On the coalgebraic ring and
BousfieldKan spectral sequence
for a Landweber exact spectrum
Martin Bendersky*& John R. Huntonyz
5 December 2001
Abstract
We construct a BousfieldKan (unstable Adams) spectral sequence
based on an arbitrary (and not necessarily connective) ring spectrum E
with unit and which is related to the homotopy groups of a certain unsta
ble E completion X^Eof a space X. For E an Salgebra this completion
agrees with that of the first author and R. Thompson [7]. We also es
tablish in detail the Hopf algebra structure of the unstable cooperations
______________________________
*Hunter College and the Graduate Centre, CUNY, New York, USA. Email: mben
ders@shiva.hunter.cuny.edu
yDepartment of Mathematics and Computer Science, University of Leicester, Un*
*iversity
Road, Leicester, LE1 7RH, England. Email: J.Hunton@mcs.le.ac.uk
zCorresponding author. Tel.: ++44116 252 5354; fax: ++44116 252 3915.
1MSC: 55P60, 55Q51, 55S25, 55T15. Secondary: 55P47.
2Keywords: BousfieldKan spectral sequence, unstable Adams spectral sequence*
*, Hopf
rings, coalgebraic algebra, Landweber exact, unstable cooperations, unstable co*
*mpletions
1
(the coalgebraic module) E*(E_*) for an arbitrary Landweber exact spec
trum E, extending work of the second author with M. Hopkins [15] and
with P. Turner [20] and giving basisfree descriptions of the modules of
primitives and indecomposables. Taken together, these results enable us
to give a simple description of the E2page of the Etheory Bousfield
Kan spectral sequence when E is any Landweber exact ring spectrum
with unit. This extends work of the first author and others and gives a
tractable unstable Adams spectral sequence based on a vnperiodic theory
for all n.
Introduction
An unstable Adams spectral sequence computes homotopy theoretic
information for a space X from homological information. More specifi
cally, such a spectral sequence based on a homology theory E*() seeks,
under certain hypotheses, to compute the homotopy of an appropriate
Ecompletion of X from an Ext group (in a suitable category) involv
ing E*(X). This paper identifies, for E a general ring spectrum with
unit, an unstable Ecompletion X^Eof X and an associated Etheory
BousfieldKan spectral sequence with E2term the homology of a cer
tain unstable cobar complex. When E is an arbitrary Landweber exact
spectrum [22] we obtain a more tractable description of the E2term,
and, when E additionally has the structure of an Salgebra in the sense
of [13], our completion X^Eand spectral sequence agree with those of [7].
In order to obtain the description of the E2term we prove a number of
results on the generalised homology of the spaces in the spectrum for
2
a Landweber exact theory E, that is, on the coalgebraic ring F*(E_*).
These results are of independent interest; see, for example, [14].
The first example of an unstable Adams spectral sequence based on a
theory E other than ordinary homology was that of the first author with
E. Curtis and H. Miller [5] which considered the case of a connective
theory E and concentrated in particular on the case of BP theory. This
provided a sequence that converged to the plocalisation of the unstable
homotopy of an odd dimensional sphere and identified the E2term as an
Ext group in a nonabelian category of unstable BP*(BP )coalgebras.
Using results of Wilson [25], the E2term was given a simpler, and
more computationally practical, interpretation as the homology of an
unstable cobar complex and which could be further considered [4] as the
homology of a certain subcomplex of the stable cobar complex. This
spectral sequence and subsequent variations were generalisations of that
of Bousfield and Kan [9] and we refer throughout to all these models as
BousfieldKan spectral sequences.
A more mysterious gadget however has been that of a BousfieldKan
spectral sequence based on a periodic theory E. Theories E one would
naturally wish to consider include complex Ktheory, the Johnson
Wilson theories E(n) and the Morava E and Ktheories; for techni
cal reasons one is probably going to make easiest headway with those
theories E which are also Landweber exact, as in the connective ex
ample BP successfully dealt with by [5] and subsequent papers. With
R. Thompson, the first author has developed a framework [7] to define
and study sequences based on periodic theories. The requirements on
E to set such a sequence up, to identify the E2term in a practical and
3
computable manner, and to prove convergence to an identifiable object
are however significant. In brief, convergence is proved, in appropriate
cases, to a certain `unstable Ecompletion' of the underlying space X,
where this completion is defined as Tot of a certain cosimplicial space,
and is defined only when E is represented by an Salgebra in the sense
of [13]. Of the example theories E listed above, to present knowledge
this rules out all but complex Ktheory and the Morava Etheories.
The understanding of the E2page of any of these BousfieldKan
spectral sequences involves in large part having good and well under
stood structure in the un stable Etheory cooperation algebras, that
is, in the coalgebraic ring [19] or Hopf ring [24] E*(E_*) where the E_r
denote the spaces in the spectrum for Etheory. In [7] the E2term
was identified in a practical manner under the hypotheses that each
E*(E_r) was free as an E*module and that the submodule of primitives
P E*(E_r) inject under infinite stabilisation in the stable cooperation ring
E*(E). Work of the second author and M. Hopkins [15] showed that
these hypotheses were satisfied for a Landweber exact theory E whose
coefficients were `not too large': this incuded the cases of Ktheory and
the JohnsonWilson theories E(n), but not the Salgebra examples of
the Morava Etheories. Between these two sets of requirements on E 
for convergence and for the computation of the E2term  fully satisfac
tory results in [7] were obtainable only when E was taken as complex
Ktheory.
