BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL
SEQUENCE
MARK MAHOWALD AND CHARLES REZK
Abstract.We show that the class of p-complete connective spectra with
finitely presented cohomology over the Steenrod algebra admits a duality*
* the-
ory related to Brown-Comenetz duality. This construction also produces a
full-plane version of the classical Adams spectral sequence for such spe*
*ctra,
which converges to the homotopy groups of a "finite" localization.
1.Introduction
In the paper [3], Brown and Comenetz introduced a notion of duality into stab*
*le
homotopy. In [4] Hopkins and Gross showed that this notion in certain situations
is closely connected with Spanier-Whitehead duality. In this note we wish to ex-
plore this connection and investigate it in connection with Adams spectral sequ*
*ence
considerations. In particular, we study a class of spectra which we call fp-spe*
*ctra
(Section 3). These are connective, p-complete spectra whose mod p cohomology
is finitely presented over the Steenrod algebra; that is, the cohomology of suc*
*h a
spectrum is described by a finite set of generators together with a finite set *
*of rela-
tions. This class of spectra includes the Johnson-Wilson spectra BP , connec*
*tive
K-theories, and the "higher" connective K-theory spectrum eo2. The class of fp-
spectra also includes some objects whose Bousfield Ln-localizations are the sam*
*e as
Ln localizations of finite complexes, at least in some cases. A classical examp*
*le is
the connective image-of-J spectrum, whose L1-localization is L1S0. It follows f*
*rom
calculations of Shimomura and Yabe that a -1-connective cover of L2S0 at primes
p 5 is also an fp-spectrum (Proposition 3.7).
We show that the category of fp-spectra admits a notion of duality (Theo-
rem 8.11). This duality is related to both Brown-Comenetz duality and Spanier-
Whitehead duality. The dual W X of an fp-spectrum X will be defined to be the
Brown-Comenetz dual of the fiber of the map X ! LfnX to the "finite localiza-
tion" of X (for n sufficiently large). The dual W X is itself an fp-spectrum. T*
*his
duality is related to Spanier-Whitehead duality through its action on cohomolog*
*y,
in the following sense. If H*X A* A*(n)M where A*(n) A* is a finite
sub-Hopf algebra of the Steenrod algebra, and M is a finite A*(n) module, then
H*W X A* A*(n)M , where M hom Fp(M; Fp) is the "Spanier-Whitehead
dual" of M as a finite module over A*(n).
Because of this duality, the Lfn-localization of an fp-spectrum is quite com-
putable. We show that there is a full-plane spectral sequence computing ss*LfnX,
with E2-term a "Tate cohomology" of H*X as a module over the Steenrod algebra
(Proposition 6.3 and Theorem 7.1). In all cases we know of LfnX LnX for an
fp-spectrum X.
____________
Date: May 15, 1998.
1
2 MARK MAHOWALD AND CHARLES REZK
1.1. Organization of the paper. In Section 2 we discuss modules and comodules
which are finitely presented over the Steenrod algebra. In Section 3 we define *
*the
notion of fp-spectra, and give examples. In Section 4 we discuss a duality func*
*tor
for finitely presented comodules over the Steenrod algebra, which is related to
the action of Brown-Comenetz duality on Eilenberg-Mac Lane spectra discussed
in Section 5. In Section 6 we note that an fp-spectrum admits a tower associated
to a spectral sequence whose E2-term is the "Tate cohomology" of the homology
of the spectrum, and in Section 7 show that such a tower realizes the localizat*
*ion
functor Lfn. In Section 8 we describe the duality theory of fp-spectra. In Sect*
*ion 9
we calculate some examples.
1.2. Notation. In this paper we work at one prime p at a time. We let A* denote
the mod p Steenrod algebra, and A* denote the dual mod p Steenrod algebra.
Unless otherwise indicated, all vector spaces, modules, and comodules in this
paper are graded. If V is a graded vector space over Fp, then V denotes the lin*
*ear
dual hom (V; Fp). If V is a left comodule over a graded Hopf algebra B, then V *
*is
taken to be a left comodule over B, via the canonical anti-automorphism O of B.
When dealing with graded objects, we use the following sign convention: a sign
is introduced whenever two symbols of odd degree are commuted.
2. Finitely presented modules and comodules over the Steenrod
algebra
A module M over the mod p Steenrod algebra A* is called finitely presented
if it fits in an exact sequence of modules
A* V1 ! A* V0 ! M ! 0
where Vi for i = 0; 1 are finite dimensional graded Fp-vector spaces. Likewise*
*, a
comodule N over the dual mod p Steenrod algebra A* is called finitely presented
if it fits in an exact sequence of comodules
0 ! N ! A* V0 ! A* V1
where Vi for i = 0; 1 are finite dimensional graded Fp-vector spaces. Because *
*all
finitely presented modules and comodules are of finite type, we can pass easily
between comodule and module language by taking vector space duals.
The Steenrod algebraSA* is a union of finite-dimensional sub-Hopf algebras. F*
*or
example, A* = nA*(n), where A*(n) A* is a finite dimensional sub-Hopf
algebra of the Steenrod algebra, generated as an algebra by { Sq2i| i n + 1i}f
p = 2 and by { fi; P pi| i inf}p is odd. Recall that A* is free as an A*(n)-mod*
*ule.
Lemma 2.1.
1. A module M over A* is finitely presented if and only if it is of the form
M A* E N for some finite dimensional sub-Hopf algebra E A* and
some finite dimensional E-module N.
2. Every map f :M ! M0 of finitely presented A-modules is of the form f
A*E g :A*E N ! A*E N0 for some finite dimensional sub-Hopf algebra
E A* and some map g :N ! N0 of finite dimensional E-modules.
Proof.Any finite sub-Hopf algebra E A* is contained in A*(n) for some n 0, *
* __
whence part 1 is [12, Ch. 13, Prop. 2(a)]. Part 2 follows by similar arguments.*
* |__|
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 3
Proposition 2.2.
1. The kernel and cokernel of a map of finitely presented A*-modules are fini*
*tely
presented.
2. A retract of a finitely presented A*-module is finitely presented.
3. If 0 ! M0 ! M ! M00! 0 is a short exact sequence in which M0 and M00
are finitely presented, then M is also finitely presented.
Proof.Since A* is A*(n)-free, the functor A* A*(n)(- ) is exact, and hence part
1 follows from (2.1).
To prove part 2, note that a retract N of M is the kernel of an idempotent
self-map e: M ! M. Hence part 2 follows from part 1.
The proof of part 3 is a standard result about finitely presented modules_over
any ring. |__|
Proposition 2.3.Suppose M is an A*-module and F is a finite A*-module. Then
M is finitely presented if and only if M F is.
Proof.It is immediate from (2.2) that M F is finitely presented if M is, since*
* F
admits a finite filtration whose subquotients are copies of Fp.
