( Xj)
j j
is an isomorphism, where the coproducts are taken in McI and ho(LK ME )I, re-
spectively. Furthermore, if M 2 LK DE is pro-free, then
ß*FiM ~=Fi(ß*M).
Proof.For the first statement, it suffices to check that
a a
ß*Ev iXj ~=ß*Ev i( Xj).
j j
Since Evi is a left adjoint, it commutes with coproducts, and so the result fol*
*lows
from Lemma 6.1.
For the second statement, we choose a cofibrant representative M0 for M in
LK ME . Then
a a
ß*Ev jFiM ~=ß* M0 ~= ß*M ~=Evj Fi(ß*M),
I(i,j) I(i,j)
again using Lemma 6.1.
Returning to the general case, we note that when M is closed symmetric mon-
oidal, it will act on MI.
Proposition A.3. Suppose M is a cofibrantly generated, closed symmetric mon-
oidal model category, and I is a small category. Then MI is an M-model category;
that is, MI is tensored, cotensored, and enriched over M in a way that is compa*
*t-
ible with the model structures.
Proof.Denote the symmetric monoidal product in M by K L and denote the
closed structure by KL or map (K, L). Given X 2 MI and K 2 M, we define
(X K)i = Xi K and (XK )i = XKi. We define map (X, Y ) to be the equalizer
in the diagram below.
Y Y
map (X, Y ) -! map (Xi, Yi) ' map(Xj, Yk)
i j-!k
The j -!k component of the top map is the composite
Y
map(Xi, Yi) -!map (Xj, Yj) -!map (Xj, Yk)
i
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 29
induced by the structure map of Y . The j -! k component of the bottom map is
the composite
Y
map (Xi, Yi) -!map (Xk, Yk) -!map (Xj, Yk)
i
induced by the structure map of X.
To check that this action is compatible with the model structures, suppose
j :K -! L is a cofibration in M and p: X -! Y in a fibration in MI. It suf-
fices to check that the map
XL -! XK xY KY L
is a fibration in MI that is trivial if either j or p is so. But we can check *
*this
objectwise, where it follows easily from the compatibility of the model structu*
*re on
M with the monoidal structure.
Corollary A.4. If M is a cofibrantly generated, closed symmetric monoidal, stab*
*le
model category, then MI is stable for all small categories I.
This corollary implies, in particular, that ho(MIE) and ho(LK ME )I are trian-
gulated.
Proof.The category hoMI is enriched, tensored, and cotensored over hoM; since
hoM is triangulated, so is hoMI, and hence MI is stable.
Just as left adjoints preserve colimits, so left Quillen functors preserve ho*
*motopy
colimits.
Proposition A.5. Let F :M -! N be a left Quillen functor of cofibrantly gen-
erated model categories, and let I be a small category. Extend F to a functor
MI -!N Iby applying it objectwise. Then there is a natural isomorphism
hocolim((LF )X) -!(LF )(hocolimX)
for X 2 hoMI, where LF denotes the total left derived functor of F .
Proof.Note that F :MI -!N Iis still a left Quillen functor. Indeed, if U denotes
the right adjoint of F , then applying U objectwise induces a functor U :N I-! *
*MI,
right adjoint to the extension of F , that obviously preserves fibrations and w*
*eak
equivalences. Now the proposition follows by taking the left derived version of*
* the
fact that F preserves colimits.
The following corollary is then immediate.
Corollary A.6. If M is a cofibrantly generated, closed symmetric monoidal model
category, I is a small category, X 2 hoMI and K 2 hoM, then there is a natural
isomorphism
hocolim(X ^ K) -!(hocolimX) ^ K.
Here we are using the symbol ^ for the total left derived functor of the acti*
*on of
M on MI.
Another useful corollary is the following, written specifically for our situa*
*tion.
30 MARK HOVEY
Corollary A.7. Suppose X is a diagram in MIEfor some small category I. Let
hocolimX denote the homotopy colimit of X thought of as an element of hoMIE,
and let hocolimLKX denote the homotopy colimit of X thought of as an element
of ho(LK ME )I. Then there is a natural isomorphism
LK (hocolimX) -!hocolimLK X.