The main results of this paper fall into three sections. In section 1 we
study in depth the homology and generalised homology of the spaces E_r
in the spectrum for an arbitrary Landweber exact theory E, assuming
4
only that the coefficients E* are concentrated in even dimensions (an
assumption which fails only in rather artifical examples). These results
extend those of [15], removing the size restrictions on the coefficients,
proving for example (Theorem 1.4) that the algebras F*(E_r) for a wide
class of homology theories F*() are polynomial or exterior for r even
or odd respectively. However, they go further than the type of results in
[15], giving also basisfree descriptions (Theorem 1.10) of the modules of
primitives and indecomposables associated to the F*(E_r). Unusually for
results on the coalgebraic ring F*(E_*) for theories F and E, these results
give explicit descriptions of the individual Hopf algebras F*(E_r), rather
than just implicit descriptions in terms of the global object F*(E_*). We
also relate (Corollary 1.11) the modules P F*(E_r) and QF*(E_r) to the
primitives and indecomposables of the universal example MU*(MU__*).
These results are of independent interest in the study of the homology
of spectra, having applications, for example, to group cohomology
[14] and [18]. In the case of certain completed spectra, such as the
Morava Etheories and the BakerWürgler completions [E(n)[3], these
results have parallels with those of [16] where the homological effects of
completion on spectra are examined using rather different methods.
In section 2 we suppose E merely to be a ring spectrum with a unit.
For a space X we define a notion (Definition 2.2) of Ecompletion of
X, denoted X^E. If E is an Salgebra then the space X^E turns out
to be homotopy equivalent to the Ecompletion of X as defined in [7].
The Etheory BousfieldKan spectral sequence related to the homotopy
groups of this space X^Eis introduced and we identify (Theorem 2.8)
the E2page as the homology of an unstable cobar complex.
5
The results of section 2 are very general but as they stand offer
small hope for specific computation. In section 3 we build on them in
the special case of a Landweber exact ring spectrum (with unit), using
the work of section 1 on the coalgebraic ring for such a spectrum. The
main result here is a `change of rings' theorem that identifies (Theo
rem 3.1) the E2page of the Etheory BousfieldKan spectral sequence
of section 2 as an Ext group in a convenient, moreover abelian, cate
gory. This applies to spaces X such as torsion free Hspaces and odd
dimensional spheres. We note also (Remark 3.11) that a similar result
holds for spaces such as S2n+1, though care is needed for such exam
ples as, by the work of section 1, the relevant Hopf algebras E*(E_2r)
in the computation are not primitively generated. Taken together, the
results of this article allow for the construction and description of an
unstable Adams spectral sequence based on a vnperiodic theory for any
positive integer n, extending the framework of [7] which established the
v1periodic case.
Notation The convention we use for denoting spaces, spectra, etc. re
lated to a theory E is as follows. For a theory E we write E*() and
E*() for the generalised Ehomology and cohomology, E for the asso
ciated spectrum when we wish to consider it as an explicit object in the
stable category, and E_rand E_*for the spaces in the spectrum and for
the spectrum itself. Thus the space E_rrepresents the cohomological
functor Er() in the sense that Er(X) = [X, E_r] for any space X. The
E_rare related by equivalences E_r+1' E_r.
6
1 The coalgebraic module F*(E_ *)
Throughout this section we shall assume that E_* is an spectrum
representing a Landweber exact cohomology theory [22]. Such theo
ries include the examples of complex cobordism MU and the Brown
Peterson theories BP [1], the JohnsonWilson theories E(n) [21] and
their Inadic completions [E(n)[3] as well as Morava Etheory, complex
Ktheory, various forms of elliptic cohomology [23] and their comple
tions. For simplicity in the statement of our results, we assume the
coefficients E* are concentrated in even degrees; this is satisfied by all
standard examples including all those just mentioned. As E is neces
sarily a module spectrum over MU , the mod p homology H*(E_*; Fp)
will be a coalgebraic module over both H*(MU__*; Fp) and Fp[MU*] in
the sense of [19]; if, as will in fact generally be the case, E is a ring
spectrum, H*(E_*; Fp) will be a coalgebraic ring (Hopf ring) and a coal
gebraic algebra over these objects as well. We assume the notation and
results on H*(MU__*; Fp) to be found in [24] and the notions of coal
gebraic algebra as in [19, 24]. If E is a plocal spectrum then similar
statements hold on replacement of MU by BP .
__
The work of [19] establishes in particular a tensor product in
the category of Fp[MU*] coalgebraic modules; note that this is quite
distinct from the tensor product of the underlying Fp coalgebras. The
main theorem of [20] tells us
__ *
Theorem 1.1 H*(E_*; Fp) ~=H*(MU__*; Fp) Fp[E ].
Fp[MU*]
Corollary 1.2 H*(E_2r+1; Fp) is an exterior algebra.
7
Proof Consider the indecomposable quotient QH*(E_2r+1; Fp). Un
__
winding the definition of , elements in this quotient are represented
__
by sums of O products of elements of the form q x where q represents
an indecomposable in an odd MU space and x 2 Fp[E*] = H0(E_*; Fp)
(this follows from the fact that E* is concentrated in even dimensions).
As QH*(MU__s; Fp) lies in odd homological dimensions if s is odd, [24],
we conclude that QH*(E_2r+1; Fp) lies in odd homological dimension.