Suppose M F is finitely presented. Since F is a finite module, we can choose*
* a
"pinch" map ss :F ! dFp to a "bottom-dimensional cell" of F , and we can write
i: F ! F for the kernel of ss. Then there is an exact sequence
M F F -1iss---!M F -1ss-!M ! 0
which exhibits M as a cokernel of a map between finitely presented modules,_and_
the result follows from (2.2). |__|
Remark 2.4.Note that (2.1), (2.2), and (2.3) dualize to similar statements about
finitely presented comodules. We will not state the dual form of these results,
although we will make use of them in what follows.
2.5. Homological algebra for finitely presented comodules. Henceforth we
concentrate on finitely presented comodules. We let Mfpdenote the category of
finitely presented comodules over A* . By (2.2) we see that Mfp is an abelian
category.
Proposition 2.6.The dual Steenrod algebra A*, viewed as a A*-comodule, is both
projective and injective in Mfp, and Mfphas enough projectives and injectives.
Proof.It is clear that A* is injective in the full category of A* comodules, and
hence A* is injective in Mfpand there are enough injectives in Mfp. To prove th*
*at
A* is projective, consider a surjection M ! M0 of finitely presented comodules.*
* By
(2.1), this map is extended up from a surjection N ! N0 of finite A*(n)-comodul*
*es.
Since hom A*(A* ; M) homA*(n)(A* ; N) and A* is A*(n)-free, any map A* ! M0
can be lifted to a map to M. Furthermore, we can always produce enough maps *
* __
from A* to a finitely presented comodule, and thus Mfphas enough projectives. *
*|__|
Remark 2.7.It is known that the Steenrod algebra is injective as an A*-module
over itself [12, p. 201]. It would be interesting to know whether A* is project*
*ive as
a comodule over itself, without the restriction to the finitely presented categ*
*ory.
4 MARK MAHOWALD AND CHARLES REZK
Given a finitely presented comodule M, one can define its Tate cohomology
as follows. By (2.6) we can choose injective and projective resolutions
0 ! M ! C0 ! C1 ! C2 ! : : :
and
: :!:C-3 ! C-2 ! C-1 ! M ! 0
by finitely generated free A*-comodules. By gluing the ends together, we obtain*
* an
unbounded complex
: :!:C-2 ! C-1 ! C0 ! C1 ! : :;:
which in each degree is injective in the category of comodules. For s 2 Z define
HsTate(M) = Hs [homA*(Fp; Co)]:
Of course, if we can write M A* A*(n)N for some A*(n)-comodule N, and we
choose resolutions 0 ! N ! Do and D-o ! N ! 0 of N by finite free A*(n)-
comodules, then we see that
HsTate(M) Hs hom A*(n)(Fp; Do):
These groups are the same as morphisms in the stable category of A*(n)-modules,
as is shown in [7, Sec. 9.6].
3. fp-spectra
In this section we define the notion of fp-spectra, and produce several examp*
*les.
Recall that we work in the category of p-local spectra.
We first note the following theorem of Mitchell.
Theorem 3.1 (Mitchell).[13] For each n there exists a non-trivial finite complex
F such that H*F is A*(n)-free.
If X is a spectrum, say that ss*X is finite if sskX = 0 for all but finitely *
*many
k 2 Z, and is a finite group otherwise.
Proposition 3.2.Suppose X is a connective, p-complete spectrum. Then the fol-
lowing are equivalent.
1. H*X is finitely presented as a comodule over the Steenrod algebra.
2. H*X A* A*(n)M for some n 0 and some finite A*(n)-comodule M.
3. There exists a non-trivial finite complex F such that X ^ F is a finite we*
*dge
of suspensions of mod p Eilenberg-Mac Lane spectra.
4. There exists a non-trivial finite complex F such that ss*(X ^ F ) is finit*
*e.
Proof.The equivalence of 1 and 2 is just (2.1). Likewise, 4 is immediate given *
*3.
To show that 2 implies 3, we let F be as in (3.1), with H*F free over A*(n).
Thus H*(X ^ F ) is free over the Steenrod algebra on a finite set of generators,
whence X ^ F is a wedge of mod-p Eilenberg-Mac Lane spectra HFp.
To show that 4 implies 1, note that if ss*Y is finite for a connective spectr*
*um
Y , then Y can be built from finitely many copies of HFp, whence H*Y is finitely
presented by (2.2). Thus if ss*(X ^ F ) is finite, then H*(X ^ F ) H*X H*F is*
* __
finitely presented, and hence H*X is finitely presented by (2.3). *
*|__|
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 5
We call a spectrum X an fp-spectrum if it is connective, p-complete, and
satisfies any of the four equivalent statements of (3.2). Let C denote the cla*
*ss
of all fp-spectra. This class includes the Eilenberg-Mac Lane spectra HZ=pn and
HZp, the p-completed Johnson-Wilson spectrum BP , which has ss*BP
Zp[v1; : :;:vn], and connective Morava K-theories. Non-trivial suspension spect*
*ra,
and in particular finite complexes, are not fp-spectra.
Recall that a finite complex F is of type n if K(0)*F . . .K(n - 1)*F 0
and K(n)*F 6 0, where K(m) denotes the mth Morava K-theory. Define the
fp-type of an fp-spectrum X by
fptype(X) = min{(type(F ) - 1) such that ss*(X ^ F ) is}finite:
By the thick subcategory theorem [6], if fptype(X) = n then ss*(X ^ F ) is fi-
nite for all F of type > n. Thus, fptype(HFp) = -1, fptype(HZp) = 0, and
fptype(BP ) = n. Furthermore, if H*X A* A*(n)M, then fptype(X) n.
Let Cn denote the class of fp-spectra of type n. Then Cn is a subcategory of
the category of spectra, and Cn Cn+1.
Proposition 3.3.The classes C and Cn for n -1 are thick subcategories of the
homotopy category of spectra.
Proof.That C is a thick subcategory is an immediate consequence of criterion 4 *
*of
(3.2). Alternately, this follows from criterion 1 of (3.2) together with (2.2).*
*_The
proof for Cn is similar. |__|
For example, C-1 is the class of all spectra X with ss*X finite; as a thick sub*
*category
it is generated by HFp. Likewise, C0 is the class of all fp-spectra which are f*
*inite
Postnikov towers; as a thick subcategory it is generated by HZp. The class C1
contains the p-completed connective K-theory spectra bo and bu, along with their
connective covers. Thus C1 contains the image of J spectrum, since J fib(bo !
bspin). The class C2 contains eo2, the connective version of the "higher real *
*K-
theory" spectrum EO2 of Hopkins and Miller.
3.4. Ln-localization and fp-spectra. Let Ln denote Bousfield localization with
respect to the wedge K(0) _ . ._.K(n) of Morava K-theories. A spectrum W is
Ln-local if LnW W .