Proof.The identity is a left Quillen functor from ME to LK ME ; its total left
derived functor is LK . The result now follows from Proposition A.5.
Now, in the usual stable homotopy category, it is of course well known that h*
*o-
mology commutes with sequential colimits. In fact, this is true for filtered ho*
*motopy
colimits as well, but only in exceptionally nice model categories.
To state this more precisely, recall that an object A in a category C is call*
*ed
finitely presented with respect to a subcategory D if the natural map
colimC(A, Xi) -!C(A, colimXi)
is an isomorphism for all filtered diagrams {Xi} in D. For example, the argument
of [Hov99 , Proposition 2.4.2] shows that compact topological spaces are finite*
*ly
presented with respect to the closed T1 inclusions, and in particular, with res*
*pect
to the Serre cofibrations. It then follows, with some work, that the domains and
codomains of the generating cofibrations and trivial cofibrations of S-modules *
*MS
or R-modules MR for R an S-algebra are also finitely presented with respect to *
*the
cofibrations in their model structures. See, for example, Lemma 2.3 of [EKMM97 *
* ].
Theorem A.8. Suppose M is a pointed simplicial cofibrantly generated model
category in which the domains and codomains of the generating cofibrations are
cofibrant, and the domains and codomains of the generating trivial cofibrations*
* are
finitely presented with respect to the cofibrations. For any cofibrant A such t*
*hat A
and A x I are finitely presented with respect to the cofibrations, and for any *
*filtered
diagram {Xi}, the natural map
colimho M(A, Xi) -!ho M(A, hocolimXi)
is an isomorphism. In particular, homotopy commutes with filtered homotopy col-
imits in MR , for any S-algebra R.
Note that Theorem A.8 does not imply that homotopy commutes with filtered
homotopy colimits in LK ME , because in the process of Bousfield localization, *
*one
loses control of the generating trivial cofibrations.
Before proving this theorem, we explain why we require that the domains and
codomains of the generating cofibrations be cofibrant.
Lemma A.9. Suppose M is a cofibrantly generated model category in which the
domains and codomains of the generating cofibrations are cofibrant, and I is a *
*small
category. Then for any cofibrant object X of MI, the structure maps Xi-! Xj are
cofibrations.
Proof.The generating cofibrations of MI are the maps Fif, where f :K -!L is a
generating cofibration of M and i 2 I. We can write X as a retract of a transfi*
*nite
composition
0 oe1
* = X0 oe-!X1 -! . . .
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 31
where each map oeffis a pushout of a map of the form Fif. We prove by transfini*
*te
induction on ff that the structure maps of Xffare cofibrations. For the success*
*or
ordinal step, given j -!k in I, we have a map from the pushout square
(FiK)j --Fif--!(FiL)j
?? ?
y ?y
Xffj ----! Xff+1j
to the pushout square below.
(FiK)k --Fif--!(FiL)k
?? ?
y ?y
Xffk ----! Xff+1k
This map is a cofibration in all of the corners except possibly the lower right*
* corners,
by induction and the fact that K and L are cofibrant. Furthermore, the map
(FiK)k q(FiK)j(FiL)j -!(FiL)k
is a coproduct of some copies of L with a coproduct of copies of f :K -! L, so
is a cofibration. Lemma 7.2.15 of [Hir03] then implies that Xff+1j-!Xff+1kis a
cofibration.
Now let fi be a limit ordinal. We claim that the map of fi-sequences
X0j ----! X1j ----! . . .
?? ?
y ?y
X0k ----! X1k ----! . . .
which is a degreewise cofibration by the induction hypothesis, is in fact a cof*
*ibration
in the model structure on fi-sequences. This model structure is a special case *
*of the
model structure on diagrams, and is described in [Hov99 , Section 5.1]. In part*
*icular,
to show this map is a cofibration, we must show that the map
q :XffkqXffjXff+1j-!Xff+1k
is a cofibration. Since Xff-! Xff+1is a pushout of Fif for some generating cofi-
bration f :K -!L and some i 2 I, we see that q is isomorphic to the map
Xffkq(FiK)k[(FiK)k q(FiK)j(FiL)j] -!Xffkq(FiK)k(FiL)k.