Thus any finite dimensional subHopf algebra of H*(E_2r+1; Fp) lies
in a finite dimensional subHopf algebra generated by odd dimensional
elements, and so is an exterior algebra. As H*(E_2r+1; Fp) is the colimit
of its finite dimensional subalgebras, the result follows.
Corollary 1.3 H*(E_2r; Fp) is a polynomial algebra and homology sus
pension induces an isomorphism QH*(E_2r; Fp) ~=QH*(E_2r+1; Fp).
Proof This is an immediate consequence of Corollary 1.2 and the ho
mology EilenbergMoore spectral sequence [12]
Cotor H*(E_2r+1;Fp)(Fp, Fp) =) H*(E_2r; Fp).
As H*(E_2r+1; Fp) is exterior, the E2page is already polynomial and
is concentrated in even degrees. The sequence thus collapses and the
result follows.
We require knowledge of F*(E_*) for more general theories F than
just mod p homology; for example we need results with F = E for the
unstable homotopy spectral sequences later, but other examples are of
importance too. For the remainder of this article we shall assume that F
is a plocal ring spectrum, with coefficients torsion free and concentrated
8
in even dimensions. We shall have occasion also to consider a version of
a theory F with coefficients reduced mod p; such homology of a space
X will be denoted F*(X; Fp).
Theorem 1.4 Suppose E_*and F are as above. Then F*(E_s) is a free
F* module, with algebra structure polynomial for s even and exterior for
s odd.
Proof Begin with the case F = HZ(p). As H*(E_2r; Fp) is polynomial,
and in even dimensions, its generators lift to polynomial generators of
the torsion free algebra H*(E_2r; Z(p)). From this we can deduce that
the homology of the odd spaces H*(E_2r+1; Z(p)) are torsion free, exterior
algebras, generated by the suspensions of generators of H*(E_2r; Z(p)).
The result for general F follows by a collapsing AtiyahHirzebruch spec
tral sequence argument.
Corollary 1.5 For E_* and F as above, F*(E_*) is a F*(MU__*) coal
gebraic module; if E is a ring spectrum, F*(E_*) is also a coalgebraic
ring.
Proof This result is a standard and formal argument (see, for example,
[24]) and follows as soon as a Künneth theorem
F*(E_rx E_s) ~=F*(E_r) F*(E_s)
F*
is established. This holds by the freeness result of (1.4).
For E a ring spectrum, recall the algebraic model coalgebraic rings
F*R(E_*) and F*Q(E_*) constructed in [24] and [17] respectively. The for
mer, F*R(E_*), is the free F*[E*] coalgebraic algebra generated by certain
9
classes arising from the complex orientation on E, modulo specific re
lations arising from the interaction of the E and F formal group laws;
the latter, F*Q(E_*), can be constructed as a certain subcoalgebraic ring
of the rational object F Q*(EQ_*). There are natural maps
F*(E_*) fi F*R(E_*) ! F*Q(E_*).
Corollary 1.6 For E and F as above, there are isomorphisms of coal
gebraic rings
F*(E_*) ~=F*R(E_*) ~=F*Q(E_*).
Proof That ø is an isomorphism follows from [20] and the correspond
ing result for MU, [24]. This, together with the fact that F*(E_*) is
torsion free by Theorem 1.4, gives the second isomorphism using [17]
corollary 6.3.
We seek now to describe the modules of primitives P F*(E_s) and
indecomposables QF*(E_s) for the Hopf algebras F*(E_s). One of the
main ideas of [17] is that when F*(E_*) is torsion free (as here), simple
descriptions of its algebra structure can be obtained by identifying its
image in the rational coalgebraic ring F Q*(EQ_*). The following result
allows analagous descriptions of P F*(E_s) and QF*(E_s) by embedding
these modules in the stable object F*(E).
Proposition 1.7 For E and F as above, homology suspension induces
monomorphisms
P F*(E_s) ! F*s(E)
QF*(E_s)  ! F*s(E).
10
Proof The proof is essentially given by the following commutative di
agram
P F*(E_s) '! QF*(E_s)? ffs! F*s(E)?
?? ?
y Qæ* ?yæ*
QF Q*(E_s) ffs!F Q*s(E).
The map ' is the natural map from primitives to indecomposables; as
F*(E_*) is torsion free, by (1.4), this is an inclusion. Infinite homology
suspension from the sth space is denoted by oes and æ indicates the
rationalisation map F ! F Q. Then Theorem 1.4 tells us that the left
hand vertical map Qæ* is a monomorphism, and the analysis of rational
coalgebraic rings in [17] shows that the suspension oes: QF Q*(E_s) !
F Q*s(E) is also monic.
This result allows us to give a basis free description of P F*(E_s) and
QF*(E_s) as a submodule of the stable module F*(E). This generalizes
the construction of [4], definition 2.13. First though we need to recall
some standard notation for elements in coalgebraic rings; see [24] for
further details.
Recall there are classes bt 2 H2t(MU__2). When localized at a prime
p (as here) it is customary to denote the class bpsby b(s)2 H2ps(MU__2).
We shall use this notation throughout, reserving the names bt (without
brackets) for elements of the stable module F*(E), as below. There is
also the suspension element e 2 H1(MU__1) with the relation eOe = b(0).