Proposition 3.5.Let W be an Ln-local spectrum such that for each k 2 Z the
homotopy group sskW has the form
sskW Fk Zakp (Q=Z(p))bk Qckp;
where Fk is a finite p-group, ak = 0 = ck for all sufficiently small k 0, and
bk = 0 = ck for all sufficiently large k 0.
Then there exists a map f :X ! W such that X is an fp-spectrum of fp-type n
and LnX ! LnW W is a weak equivalence.
Proof.Consider the connected cover Y = W (-N; : :;:1), where N is chosen so
that ak = ck = 0 for k < -N. If F is a finite complex with bottom cell in dimen*
*sion
0 and top cell in dimension d, then the map Y ^ F ! W ^ F is an isomorphism
on ssk for k > d - N, as can be seen by comparing the Atiyah-Hirzebruch spectral
sequences computing Y*F and W*F .
If F is a type (n + 1) complex, then W ^ F *, and so Y ^ F has non-trivial
homotopy in only a finite range of dimensions (-N; : :;:d-N), and each homotopy
6 MARK MAHOWALD AND CHARLES REZK
group is finite. Thus we have found a connective spectrum Y which satisfies cri*
*terion
4 of (3.2), and furthermore LnY W , since the fiber of Y ! W is coconnective
with torsion homotopy and thus is killed by Ln.
In order to get a p-complete spectrum, it suffices to replace any copies of Q*
*=Z(p)
or Qp in the homotopy of Y by a finite torsion group or a copy of Zp respective*
*ly;
by hypothesis there are only finitely many such copies to worry about. Note that
[-iHA; HQ=Z(p)] is a finite torsion group if A is a finitely generated Zp-module
and i > 0; thus by induction on the Postnikov tower of Y we can see that we can
always find a map
_r
Y ! niHQ=Z(p)
i=1
which is surjective on homotopy, and so that the fiber X of this map is p-compl*
*ete._
Then since LnHQ=Z(p) * we see that X is the desired spectrum. |__|
Remark 3.6.In view of Conjecture (7.3) and (8.9), it seems likely that the conv*
*erse
of (3.5) should hold. That is, if X is an fp-spectrum with fptype(X) = n, then *
*we
expect that sskLnX has the form given in (3.5).
It is interesting to know when the Ln-localization of a finite complex F is a*
*lso
the Ln-localization of an fp-spectrum X. One can say the following.
Proposition 3.7.If F is the p-completion of a finite complex, then LnF is the
Ln-localization of an fp-spectrum X in the following cases,
1. n = 0 or n = 1 (at any prime),
2. n = 2 if p 5, or
3. for any n and any prime p if F is a type n complex.
Proof.It suffices to show in each case that the hypotheses of (3.5) are satisfi*
*ed.
The cases n = 0 and n = 1 are well known. In fact, for n = 0 take X = HZp^F ,
and for n = 1 take X = J ^ F , where J is the connective image-of-J spectrum.
If n is arbitrary, but F is a type n complex, then the hypotheses of (3.5) ho*
*ld.
This is a consequence of the fact that the cohomology of the Morava stabilizer
algebra is a finitely generated algebra (see [15, Thm. 6.2.10]), together with *
*Hopkins
and Ravenel's demonstration of a horizontal vanishing line at the E1 -term of t*
*he
Adams-Novikov spectral sequence of LnF (see [16, Section 8.3]). These imply that
sskF is finite for all k.
When n = 2 and p 5, one can take a spectrum of the form Y ^ F , where Y is
an fp-spectrum such that L2Y L2S0. We show that the hypotheses of (3.5) hold
for L2S0 at p 5.
First, we note that the hypotheses of (3.5) hold for L2M(p), the localization*
* of
the mod p Moore space. This is a consequence of calculations of Shimomura [18],
as we explain below. There is a diagram
L2M(p) ________//L1M(p)_______//L2M(p; v11)
| | |
| | |~
fflffl| fflffl| fflffl|
LK(2)M(p) ____//_L1LK(2)M(p)___//L2M(p; v11)
in which the left-hand square is a pull-back square; this is because all the ob*
*jects in
it are L2-local, and the square is a pull-back after smashing with K(0)_K(1)_K(*
*2).
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 7
Since L1M(p) L1(v-11M(p)) L2(v-11M(p)), the top row is a cofiber sequence,
and thus so is the bottom row.
Shimomura computes the E2-term of the Adams-Novikov spectral sequence for
L2M(p; v11). Using Shimomura's calculation one may (with careful analysis) read
off that each group sskL2M(p; v11) must be finite; see the presentation of the *
*results
of this calculation given in [17]. Since sskL1M(p) is known to be finite, this *
*shows
that sskL2M(p) must be finite.
Hovey and Strickland [8, Thm. 15.1] actually carry out the careful analysis to
show that each group sskLK(2)M(p), k 2 Z, is finite, so we will derive what we
need from their results. To derive the finiteness of sskL2M(p), it suffices to *
*show
that L1LK(2)M(p) has finite homotopy groups. This spectrum is equivalent to
v-11LK(2)M(p), since the v1-self map of M(p) is trivial on K(2)*M(p). It happens
that ss*LK(2)M(p) decomposes as a finite sum of copies Fp[v1] plus a summand
which is v1-torsion. Thus ss*v-11LK(2)M(p) is a finite sum of copies of the fo*
*rm
Fp[v1 ], which is clearly a finite group in each dimension.
When n = 2 and p 5, the hypotheses of (3.5) hold for L2S0; this is a conse-
quence of the above remarks together with the calculation of Shimomura and Yabe
[19] of ss*L2S0 at p 5. They show that the homotopy of L2S0 consists of a free
summand in dimension 0, summands of the form Q=Z(p)in stems -3, -4, and -5,
together with a summand T consisting of non-infinitely divisible torsion. The a*
*bove
remarks on the finiteness of sskL2M(p) imply that the summand T of ss*L2S0_is
finite in each stem. |__|
We are led to make the following conjectures.
Conjecture 3.8. The hypotheses of (3.5) hold for any Ln-localization of a finite
complex, for any n 0.
Conjecture 3.9. For every finite complex F there exists an fp-spectrum X such
that LnF LnX.
Of course, Conjecture (3.8) implies Conjecture (3.9) as we have shown above.
There is also reason to believe that Conjecture (3.9) would imply Conjecture (3*
*.8);
see section (7.2).
3.10. Adams towers for fp-spectra. For a spectrum X we can construct an
Adams tower. This is a tower of spectra : :!:Xs+1 ! Xs ! : :w:ith X0 = X
and with fiber sequences Xs+1 ! Xs ks-!-sHVs, where ks is injective on mod p
homology; hence there is a resolution
0 ! H*X ! H*HVo;
and an Adams spectral sequence with Es;t2= Exts;tA*(Fp; H*X).