This map is a cofibration by another application of [Hir03, Lemma 7.2.15]. Thus
X*j-!X*kis a cofibration of fi-sequences. Since the colimit functor is a left Q*
*uillen
functor, we conclude that the map Xfij-!Xfikis a cofibration, completing the li*
*mit
ordinal step of the induction.
Proof of Theorem A.8.To compute hoM(A, hocolimXi), we can assume X is cofi-
brant and fibrant in MI. This means that every Xiis fibrant and that hocolimXi~=
colimXi in hoM. Because the domains and codomains of the generating trivial
cofibrations are finitely presented with respect to the cofibrations, it follow*
*s that
colimXi is fibrant. Hence
hoM(A, hocolimXi) ~=M(A, colimXi)=(~),
32 MARK HOVEY
where ~ denotes the (left or right) homotopy relation. Since both A and A x I a*
*re
finitely presented with respect to the cofibrations, this is in turn isomorphic*
* to
colimM(A, Xi)=(~) ~=colimhoM(A, Xi),
as required.
As a practical matter, since homotopy colimits are closely related to colimit*
*s, it
should be easy to compute maps out of them. Bousfield and Kan [BK72 , p. 336]
give a spectral sequence
Es,t2= limshoM( tXi, N) ) hoM( t-shocolimX, N).
To be accurate, Bousfield and Kan construct this spectral sequence for homotopy
colimits of simplicial sets, but they get it from the corresponding spectral se*
*quence
for the homotopy groups of a homotopy limit of simplicial sets [BK72 , p. 311] *
*by
using the relationship
ho M( tXi, N) ~=ßtmap (Xi, N),
and this relationship will work in any simplicial model category. The convergen*
*ce
of the Bousfield-Kan spectral sequence is delicate, but it does converge strong*
*ly
when the E2-term is 0 for s s0 for some integer s0 [BK72 , p. 263].
In particular, we get the following proposition.
Proposition A.10. Suppose M is a simplicial model category, and I is a small
category for which the inverse limit functor
lim: Ab I-! Ab
has only finitely many nonzero right derived functors, so that lims= 0 for all *
*s s0.
Let 0 1 k-1
X0 f-!X1 f-!. .f.---!Xk
be a sequence of maps in MI with k > s0 such that fjiis null for all i 2 I and *
*all
j k - 1. Then the induced map hocolimX0 -!hocolimXk is null.
In particular, this proposition applies to any directed set with cardinality *
* @k
for some k < 1 by [Jen70].
Proof.It suffices to show that the induced map
ho M(hocolimXk, W ) -!ho M(hocolimX0, W )
is null for all W . Applying the Bousfield-Kan spectral sequence above, which
converges strongly under our hypothesis since it has only finitely many differe*
*ntials,
we see that if a map in hoM(hocolimXk, W ) is detected in filtration j, then its
image in hoM(hocolimX0, W ) has filtration at least j + k. Since j + k > s0, the
result follows.
A.2. Sequential colimits are homotopy colimits. In this section, we establish
that sequential colimits defined in the usual way in a stable homotopy category
such as LK DE are examples of homotopy colimits. This is an unsurprising but
somewhat technical result, proofs of which we learned from Neil Strickland and
Stefan Schwede. The result is probably well-known, but we do not know a referen*
*ce
for it.
Suppose we have a sequence
M0 -!M1 -!M2 -!. . .
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 33
in LK DE . The usual way to take the colimit of suchWa sequenceWin a triangulat*
*ed
category is by taking the cofiber of the map g : Mi -! Mi which takes Mi to
Mi_ Mi+1 by the negative of the identity to Mi and the structure map to Mi+1.
This is called the sequential colimit in [HPS97 ]; let us denote it by scolimMi.
On the other hand, by choosing cofibrant and fibrant representatives Xi in
LK ME for Mi, we can get a diagram
X0 f0-!X1 f1-!X2 f2-!. . .
in LK ME .