For v a homogeneous element of MU*, say v 2 MUv= ß0(MU__v),
we have the element [v] 2 H0(MU__v), its Hurewitz image; note that
v 2 MUv= MUvand v > 0. By [24], H*(MU__*; Z(p)) is generated
as a coalgebraic ring by the classes [v], e and b(s). The algebraic models
11
F*R(E_*) are by definition generated by the analogous elements [v], e
and b(s)with the v homogenous elements of E*, and by Corollary 1.6
we know the corresponding classes also generate F*(E_*).
The suspension homomorphisms, oes: F*(E_s) ! F*s(E) send b(i)
to bi2 F2pi2(E), kill * products and take O products to multiplication
in F*(E); note in particular that oe2: b(0)7! 1. For v a homogeneous
element of E*, we denote also by v the image of [v] under suspension
in Fv(E); this is additionally the image of v in F*(E) under the right
unit F* ! F*(E).
We shall denote the free E* module generated by a class 'n in dimen
sion n by Mn, or, when we need to indicate the spectrum considered,
by MEn. This is useful for keeping track of the domain of the stabiliza
tion map, oes: F*(E_s) ! F*s(E) and is accomplished by defining the
range of oes to be F*s(E) Ms. In this notation b(i)2 F2pi(E_2) maps
E*
to bi '2 while a class such as b(i)O[v] 2 F2pi(E_2v) maps to bi v'2v.
Notice that the suspension homomorphisms now preserve dimension.
For each finite sequence of nonnegative integers I = (i1, i2, . .,.in)
we write bI for the stable element
bI = bi11bi22. .b.inn2 F*(E).
The length of I is the integer l(I) = i1 + . .+.in. Write bOI for the
unstable element bOi1(1)O . .O.bOin(n)2 F*(E_2l(I)). Of course oe2l(I):bOI 7! b*
*I.
More generally, any element of the form (bOr(0)O bOI+ decomposables)
suspends to bI.
Definition 1.8 Let M be a free, graded E*module. Write UF (M) for
the subF*module of F*(E) M spanned by all elements of the form
E*
12
bI m where 2l(I) < m.
Definition 1.9 Let M be a free, graded E*module. Write VF (M) for
the subF*module of F*(E) M spanned by all elements of the form
E*
bI m where 2l(I) 6 m.
The special case of the next result for E = F = BP was proved in
[4, 5].
Theorem 1.10 The image of the suspension homomorphism, oes: QF*(E_s) !
F*s(E) Ms ~=F*(E) lies in V (Ms) and
E*
oes: QF*(E_s) ! V (Ms)
is an isomorphism.
Furthermore the image of oesPF*(E_s)lies in U(Ms) and
oes: P F*(E_s) ! U(Ms)
is an isomorphism.
Proof We start with the identification of the image of the indecompos
ables QF*(E_s) with VF (Ms). As oes on QF*(E_s) is monomorphic this
will prove the first statement. We begin with the case where s is even.
By Corollary 1.6 we know that any element of QF*(E_s) can be
written as a F*linear sum of elements of the form bOr(0)O bOIO [v] with
r > 0. Such an element suspends to bI v's. The condition that an
element bOr(0)O bOIO [v] lies in the F homology of the sthspace E_sis that
2r + 2l(I)  v = s, thus 2l(I) 6 v + s = v's and so the image of oes
lies in VF (Ms).
13
Conversely, if bI v's lies in VF (Ms) then 2l(I)  v 6 s and so
bI v's = oes(bOr(0)O bOIO [v]) where 2r = s + v  2l(I) > 0. Hence oes is
onto VF (Ms) and the isomorphism for even spaces is shown.
The result for odd spaces is very similar; note that circle multiplica
tion by e induces a one to one correspondence between QF*(E_2t) and
QF*(E_2t+1).
The result for primitives is again similar and follows immediately
after making the observation that P F*(E_s) for even s is the F*linear
span of elements of the form bOr(0)O bOIO [v] with r > 0. Also, for odd s
there is an isomorphism P F*(E_s) ~=QF*(E_s).
The F*module QF*(E_s) is not as it stands an E* module, but may
be modified to be so. Looking at all the spaces together, the bigraded
object QF*(E_*) is an E* module under the action x v 7! x O [v] for
x 2 QF*(E_*) and v 2 E*. (Verification that x O [v + w] = x O [v] + x O [w]
in QF*(E_*) is left as an exercise in coalgebraic modules: see the axioms
listed in [24].)
We may modify the construction of VF (Ms) so as also to carry the
action of E* by considering the corresponding bigraded object VF (M*)
equipped with the action (y v's) w 7! y vw'sw. Define a
global suspension map oe :QF*(E_*) ! VF (M*) as oes on the compo
nent QF*(E_s). With these definitions and the previous result it may
easily be checked that oe is a F*E* bimodule isomorphism. Similar
constructions may be made and results established for the objects of
primitives P F*(E_*) and UF (M*).
Our second description of the modules of primitives and indecom
posables for F*(E_s) can now be given in terms of a simple relation to
14
those of the universal theories. As in the underlying philosophy of [24],
etc., this requires us to consider all spaces E_stogether.
Corollary 1.11 Let E and F be as above. Then there are isomor
phisms
QF*(E_*) = F* QMU*(MU__*) E* = QF*(MU__*) E*
MU* MU* MU*
P F*(E_*)= F* P MU*(MU__*) E* = P F*(MU__*) E*
MU* MU* MU*
where means tensor product of modules in the standard sense. As
suming E is plocal, analgous results hold on replacing MU by BP .