If X is an fp-spectrum, we can choose an Adams tower in which each Vs is a
finite dimensional graded vector space. We call such an fp-Adams resolution.
In fact, if H*X A A*(n)M for some finite A*(n)-module M, then the chain
complex H*HVo is induced from a resolution of M by A*(n)-comodules.
Given an Adams tower {Xs}, we may produce another tower
: :!:Xs0! Xs-10! : :!:X10! X00! *
by taking cofibers Xs+1 ! X ! Xs0. The fibers in this tower are -sHVs !
Xs0! Xs-10. Note that in the E2-term of the spectral sequence for this tower, e*
*ach
8 MARK MAHOWALD AND CHARLES REZK
horizontal line Es;*2is finite dimensional, and thus the spectral sequence sati*
*sfies
the complete convergence condition of [2, p. 263]. Thus the homotopy inverse li*
*mit
holimsXs0does not depend on the choice of fp-Adams resolution, and by a result
of Bousfield [1, Prop. 5.8 and Thm. 6.6] is equivalent to X, since X is p-compl*
*ete
and connective.
4. Duality for finitely presented comodules
In this section we describe a duality functor "Ion the category of finitely p*
*resented
comodules. This functor was essentially introduced by Brown and Comenetz in [3].
Our interest in this functor stems from the fact that it generalizes the notion*
* of
"Spanier-Whitehead duality" of finite comodules over A*(n), in which a comodule
is dual to its vector space dual. It will be needed in later sections to study *
*duality
on fp-spectra.
4.1. Construction of functors J" and I". To motivate the construction, note
that if X is a p-complete spectrum and H denotes the mod p Eilenberg-Mac Lane
spectrum, then the graded vector space [H; X]* is in a natural way a module over
the Steenrod algebra. This object is approximated by the edge map [H; X]* !
homA* (H*H; H*X) of the Adams spectral sequence. This algebraic approximation
itself admits an action by the Steenrod action, and we will call this module "J*
*(H*X).
The vector space dual to "J(H*X) will be "I(H*X), which admits a coaction by the
dual Steenrod algebra.
Recall that if we regard the dual Steenrod algebra A* as a left-comodule over
itself, then each element a 2 A* of the Steenrod algebra induces a map A* ! A*
of left A*-comodules via
X
a . z = (-1)|a||z|z0;
P
where z 2 A*, z0 z00is the diagonal of z in A* A*, and represents the
usual pairing of A* and A*. In fact, there is an isomorphism of algebras
(4.2) hom A*(A* ; A*) A*:
We reserve the notation a . z for this action. Note that this action gives A* *
*the
structure of a left A*-module; however, this is not identical to the usual left*
* action
of the Steenrod algebra on H*H.
We define a functor "Jfrom left A*-comodules to left A*-modules by
J"(M) = homA*(A* ; M):
This has a natural right A*-action induced by pre-composition of comodule maps,
using (4.2), which is made into a left A*-action using O; if z 2 A*, a 2 A*, and
f 2 "J(M), the left action can be written
(a . f)(z) = (-1)|a||f|f(Oa . z):
Note that if M is finitely generated then "J(M) is bounded below (where we use
cohomological grading for "J(M)).
We define a functor "Ifrom finitely generated left comodules to left comodules
by
"I(M) = (J"(M)) homA*(A* ; M):
Since "J(M) was bounded below, "I(M) is bounded below, and thus receives a left
comodule structure in the usual way. This structure is characterized as follows*
*: if
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 9
P
u 2 "I(M) and u 7! u0 u002 A* "I(M) is the comodule action on u, and if
a 2 A* and f 2 "J(M), then
X 00
__ = (-1)|a||u<|u0; a>:
In comparison, Brown and Comenetz [3] define a functor cp on the category of
A*-modules; their functor is defined by
cp(M) homA*(M; A*);
with an appropriate A*-action. Thus, our "Iis just a comodule version of their *
*cp.
4.3. Action of J"and "Ion free comodules. Let Mfree Mfpdenote the full
subcategory of comodules of the form A* V , where V is a finite vector space. We
want compute the action of "Jand "Ion this subcategory.
The following describes the category Mfree.
Proposition 4.4.For finite V and W , homA* (A* V; A*W ) A*hom (V; W ),
where a oe 2 A* hom(V; W ) corresponds to the morphism of comodules
z v 7! (-1)|oe||z|a . z oe(v); z 2 A*; a 2 A*; v 2 V; oe 2 hom(V; W );
and composition of maps is given by (a oe) O (b o) = (-1)|oe||b|ab oeo.
Proof.Straightforward. |___|
Proposition 4.5.There is a natural isomorphism
"J(A* V ) homA*(A* ; A* V ) A* V
of left A*-modules, where a v 2 A* V corresponds to the map defined by
z 7! (-1)|z||v|Oa . z v; z 2 A*; a 2 A*; v 2 V:
Given a map a oe :A* V ! A* W of comodules, the induced map J"(a
oe): A* V ! A* W sends b v 7! (-1)|b|(|a|+|oe|)b Oa oe(v).
Proof.Straightforward. |___|
Corollary 4.6.There is a natural isomorphism
"I(A* V ) (A* V ) A* V
of left-comodules. The induced map "I(a oe): A* W ! A* V sends
z w 7! (-1)|oe||z|Oa . z oe(w ); z 2 A*; a 2 A*; w 2 W ;
where oe2 hom(W ; V) is the adjoint to oe 2 hom(V; W ). In other words, "I(a o*
*e) =
Oa oe.
Remark 4.7.In terms of bases vj and wifor V and W we can view maps A*V !
A* W as matrices (aij) with entries in the Steenrod algebra, acting by
X X
zj vj 7! aij. zj wi:
j i;j
Hence, the induced map "I(aij): A*W ! A*V corresponds to the matrix (Oaji)
in terms of the dual bases vjand wi of V and W .
10 MARK MAHOWALD AND CHARLES REZK
4.8. Properties of the duality functor.
Proposition 4.9.The functor "Irestricts to a functor "I:Mopfp! Mfp, and is
exact on Mfp. Furthermore, there is a natural isomorphism M ! "I"IM for objects
in Mfp.
Proof.That "Iis exact on the subcategory of finitely presented comodules follows
from (2.6). By (4.6) the functor "Itakes finitely generated free comodules to t*
*he
same, and thus by exactness "I(Mfp) Mfp.
One can construct a natural isomorphism M ! "I"IM when M A* V , since
by (4.6) "I"I(A* V ) is tautologically isomorphic to A* V . This isomorphism*
*__
extends by exactness to all finitely presented comodules. |*
*__|
Proposition 4.10.Let M be a finite A*(n)-comodule. Then
"I(A* A*(n)M) A* A*(n)a(n)M
as left comodules, where a(n) is the dimension of the "top cell" of A*(n).