Theorem A.11. In the above situation, there is an isomorphism scolimMi -!
hocolimXi in LK DE .
We learned the proof below from Stefan Schwede; Neil Strickland also provided*
* a
proof to the author. Schwede's proof works in any simplicial stable model categ*
*ory.
The model categories ME and LK ME are in fact topological [EKMM97 ], so in
particular simplicial.
To prove this theorem, we need a model of the homotopy colimit. This means
we must replace our sequence Xi by a cofibrant object in the model structure
on (LK ME )I; that is, by a sequence of cofibrations. Let I denote the simplici*
*al
interval [1]. We define Sm as the pushout in the diagram below.
Pm -gm---!Qm
? ?
(A.12) hm ?y ?y
Rm ----! Sm
Wm-1 Wm-1 Wm
In this diagram, Pm = j=0 (Xj_ Xj), Qm = j=0 (Xjx I), and Rm = j=0Xj.
The map gm is just inclusion into the two ends of the cylinder, which is a cofi*
*bration
since each Xj is cofibrant. The map hm is the inclusion on the left summand of *
*Xj
and the given map fj: Xj -!Xj+1 on the right summand of Xj.
Lemma A.13. There is a commutative diagram
S0 --i0--!S1 --i1--!S2 --i2--!. . .
?? ? ?
y ?y ?y
X0 ----!fX1 ----! X2 ----! . . .
0 f1 f2
in which the vertical maps are weak equivalences and the ij are cofibrations. *
*In
particular, colimSm is a model for hocolimXm .
Proof.There is an obvious map from the square defining Sm to the square defining
Sm+1 , which is a cofibration at each spot since the Xj are cofibrant. Examinat*
*ion
shows that the map
Qm qPm Pm+1 -! Qm+1
W m-1
is the coproduct of the cofibration Xm _ Xm -! Xm x I with j=0(Xj x I). In
particular, it is a cofibration, and so [Hir03, Lemma 7.2.15] implies that the *
*induced
map im :Sm -! Sm+1 is a cofibration.
34 MARK HOVEY
Now, Sm is also the pushout in the diagram below.
W m-1 W m-1
j=0(Xjx?{0}) ----! j=0(Xjx?I)
?y ?y
Xm ----! Sm
Here the left-hand vertical map sends Xj to Xm by a composite of the maps fi.
The resulting map Xm -! Sm is then a weak equivalence, as the pushout of a triv*
*ial
cofibration. This map is not part of a map of sequences, but we can construct a
map Sm -! Xm by defining it to be the identity on Xm and the composite
Xjx I -!Xj fj-!. .f.m-1---!Xm
on XjxI, where the first map above is induced by I -!{0}. This map Sm -! Xm is
also a weak equivalence, since it is a left inverse to the weak equivalence Xm *
*-! Sm ,
and it is part of a map of sequences.
W 1
Now let Y denote j=0Xj, and let f :Y -!Y denote the coproduct of the
maps fm . By taking the colimit of the pushout squares in A.12, we get the push*
*out
square below.
Y _ Y ----! Y x I
? ?
(1,f)?y ?y
Y ----! colimSm
In the homotopy category LK DE , this induces a map of exact triangles
Y _ Y ----! Y x I ----! Y ---h-! Y _ Y
?? ?? flfl ??
(A.14) (1,f)y y fl y (1, f)
Y ----! colimSm ----! Y ----!æ Y
We must identify the map h.
Lemma A.15. If Y is an object in a stable simplicial model category M, the map
h: Y -! Y _ Y in hoM induced by the cofibration sequence
Y _ Y -! Y x I -! Y
is the negative of the identity on the first factor and the identity on the sec*
*ond
factor.
Proof.It suffices to check this for Y = S0+in the category of pointed simplicia*
*l sets;
one gets the general case by smashing with Y . This is then easy to check.
It then follows that the map æ: Y -! Y in A.14 is the shift- 1 map, and
therefore that colimSm is the sequential colimit of the Yi's, completing the pr*
*oof
of Theorem A.11.
SOME SPECTRAL SEQUENCES IN MORAVA E-THEORY 35
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org