Proof We prove the first line, concerning the indecomposable functor:
the proof of the version involving the primitives is essentially identical.
Note also that the equality
F* QMU*(MU__*) E* = QF*(MU__*) E*
MU* MU* MU*
follows immediately since each MU*(MU__s) is a free (left) MU* algebra
and hence F* QMU*(MU__*) = QF*(MU__*).
MU*
We show that QF*(E_*) = QF*(MU__*) E*. By Theorem 1.10
MU*
it suffices to show that VF (ME*) = VF (MMU*) E*. Of these, the
MU*
left hand side is the (bigraded) subF*module of F*(E) ME*spanned
E*
in grading s by elements bI m's satisfying 2l(I) m's. As E is
Landweber exact, and, by definition, ME* and MMU* are free over E*
and MU* respectively in each grading,
F*(E) ME*= F*(MU) E* ME*= F*(MU) MMU* E*.
E* MU* E* MU* MU*
15
Under this equivalence, the element bI m's 2 VF (ME*) F*(E) ME*
E*
is then identified with
bI 's m 2 VF (MMU*) E* F*(MU) MMU* E*.
MU* MU* MU*
This is also onto VF (MMU*) E* as the map MU ! E induces a left
MU*
inverse.
The constructions UF and VF may be extended to other E* modules
M. For an arbitrary nonnegatively graded left E*module M let
F1f!F0 ! M ! 0
be exact with F0 and F1 free over E*. Then UF may be extended to M
by defining
UF (M) = coker(UF (f): UF (F1) ! UF (F0)).
VF is similarly extended to such E*modules.
Proposition 1.12
VF (Ms Z=p) ~= QF*(E_s; Fp)
UF (Ms Z=p) ~= Im(P F*(E_s; Fp) ! QF*(E_s; Fp)).
Proof Since F*(E_s) is a free algebra, there is a diagram with rows short
exact
0 ! QF*(E_s) xp! QF*(E_s) ! QF*(E_s; Z=p) ! 0
k k
0 ! VF (Ms) xp! VF (Ms) ! VF (Ms Z=p) ! 0.
Hence there is an induced isomorphism QF*(E_s; Fp) ! VF (Ms Z=p).
A similar proof gives the second isomorphism.
16
Remark 1.13 Since in practice our cohomology theories tend to be
Z(p)local it can be advantageous to use BP generators. The generators
hi = c(ti), where the ti are the standard generators for BP*(BP ) and
c denotes the canonical antiisomorphism, have proven to be useful for
unstable calculations. Following ([5], 8.5) we may replace the generator
bi with hi in Theorem 1.10.
We conclude this section with an example to help clarify these defi
nitions.
Example 1.14 Let F = E = BP . We claim that ph1 '1 defines
a nonzero element in VBP (M1 Z=p) = QBP*(BP_1; Fp), but which
suspends to zero in QBP*(BP_2; Fp). To see that ph1 '1 is indeed
in VBP (M1 Z=p), note that the right action formula tells us that
ph1 = v1 . 1  1 . v1. Thus
ph1 '1 = v1 '1  1 v1 . '1,
an element of VBP (M1). This element is not divisible by p in VBP (M1),
and so is not zero in VBP (M1 Z=p). On the other hand, h1 '2 is an
element of VBP (M1) and so ph1 '2 is pdivisible in VBP (M1) and thus
is zero in VBP (M1 Z=p). In general, VF (Ms Z=p) is not a submodule
of F*(E): when working mod p the unstable classes do not necessarily
inject into the stable module.
In a similar fashion the right action formula for ph1 can be used to
show that phn1is a nonzero element in VBP (M2n1 Z=p) but which
suspends to zero in VBP (M2n Z=p).
17
Example 1.15 Consider the Araki generators wi2 BP2pi2, as in [2],
X pi
pmn = mi(wni); w0 = p.
06j6n
We prefer the Araki generators to the Hazewinkel generators because
of the integral form of Ravenel's formulæ
F*X i F*X i
hpj. wi= wpj. hi.
Here F c(fli) is the formal group sum, c is the canonical antiisomorph
P F*
ism and F*fli = c( F c(fli)). (i.e., looks like the usual formal
group law, but the formal group coefficients act on the right). It is easy
i * pi
to check that the Ravenel formulæ imply that F*hpj. wi= F wj . hi
hold also in VF (MEs) and VF (MEs Z=p) with s > 2. There are similar
formulæ involving the Hazewinkel generators, but they are only true
stably mod p. We do not know if the Hazewinkel generators satisfy
similar, mod p formulæ, unstably.
If E = E(1), Adams' summand of plocal Ktheory, or equivalently
the first JohnsonWilson theory and we take F to be H, integral ho
mology, these formulæ reduce to
_ F* !
X p
hj . w1 's = 0 inVH*(Ms Z=p) ifs > 2.
Using the grading, this implies that hpj. w1 's = 0 here. (Notice that
hpj.w1 '2 = hpj w1.'2 and w1.'2 has degree 2p so this class is defined.)
18
2 The BKSS for Etheory
Let S be the category of pointed CW complexes and suppose E is a
ring spectrum with unit. Associated to E is a functor TE :S ! S given
by sending X to 1 (E ^ 1 X). There are natural transformations
OE: 1S ! TE and ~: TE2= TE O TE ! TE induced by the unit and the
multiplication in E respectively and these make (TE , OE, ~) a triple up
to homotopy. See, for example, [6, x2], [7, x4] and [8] for details of
the notions of triple, cotriple, their associated categories and derived
functors, as used in this and the next section.