Proof.There is an exact sequence
0 ! M ! A*(n) V -(aij)--!A*(n) W
of A*(n)-comodules, where aij2 A*(n). After applying vector space duals we can
identify the resulting sequence with
0 -a(n)M A*(n) V -(Oaji)---A*(n) W
by "Poincare duality" of A*(n) [12, Ch. 12.2]. The result follows from (4.7)_af*
*ter
extending up to A*. |__|
5.Brown-Comenetz duality
Recall that the functor
X 7! hom(ss0X; Q=Z)
is a generalized cohomology theory satisfying the wedge axiom, and hence is rep*
*re-
sented by a spectrum I. We write IY = F(Y; I) for the function spectrum, whence
IY is the spectrum representing the functor
X 7! IY 0(X) = hom(Y0X; Q=Z):
The spectrum IY is called the Brown-Comenetz dual of Y .
We write DX F(X; S0) for the Spanier-Whitehead dual of X. Note that if X is
any spectrum and F is a finite complex, then the natural map IX ^DF ! I(X ^F )
is an equivalence.
There is a natural double-dual map X ! IIX. If Y is a spectrum such that
each homotopy group sskY is finite, then Y IIY via this map. Thus, given such
a Y and given any spectrum X, there is a natural isomorphism
[X; Y ] [IY; IX]:
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 11
5.1. Eilenberg-Mac Lane spectra. Let H HFp denote the mod p Eilenberg-
Mac Lane spectrum. Then IH H. In fact, by the universal coefficient theorem,
IH*X hom(H*X; Q=Z) hom(H*X; Fp) H*X:
Lemma 5.2. Under the above identification the map
I :[H; H]* ! [IH; IH]* [H; H]*
sends a 2 A* to Oa.
Proof.This is [3, Thm. 1.9(d)]. |___|
More generally, let V denote a finite dimensional graded Fp-vector space, and
let V = hom (V; Fp) denote its vector space dual. Let HV denote the generalized
Eilenberg-Mac Lane spectrum with ss*HV = V , whence HV*X H*X V .
Proposition 5.3.There is an equivalence IHV HV , and we have isomorphisms
H*(IHV ) "J(H*HV )
of A*-modules and
H*(IHV ) "I(H*HV )
of A*-comodules which are natural in HV .
Proof.This is immediate from (4.6) and (5.2); alternatively, it follows_from [3,
Thm. 1.3]. |__|
5.4. Algebraic approximation. We note that the functor "Iof Section 4 serves as
an algebraic "approximation" to H*IX, at least when H*X has finitely presented
homology.
Proposition 5.5.For each spectrum X with finitely presented homology, there is
a map
X :"I(H*X) ! H*IX
which is natural in X. Furthermore, this map is an isomorphism when ss*X is
finite.
Compare with [3, Thm. 1.13].
Proof.Given X we can choose an Adams resolution
X ! HV0 ! HV1 ! : :;:
in which V0 and V1 are finite dimensional vector spaces, and the sequence
0 ! H*X ! H*HV0 ! H*HV1
is exact. Applying I to the first diagram gives maps
IX HV0 HV1
and a sequence
H*IX "I(H*HV0) "I(H*HV1);
not necessarily exact. We let X be the induced map
i j
: "I(H*X) = Cok "I(H*HV1) ! "I(H*HV0) ! H*IX:
To see that X is independent of the choice of resolution and is natural, use a
map between Adams resolutions.
12 MARK MAHOWALD AND CHARLES REZK
The map is by construction an isomorphism for X = HFp. By the exact-
ness property of (4.9), we see that X is an isomorphism for any X in the thick*
* __
subcategory generated by HFp, which are precisely the X with finite homotopy. *
*|__|
6.Geometric realization of Tate cohomology
In this section we note that one can construct for each fp-spectrum a Z-index*
*ed
Adams tower; this is a tower which extends both above and below X, whose layers
are finite mod p generalized Eilenberg-Mac Lane spectra, and which leads to a
spectral sequence whose E2-term is the Tate cohomology of H*X. We give several
constructions, starting with the most general.
6.1. Construction of the tower.
Lemma 6.2. Let X be an fp-spectrum. Then
[H; X]* homA*(A* ; H*X) "J(H*X):
Proof.Choose an fp-Adams tower {Xs} for X. Then there is a spectral sequence
Es;t1= [H; HVs]t=) [H; X]t-s. We claim that
1. Es;t2 0 for s > 0, so that E0;t2 E*;t1, and
2. E0;t2 homA*(A* ; H*X).
Since X is connective and p-complete, the first claim implies that the spectral
sequence converges, and thus E0;t2 [H; X]t.
To prove the claim, recall that the resolution 0 ! H*X ! H*HVs is extended
up from a resolution 0 ! M ! C(s) of A*(n)-modules. Now
[H; HVs]* homA*(A* ; A*A*(n)C(s)) homA*(n)(A* ; C(s)):
Since as an A*(n)-comodule A* A*(n) A*==A*(n) we see that A* is projective
as an A*(n)-comodule, and thus the sequence
0 ! homA*(A* ; H*X) ! homA*(A* ; H*HVo)
is exact as desired. |___|
Thus, given an fp-spectrum X, we may construct a tower "realizing" the Tate
cohomology by constructing fp-spectra X-s for s 0 inductively as follows. Firs*
*t,
let X0 = X. Since H*X-s A* A*(n)M(-s) for some finite module M(-s), we
can choose a surjection C(-s - 1) ! M(-s) from a free A*(n)-comodule. This
may be extended to a surjection A* A*(n)C(-s - 1) ! A* A*(n)M(-s), which
in turn by the above lemma is realized by a map of spectra sHV-s-1 ! X-s. If
we take the cofiber
sHV-s-1 ! X-s ! X-s-1
we get another fp-spectrum X-s-1; iteration produces an infinite sequence : :!:
X-s ! X-s-1 ! : :.:If we put this sequence together with an fp-Adams tower
for X, we get a tower {Xs}s2Z, which we call a Z-indexed Adams tower for X.
Note that given a map f :X ! Y of fp-spectra and given Z-indexed Adams
towers {Xs} and {Ys}, we can extend f to a map of towers, by the "dual" of the
usual argument, using (6.2). In particular, given any two Z-indexed Adams towers
for X we can produce a map between them.
For such a tower, let bX= hocolims!1 X-s. There is a full-plane Adams-type
spectral sequence Es;t2 Hs;tTate(H*X) =) sst-sbXapproximating the homotopy of
bX, where Hs;tTateis the Tate cohomology of (2.5).
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 13
Given such a tower {Xs}, write Xqp= cof(Xq+1 ! Xp) for -1 < p q < 1.
Then let X-11= cof(X ! bX) hocolims!1 X-1-s. A straightforward convergence
argument shows that the homotopy spectral sequence of this colimit converges, a*
*nd
hence X-11does not depend on the choice of the Z-indexed tower.