If E is an Salgebra in the sense of [13] (for example, Ktheory),
it is shown in [7] that (TE , OE, ~) is in fact a triple on the category S.
Following [10] there is then a cosimplicial space, TE X, with coface maps
and codegeneracies denoted di and sj respectively. The completion of
X with respect to E is taken as
X^E= Tot(TE X).
The E2page of the BousfieldKan spectral sequence associated to X^E
is identified [7] with the homology of the unstable cobar complex,
Es2(X) = ßsß*TE X = Hs(ß*TE X, @),
where ß*TE X is considered as a cochain complex with coboundary map
@ = (1)iß*di.
We wish however to be able to consider an `Ecompletion' of a space
X and a corresponding Etheory BousfieldKan spectral sequence when
ever E is an arbitrary ring spectrum with a unit. In this section we use
the results of [11] to construct (2.2) a space X^Efor any such E, and
19
prove it to be homotopic to the construction in [7] if E is an Salgebra.
In Theorem 2.8 we identify the E2term of the Etheory BousfieldKan
spectral sequence as an unstable cobar complex.
We recall the notion [11] of a restricted cosimplicial space, i.e., a
öc simplicial space" without the codegeneracies.
Definition 2.1 Suppose (T, OE) is an augmented functor on S, i.e., a
functor T :S ! S equipped with a natural transformation OE: 1S ! T .
Let X be a space in S. Define the restricted cosimplicial space bTX to
be the restricted cosimplicial resolution with respect to T given by
(bTX)k = T k+1X
in codimension k, and coface maps given by
i k k TiffiTkik+1
((bTX)k1d! (bTX) ) = (T X ! T X).
We may describe a restricted cosimplicial space as a diagram in
S as follows. Let restdenote the restricted simplicial category, that
is the category whose objects are finite ordered sets [n] = {0, 1, . .,.n}
(n > 0) and whose morphisms are strictly monotone maps. A restricted,
unaugmented, cosimplicial space, Crestis equivalent to a functor
Crest: rest! S.
In particular bTEX 2 S rest.
The full simplicial category, , is the category whose objects are the
sets [n] and whose morphisms are all weakly monotone maps. Then a
cosimplicial space is a functor
C: ! S.
20
So C 2 S .
Let J : rest! be the inclusion functor. Then there is a natural
transformation
J* :S ! S rest,
essentially the forgetful functor from cosimplical spaces to restricted
cosimplical spaces.
Definition 2.2 For a general ring spectrum with unit E, define X^E,
the Ecompletion of X, to be holimbTEX.
Strictly speaking, this definition only requires E to have a unit. How
ever, we shall need E to have a ring structure directly after the next
definition, which introduces an object lying between a cosimplicial space
and a restricted cosimplicial space.
Definition 2.3 A modified cosimplical space is a restricted cosimpli
cial space with codegeneracies that satisfy cosimpliciallike identities
djdi = didj1 i < j
sjdi ' disj1 i < j
' id i = j, j + 1
' di1sj i > j + 1
sjsi ' si1sj i > j
where the first identity is the usual cosimplicial identity, but the rest are
required to hold only up to homotopy.
Remark 2.4 If E is a ring spectrum with unit, then, for X 2 S, the
triple (TE , OE, ~) induces a modified cosimplicial space which we also
21
denote by TE X. Clearly any cosimplicial space C is also a modified
cosimplicial space and so if X is an Salgebra the two objects denoted
TE X agree.
Remark 2.5 Corollary 3.9 of [11] proves that Tot(C) = holim(Crest)
when C = J*Crest. In particular, if E is an Salgebra, the completion
X^Edefined in [7] agrees with that of Definition 2.2.
Remark 2.6 It is not possible to apply Tot to modified cosimplicial
spaces. However, after applying ß* we obtain a cosimplicial group
ß*TE X which we view as a diagram ß*TE X 2 A , where A is the
category of abelian groups. Applying ß* to bTEX gives an object in
A restwhich is J*(ß*TE X).
For a wide class of diagrams X_2 SI Bousfield and Kan [10], XI 7.1,
define a spectral sequence related to the groups ß*holimX_.
Definition 2.7 For X 2 S and E a ring spectrum with unit, define
E*,*r(X), the Etheory BousfieldKan spectral sequence of X, as the
BousfieldKan spectral sequence for bTEX 2 S rest.
Theorem 2.8 Es,*2(X) is isomorphic to the homology of the unstable
cobar complex. That is to say Es,*2(X) = ßsß*TE X
Remark 2.9 Recall the cohomotopy ßsA_of a cosimplicial abelian group
A_is defined [10], X 7.1, as the cohomology Hs(ch(A_), @) where (ch(A_), @)
P
is the cochain complex given by ch(A_)n = A_nand @ = (1)idi.
Proof of (2.8) Let I be either or rest. For X_2 SI the E2page is
given by
Es,t2= limsßtX_
22
([10] page 309). Since ß*TE X is a cosimplicial group, limsß*TE X =
ßsß*TE X ([10], XI 7.3 (i)) and it suffices to show that
limsß*bTEX = limsß*TE X.
For any fixed n, denote by K_I(n) 2 SI the diagrams of Eilenberg
Mac Lane spaces K(A, n) which correspond to ß*TE X 2 A and
ß*bTEX 2 A restfor the respective I (see [10], XI 7.2). Then for s 6 n
(again from [10], XI 7.2)
limsß*TE X = ßns holimK_(n)
limsß*bTEX = ßns holimK_ rest(n)
However, J : rest! is left cofinal ([11] page 193). Thus
J* :holimK_ (n) ! holimK_ rest(n)
is a homotopy equivalence. Since n was arbitrary, it follows that limsß*TE X =
limsß*bTEX for all s.
3 The Unstable Cobar Complex for E
theory
Section 2 identifies the E2page of the BousfieldKan spectral sequence
for a ring spectrum with unit E as the homology of the cochain complex
ch(ß*TE X). However, for practical purposes, as in [5, 7], etc., it is
important to be able to reinterpret this in terms of a more manageable
object, in practice as the homology of a subcomplex of the stable cobar
23
complex, i.e., as an Ext group over a more convenient (in particular,
abelian) category.
We suppose for this section that E is a Landweber exact ring spec
trum with unit and (largely for convenience) that E is plocal with
coefficients E* concentrated in even dimensions. Let M be the cate
gory of free, graded E*modules. Drawing on the results of [5, 6, 7]
and those of sections 1 and 2, we introduce a certain associated abelian
category U. Our main theorem is the following.
Theorem 3.1 Suppose E is a Landweber exact ring spectrum with unit.
Suppose M 2 M has E*module generators only in odd degrees and
suppose X is a space with E*(X) ~= (M) as coalgebras, where (M) is
the E*Hopf algebra defined by letting M be the submodule of primitives,
i.e., (M) is the exterior algebra on M. Then the E2term of the E
theory BousfieldKan spectral sequence of X can be identified as
Es,t2(X) ~=ExtsU(E*(St), M).
Example 3.2 Spaces X satisfying the hypotheses of the theorem in
clude torsion free Hspaces and odd dimensional spheres.
We begin by defining functors G and U :M ! M. Here and below we
draw on a number of the results of section 1 with F = E, i.e., in this
section we deal only with the coalgebraic ring E*(E_*).
Definition 3.3 For a free E*module M define
(a) G(M) to be E*(EM__0), where EM denotes the spectrum realizing
the homology theory E*() M.
E*
(b) U(M) to be P G(M), the primitive elements in G(M).
24
Both G and U are functorial; they take values in M, the category of
free E* modules, by the results of section 2.
Remark 3.4 (a) As M is a free E*module, it is helpful to observe that
EM__0= 1 EM is a product of spaces in the spectrum associated to
E indexed by a set of generators of M. In particular, if {gi} are a set
of E* generators of M with gi in dimension gi,
_ !
` Y
EM__0= 1 giE = E_gi.
i i
W Q
Moreover, with this notation, M ~=ß* i giE = ß* iE_gi.
(b) Note that G is closely related to the functor TE :S ! S of
section 2. For a space X 2 S with E*(X) 2 M, there is an isomorphism
G(E*(X)) ~=E*(TE (X)).
(c) Note also that U(M) is identical to the construction UE (M) of
section 1. There is of course a similar functor V :M ! M based on the
indecomposable quotient of G(M) and given by the construction VE (M)
of section 1, but it will play no part in the proof of Theorem 3.1.
Proposition 3.5 The unit and product in E respectively induce natural
transformations
ffiG :G ! G2 fflG :G ! I
making (G, ffiG , fflG ) a cotriple on the category M. There are similar
natural transformations ffiU :U ! U2, fflU :U ! I making (U, ffiU , fflU )
also a cotriple on M and a subcotriple of (G, ffiG , fflG ).
25
Proof The proof is essentially as in sections 6 and 7 of [5]; moreover,
with the first observations of Remark 3.4 the maps ffiG and fflG , for
example, may be written explicitly. Alternatively, for Landweber exact
E, given the definition (1.8) and Theorem 1.10, the result on (U, ffiU , fflU )
also follows from the coaction formulæ for the bi.
Remark 3.6 As usual the cotriples define categories G and U of G,
respectively U, coalgebras: writing C for either G or U, recall that a C
coalgebra in M is an object M 2 M with a map _ :M ! CM such
that
fflC _ = IdM : M ! M and ffiC _ = (C_)_ :M ! C2M
(see [5, x5] for details).
In particular, recall that if M 2 M then CM is naturally a C
coalgebra with map _ on CM ! C2M given by ffiC . There are adjoint
functors
C!
M C

J
where J denotes the forgetful functor. The adjunction gives natural
isomorphisms
Hom C(D, CM) ~=Hom M (D, M)
for any D 2 C (where we identify D with its image under the forgetful
functor).
Strictly speaking, we shall abuse notation and write C not only for
the functor M ! C above, but also for the functor JC :M ! M of
the cotriple (C, ffiC , fflC ) on M and for the other composite, CJ :C ! C,
the functor of the adjoint triple (C, ~C , jC ) on C, as in [5, x5].
26
For C = G or U and objects W 2 C we recall the notions of cosimpli
cial resolution over C, as in [8, 2.5] and [6, 2.2] and the resulting derived
functors ExtC(E*, W ).
Definition 3.7 A cosimplicial resolution, N, over C, of W 2 C consists
of objects Nn 2 C for n > 1 and, for every pair of integers (i, n) with
0 6 i 6 n, coface and codegeneracy maps (in C)
di:Nn1 ! Nn , si:Nn+1 ! Nn
satisfying the usual cosimplicial identities (cf. 2.3) and such that
(a)N1 = W ;
(b)for n > 0 there is an Mn 2 M with Nn = CMn;
(c)Hn(JN) = 0 for n > 1.