The above remarks are summarized in the following proposition.
Proposition 6.3.
1. For each fp-spectrum X there exists a Z-indexed Adams tower {Xs}.
2. Given Z-indexed Adams towers {Xs} and {Ys} for X and Y , and a map
f :X ! Y , there exists a map {fs:Xs ! Ys} of towers which extends f.
3. The colimit bX= hocolims!1 X-s of a Z-indexed Adams tower depends only
on X, and not on the choice of tower.
6.4. Alternate construction of the tower. Let X be an fp-spectrum, and let
F be a finite complex such that H*(X) H*(F ) is a free A*-comodule. Suppose
also a map S0 ! F which is injective on homology. Such an F always exists; if
H*X is an extended A*(n)-comodule, let F be a Mitchell complex of type n as in
(3.1), and S0 ! F the inclusion of a bottom cell.
Form a fiber sequence F ! S0 ! F , so that by Spanier-Whitehead duality we
get a dual fiber sequence DF ! S0 ! DF . We obtain a Z-indexed Adams tower
with (
F (s) if s 0
Xs = X ^
X ^ (DF )(-s) if s < 0,
where the maps in the tower are induced by fiber sequences X^F (s+1)! X^F (s)!
X ^ F (s)^ F and X ^ DF (s)^ DF ! X ^ DF (s)! X ^ DF (s+1).
In those cases when there actually exists a complex F with H*F A*(n), and
H*X is an extended A*(n)-module, then this complex realizes the unbounded chain
complex obtained by gluing together the bar complex for M and the co-bar complex
for M as an A*(n)-comodule.
6.5. An explicit construction for bo. Recall that H*bo A* =A* (Sq1; Sq2).
Consider the minimal Adams tower {bos} for bo at p = 2; write H*bos A*A*(1)
M(s).
Let R(n) denote the cofiber of P n! S0 for n 0; thus R(0) S0. Recall from
[9] that
M(4s) H*R(8s):
We define a map R(8s + 8) ! R(8s) as follows. Let p: R(8s + 8) ! P88s+8s+1
denote the pinch map obtained by taking the quotient of R(8s) ! R(8s + 8). Then
p O (16) ~ 0, where (16) is the degree 16 map on R(8s + 8) ! R(8s + 8), and so *
*(16)
lifts to a map fs:R(8s + 8) ! R(8s). The map fs has degree 16 on the bottom
cell.
Likewise, we may consider the Spanier-Whitehead dual map
f-s = Dfs-1:DR(8s - 8) ! DR(8s):
This map has degree 16 on the top cell. Note that f-1 :S0 ! DR(8s), so we can
put all the fs for s 2 Z together into a Z-indexed tower
: :f:2-!R(16) f1-!R(8) f0-!R(0) S0 f-1--!DR(8) f-2--!DR(16) f-3--!::: :
14 MARK MAHOWALD AND CHARLES REZK
Proposition 6.6.There is a Z-indexed Adams tower {bos} for bo with
(
bo4s bo ^ R(8s) if s 0,
bo ^ DR(-8s) if s < 0,
and the map bo4s+4! bo4s is bo ^ fs.
Proof.Set bo4s as above, and construct a the start of a minimal Adams tower
bo4s+3! bo4s+2! bo4s+1! bo4sover bos. It is not hard to see that R(8s + 8) !
bo ^ R(8s) bo4slifts to bo4s+3, so that we can extend to a map bo ^ R(8s + 8) !
bo4s+3; likewise, DR(8s - 8) ! bo ^ DR(8s) bo-4s lifts to bo-4s+3, so we can
extend to a map bo ^ DR(8s - 8) ! bo-4s+3. The resulting tower is an Adams_
tower. |__|
Here is a chart which presents the Adams E2-term for bo ^ DR(16).
r| r r r
r| r| r|| r
r|| r|| r|r
r| r| r|
r| r|
r|| r
r|r
r|
r|
r r|
r r||
r r|
r r| r|
r r|| r|| r||
r r| r| r|
r r| r| r|
A chart for Exts;tA(1)(H*(DR(16)); Z=2) for s < 16.
7.Finite localization
There exists a functor Lfnand natural map X ! LfnX, called finite localiza-
tion. It is Bousfield localization with respect to the wedge Tel(0) _ . ._.Tel(*
*n),
where Tel(k) denotes the vk-telescope on some chosen type k finite complex. The
functor Lfnis characterized by the following properties [11].
1. The fiber CfnX = fib(X ! LfnX) is a homotopy colimit of some diagram of
type-(n + 1) finite complexes.
2. There are no essential maps from a type (n + 1) finite complex to LfnX (i.*
*e.,
LfnX is Lfn-local).
3. LfnX is smashing; i.e., LfnX X ^ LfnS0.
Note also that 1 and 2 imply that LfnX X if X is Lfn-local. Also, if X is a ty*
*pe
n finite complex, then LfnX v-1X, where v :dX ! X is a vn-self map of X.
Theorem 7.1. Let X be an fp-spectrum with fptype(X) n, and let {Xs} be a
Z-indexed Adams tower for X, with colimit bX. Then LfnX bX.
Proof.Consider the map i: X ! bX. It suffices to show that
1. Lfn(i) is an equivalence, and
2. bXis Lfn-local.
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 15
Claim 1 follows immediately from the fact that LfnHFp * and that Lfnis smashin*
*g,
since bXis obtained from X by attaching mod p Eilenberg-Mac Lane spectra.
To prove claim 2, we must show that bXadmits no essential maps from a type-
(n + 1) finite complex F , or equivalently, that Xb^ F * for any such complex.
This follows from the following:
3. For any finite complex F , {Xs ^ F } is a Z-indexed Adams tower for X ^ F ,
and bX^ F hocolimX-s ^ F .
4. If Y is an fp-spectrum with ss*Y finite, then bY * for any Z-indexed Adams
tower for Y .
The proof of claim 3 is straightforward.
To prove claim 4, apply Brown-Comenetz duality to the sequence
Y = Y0 ! Y-1 ! Y-2 ! : :!:bY:
This produces a tower
IYb! : :!:IY-2 ! IY-1 ! IY0 = IY
which is easily seen to be an Adams tower for IY by (5.5). Since IY is connecti*
*ve
and p-complete and ss*IY is finite, its Adams spectral sequence converges,_and
IYb *, whence bY *. |__|
7.2. Relation between Lfnand Ln. Let X ! LnX denote Bousfield localization
with respect to K(0) _ . ._.K(n). There is a natural map tn :LfnX ! LnX; this
map is an equivalence for X S0, and hence for all X, if and only if the Telesc*
*ope
Conjecture holds. This conjecture is true for n = 0 and n = 1, and is believed *
*to
be false for n 2.
It is reasonable to ask whether tn is an equivalence when X is an fp-spectrum.