Here J :C ! M is the forgetful functor and the homology of JN is the
homology of the cochain complex with groups JNn and boundary maps
(1)iJdi.
The Ext groups ExtC(E*, W ) are then defined as the homology of
chain complex associated to Hom C(E*, gJN), where gJNdenotes the unaug
mented complex
0 ! JN0 ! JN1 ! JN2 ! . ...
These are the derived functors of Hom C(E*, ) by [8].
Example 3.8 The C cobar complex provides a standard example of a
cosimplicial resolution. We illustrate it for C = U; the case of G is
similar.
27
For W 2 U, consider the resolution with qth module Uq+1(W ). The
maps in the U resolution are displayed in the diagram of E*modules
d0!
0 d1
W d! U(W ) ! . . .
s0
and are defined in terms of the triple (U, ~U , jU ) by
di= UijU Uni : Un (W ) ! Un+1 (W ), 0 6 i 6 n,
si= Ui~U Uni : Un+2 (W ) ! Un+1 (W ), 0 6 i 6 n.
The U cobar complex is then the complex
W @! U(W ) @!U2(W ) @! . . .
P n
where @ = i=0(1)ndi:Un (W ) ! Un+1 (W ).
The embedding of the primitives in the stable cooperations, (1.7)
and (1.10), shows that the acyclicity condition is satisfied since there is
an extra codegeneracy s1 :Uq+1(W ) ! Uq(W ) induced by the counit
in E*(E):
Uq+1(C) ! E*(E) Uq(C)ffl!1Uq(C).
In particular, again by (1.10), ExtU (E*, W ) is the homology of a
subcomplex of the stable cobar complex.
These constructions and the link between the functors G and TE of
Remark 3.4(b) allow us to rewrite Theorem 2.8 as follows.
Theorem 3.9 For E a ring spectrum with unit and X 2 S such that
E*(X) 2 M, there is a natural isomorphism
Es,t2(X) = ExtsG(E*(St), E*(X)).
28
Theorem 3.1 will now follow upon proving
Theorem 3.10 Suppose E is a Landweber exact ring spectrum with
unit. For M 2 M with generators in odd degree and (M) denoting
the exterior algebra on M with M (M) the submodule of primitives,
there is a natural isomorphism
ExtsG(E*(St), (M)) ~=ExtsU(E*(St), M).
Proof Let us write UM for the U cobar complex as in Example 3.8, i.e.,
with qth space Uq+1(M). Applying the functor () gives a complex
UM : (M) ! (U(M)) ! (U2(M)) ! . ...
Now let
Y q= G(Uq(M))
for q > 0. Since M is concentrated in odd degrees the same is true for
Uq(M). By the theorems (1.4) and (1.10) we have natural isomorphisms
G(Uq(M)) ~= (Uq+1(M))
and we can identify the complex UM as a complex
Y : (M) ! G(M) ! G(U(M)) ! G(U2(M)) ! . ...
The maps in Y are coalgebra maps and E*(E)comodule maps. By [5,
7.3] the maps are in G (note that [5, 7.3] does not require the assumption
[5, 7.7] that the homology of the spaces in the spectrum be cofree
coalgebras  this is not satisfied in general). The extra codegeneracy
in the U cobar complex passes via to an extra codegeneracy in Y,
showing Y to be acyclic. Thus Y is a Gresolution of (M).
29
The Ext groups ExtsG(E*(St), (M)) can be obtained using the com
plex Y by computing the homology of the complex
Hom G(E*(St), Y s) = Hom G(E*(St), G(Us(M))).
However, by the adjunction isomorphism mentioned in Remark 3.6 (ap
plied twice), shows
Hom G(E*(St), G(Us(M))) = Hom M (E*(St), Us(M))
= Hom U(E*(St), Us+1(M)).
Thus ExtG(E*(St), (M)) is isomorphic to the homology of the Ucobar
complex which by definition is precisely ExtU(E*(St), M).
Remark 3.11 The results of section 1 on the algebra structure of
E*(E_*) allow further results to follow. For example, suppose for M 2 M
we write oe1M for the isomorphic E*module with degrees shifted
downward by one, i.e., we let oe1Mt = Mt+1. Then Theorem 1.4
and its proof shows
oe1U(M) = QG(oe1M) .
If we take M = E*(S2n+1) then E*( S2n+1) = oe1M and an argument
similar to that for BP theory in [6], x6, shows that the complex Y used
in the proof of Theorem 3.10 may also be used to compute the E2page
of the Etheory BousfieldKan spectral sequence for S2n+1: for any
odd dimensional sphere S2n+1 there is an isomorphism
Es,t12( S2n+1) ~=Es,t2(S2n+1) .
30
Acknowledgements Both authors are pleased to thank the JapanU.S.
Mathematics Institute (JAMI), Johns Hopkins University, and its or
ganisers J.M. Boardman, D. Davis, J.P. Meyer, J. Morava, G. Nishida,
W.S. Wilson, and N. Yagita for support during Spring 2000 and at
which this research was initiated. The first author thanks also Em
manuel DrorFarjoun for helpful discussions on the material in section 2.
The second author thanks the University of Leicester for sabbatical leave
and the Leverhulme Foundation for a Research Fellowship during which
most of this paper was written.
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