We note that tn is an equivalence in the following cases, which include all cas*
*es we
know of.
1. Since BP is a BP -module spectrum, one can compute LnBP using the
chromatic tower method of [14, Sec. 6]. (We would like to thank Hal Sadofs*
*ky
for pointing this out to us.) This calculation shows in particular that th*
*e fiber
of BP ! LnBP is coconnective with torsion homotopy; thus the fiber
is killed by Lfn, and hence tn is an equivalence on LnBP , and is in fa*
*ct an
equivalence on the thick subcategory of C generated by BP .
2. For any fp-spectrum X obtained by the procedure of the proof of (3.5) the
map tn :LfnX ! LnX is an equivalence, since by construction the fiber of
X ! LnX is coconnective with torsion homotopy.
We make the following conjecture.
Conjecture 7.3. The map tn :LfnX ! LnX is an equivalence for all fp-spectra
X.
This conjecture is of interest, because it would give information on how badly
the Telescope Conjecture fails, assuming it does fail. Namely, suppose F is a t*
*ype
n finite complex and v-1F is its vn-telescope; then if the Telescope Conjecture*
* fails
for n and Conjecture (7.3) holds, it follows that ssk(v-1F ) is an infinite gro*
*up for
some k 2 Z, whereas sskLnF is finite for all k. To see this, we argue as follo*
*ws;
if each ssk(v-1F ) were a finite group, then the proof of (3.5) would apply to *
*show
that there exists an fp-spectrum X with LfnX v-1F . However, Conjecture (7.3)
would then imply that v-1F LnX LnF .
16 MARK MAHOWALD AND CHARLES REZK
Conjecture (7.3) also would imply, using (8.9) of the next section, that the *
*two
conjectures (3.8) and (3.9) discussed in section 3 are in fact equivalent state*
*ments.
8. Duality
8.1. Duality for fp-spectra of type less than n. Let
WnX = ICfnX;
where Cfn= fib(X ! LfnX). Thus Wn is a contravariant functor from spectra to
spectra. Since Lfnis smashing,
WnX F(X ^ CfnS0; IS0) F(X; WnS0):
That is, WnS0 is a dualizing complex for Wn. Furthermore, there is a natural map
X ! WnWnX; this map is adjoint to the evaluation map X ^ F(X; WnS0) !
WnS0.
Note that Wn vanishes on Lfn-local spectra. Also, if ss*X is finite, then Lfn*
*X *
and we have WnX IX.
Recall that Cn denotes the category of fp-spectra with fp-type n.
Theorem 8.2. Let X be an object in Cn.
1. There is a natural isomorphism H*WnX "I(H*X):
2. WnX is in Cn.
3. The natural map X ! WnWnX is an equivalence.
From this we obtain the following.
Corollary 8.3.The functor Wn restricts to a functor Wn :Copn! Cn, which is an
equivalence of categories.
Proof.The corollary follows from the fact that the functor Wn is "self-adjoint";
that is,
Wn :Sop AE S :(Wn)op
is a pair of adjoint functors, where S represents the homotopy category of spec*
*tra.
Part 2 of (8.2) says that Wn carries Copninto Cn, and part 3 says that the rest*
*riction_
of Wn to Cn gives an adjoint equivalence Wn :CopnAE Cn :(Wn)op of categories. *
*|__|
Proof of Theorem 8.2.If X is an fp-spectrum, choose a Z-indexed Adams tower
{Xs}, and as in (6.1) write Xpq= cof(Xp+1 ! Xq). Then
CfnX -1 hocolims!1X-1-s
by (7.1), and thus
i j
(8.4) WnX holims!1IX-1-s:
Write H*X A*A*(n)M. We can construct the bottom part of the Z-indexed
tower to realize any projective resolution of M by finite A*(n)-modules, e.g., *
*by
the resolution dual to the minimal resolution 0 ! M ! Co of M by finite A*(n)-
modules. Then (8.4) immediately implies that WnX is connective and p-complete,
since the tower {I(X-1-s)}s1 must necessarily be the tower associated with an
Adams tower for WnX which realizes Co, and so the connectivity of I(X-1-s) is
bounded below by a fixed N for all s 1.
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 17
Note that for any Y we have that HFp ^ LfnY *, whence the map H*CfnY !
H*Y is an isomorphism, so that there exists by (5.5) a natural comparison map
n(Y ): "I(H*Y ) ~-!"I(H*CfnY ) ! H*WnY:
We want to show that n(X) is an isomorphism.
Since X has a Z-indexed tower, there is a sequence
: :!:HV-2 ! HV-1 ! X
which induces an exact sequence
H*HV-2 ! H*HV-1 ! H*X ! 0
on homology. Applying Wn to this sequence gives a sequence
WnX ! WnHV-1 ! WnHV-2 ! : : :
which corresponds to an Adams resolution for WnX and hence induces an exact
sequence
0 ! H*WnX ! H*WnHV-1 ! H*WnHV-2:
It is clear that the comparison maps n(HV-s) are isomorphisms, and so n(X)
is an isomorphism, since "Iis exact. Thus we have proved parts 1 and 2 of the
proposition. *
* __
Part 3 follows from parts 1 and 2, together with (8.7), to be proved below. *
* |__|
Remark 8.5.If X is a spectrum which might not be p-complete, and Xp denotes its
p-completion, then one can show via an arithmetic square argument that CfnX
CfnXp, and hence WnX WnXp. In particular, if Xp is an fp-spectrum, we may
conclude that WnWnX Xp.
We still owe the reader one more fact.
Proposition 8.6.Suppose X and WnX have finitely presented homology. Then
there is a commutative square
n(WnX)
"I(H*WnX) _____//H*WnWnX
N OO
"n| NNNN |
I|(X) NNNN |
fflffl| NNN''|
"I"I(H*X)oo~_____H*X
Corollary 8.7.If X and WnX have finitely presented homology and the maps
n(X): "I(H*X) ! H*WnX; and n(WnX): "I(H*WnX) ! H*WnWnX
are isomorphisms, then the the map X ! WnWnX induces an isomorphism in
homology.
Proof of Proposition 8.6.Choose resolutions X ! HC0 ! HC1 and WnX !
HD0 ! HD1. This leads to a sequence
HD1 ! HD0 ! WnWnX ! HC0 ! HC1:
The diagonal map in the diagram is the induced map
Cok H*HD1 ! H*HD0 ! Ker(H*HC0 ! H*HC1) :
It is straightforward to check commutativity of the diagram. |_*
*__|
18 MARK MAHOWALD AND CHARLES REZK
8.8. A finiteness result.
Proposition 8.9.Let X be an fp-spectrum with fptype(X) = n, and let Y = LfnX
be its finite localization. Then for each k 2 Z the homotopy group sskY has the*
* form
sskY Fk Zakp (Q=Z(p))bk Qckp;
where Fk is a finite p-group, ak = 0 = ck for all sufficiently small k 0, and
bk = 0 = ck for all sufficiently large k 0.
Proof.There is a fiber sequence CfnX ! X ! LfnX, and the Brown-Comenetz
dual WnX of CfnX is an fp-spectrum by (8.2). Thus X is connective with sskX
FkZmkp, and CfnX is coconnective with sskCfnX Fk0(Q=Z(p))nk. The image and
coimage of the connecting map sskCfnX ! sskX can only be a finite torsion group;
thus, to prove the result for sskLfnX we need to show that, in a group extensio*
*n of
the form
0 ! F Zmp! M ! F 0 (Q=Z(p))n ! 0;
where F and F 0are finite p-groups, that M is as described in the statement of *
*the
proposition.
It is easy to reduce to the case when F = 0 = F 0. Then extensions are classi*
*fied
by elements of Ext((Q=Z(p))n; Zmp) hom (Znp; Zmp); if A 2 hom (Znp; Zmp) class*
*ifies
the extension then
M Cok Znp(A;I)---!Zmp Qnp
where I :Znp! Qnpis the standard inclusion. It follows that M Zmp= imA
Qnp= kerA; this can be shown by choosing a map B :Zmp ! Qnpsuch that (I -
BA): Znp! Qnpprojects to the kernel of A Q, in which case there is an exact
sequence
0 ! Znp(A;I)---!Zmp Qnp(x;y)!(x;y-Bx)-----------!Zmp= imA Qnp= kerA ! 0
which realizes the splitting of M. Now the result follows from the fact that_
Zmp= imA F Zapand Qnp= kerA (Q=Z(p))bQcp, where F is a finite p-group. |__|
8.10. Duality for fp-spectra. Recall that there exist natural transformations
Lfn+1X ! LfnX, and hence natural transformations WnX ! Wn+1X. We define
W X = hocolimn!1WnX:
If X is an fp-spectrum with fptype(X) = n, then (7.1) shows that LfmX LfnX
for m n, and thusSWm X WnX for m n. This, together with (8.3) and the
fact that C = n Cn is the homotopy category of all fp-spectra, gives
Theorem 8.11. The functor W induces an equivalence of categories W :Cop ! C,
and H*W X "I(H*X) for all X in C.
9.Calculations
In this section we compute W X in several cases, and thus implicitly compute
LfnX (and LnX, by (7.2)) for sufficiently large n.
Lemma 9.1. If X is a ring spectrum and Y is an X-module spectrum, then WnY
is also an X-module spectrum.
Proof.This is a formal consequence of the fact that WnY F(Y; WnS0). |___|
BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 19
Proposition 9.2.If X is an fp-spectrum which is a ring spectrum, H*X is self-
dual as a finitely-presented comodule (i.e., "I(H*X) dH*X for some d), and
H*X does not split over the Steenrod algebra, then W X dX.
Proof.Choose a map Sd ! W X which hits the bottom homology class, dual to
the unit in H*X. By (9.1) this map extends to a map dX ! W X of X-module
spectra, and this map is necessarily an isomorphism on mod p homology, and_hence
an equivalence. |__|
Corollary 9.3.We have that
n+1-1
1. W BP e(n)BP , where e(n) = 2p_____p-1- (n + 1),
2. W bu 4bu (at all primes),
3. W bo 6bo (at all primes), and
4. W eo2 23eo2 at p = 2 and at p = 3.
Proof.The only case which needs comment is 4. In this case it can be derived fr*
*om
the following facts [5]. At p = 2, eo2 ^ F BP <2>, where F is a certain finite
complex with H*F DA*(1), the "double" of A*(1). At p = 3, eo2^ (S0[ffe4[2ff_
e8) BP <2>_ 8BP <2>. |__|
Remark 9.4.In each of the above examples, we can read off ss*LnX from our knowl-
edge of the homotopy of X. In particular, in each case there is a wide "gap" be-
tween the first copy of Zp and the last copy of Q=Z(p)in the homotopy of LnX; if
W X dX, then
8
>0 if 1 - d < s < 0, and
:Q=Z
(p) if s = 1 - d.
There is a convenient heuristic for reading off the expected size of the "gap*
*" in
ss*LnX for ring spectraPof the above type. If ss*X Q Qp[x1; : :;:xn], then the
size of the gap is ni=1(|xi| + 1). For example, ss*eo2 Q Qp[x8; x12] at p =*
* 2
or 3, so the gap is (8 + 1) + (12 + 1) = 22. For BP the gap is the same as *
*the
dimension of the Toda complex V (n), should it exist.
Recall from (4.10) that if H*X A*A*(n)M, then "I(H*X) A*A*(n)dM ,
where d is the dimension of the "top cell" of A*(n).
Proposition 9.5.Let Jp denote the connective image-of-J spectrum completed at
the prime p.
1. For p odd, W Jp 3Jp.
2. For p = 2, there is a cofiber sequence 3HF2 ! J2^ (S0 [2e1[je3) ! W J2.
Proof.At an odd prime, Jp is the fiber of any map BP <1> ! qBP <1>, q = 2(p-1),
which sends the cohomology generator 2 HqqBP <1> to P 1 2 HqBP <1>. At
p = 2, J2 is the fiber of any map bo ! bspin which sends the cohomology generat*
*or
bspin2 H4bspin to Sq4bo2 H4bo [10]. In either case, the map in cohomology is
induced from a map of A*(2)-modules.
We leave the odd prime case to the reader. Suppose p = 2, and let F = S0 [2
e1 [j e3. Since bspin 7bo ^ DF and H*(F ^ DF ) is a direct sum over the
20 MARK MAHOWALD AND CHARLES REZK
Steenrod algebra of a spherical class in dimension 0 with a free A*(1) module on
one generator, we see that J ^ F fits in a fiber sequence
J ^ F ! bo ^ F ! 7bo _ 4HF2:
We can kill the copy of HF2 by taking the evident cofiber 3HF2 ! J ^ F ! C,
so that we obtain a cofiber sequence
C ! bo ^ F ! 7bo:
On applying W we get a fiber sequence
-1bo ! 6bo ^ DF ! W C ! bo f-!7bo ^ DF:
We can compute the action on f on cohomology, since by (4.10) it is induced from
a map of A*(2)-modules. The computation, which is straightforward, shows that_
the bottom class of 7bo ^ DF hits Sq4bo2 H4bo, and thus W C J. |__|
Corollary 9.6 (Hopkins).
I(L1S0) L1(S-1 [2 e0 [j e2):
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BROWN-COMENETZ DUALITY AND THE ADAMS SPECTRAL SEQUENCE 21
Department of Mathematics, Northwestern University, Evanston, IL 60208
E-mail address: mark@math.nwu.edu
E-mail address: rezk@math.nwu.edu
__