Homotopy Theory of Simplicial Abelian Hopf Algebras
Paul Goerss1 and James Turner
January 22, 1997
Abstract
We examine the homotopy theory of simplicial graded abelian Hopf a*
*lgebras
over a prime field Fp, p > 0, proving that two very different notions o*
*f weak equiv
alence yield the same homotopy category. We then prove a splitting resu*
*lt for the
Postnikov tower of such simplicial Hopf algebras. As an application, we*
* show how
to recover the homotopy groups of a simplicial Hopf algebra from its An*
*dreQuillen
homology, which, in turn, can be easily computed from the homotopy grou*
*ps of the
associated simplicial Dieudonne module.
This paper is divided into two parts. The first and larger part, is also t*
*he more
abstract; in it we undertake a thorough examination of the homotopy theory of s*
*implicial
graded abelian Hopf algebras over Fp, p > 0. The second part is a calculational*
* application,
relying heavily on the first part and intended partly to demonstrate that the h*
*omotopy
theory of simplicial Hopf algebras deserves consideration. In this second part*
* we show
that the homotopy groups of a simplicial abelian Hopf algebra support a very ri*
*ch and
rigid structure. This has implications for the cohomology spectral sequence of*
* a variety
of cosimplicial spaces. (See, for example, the work of the second author [20], *
*or Dwyer's
spectral sequence as explained in [2, x4].)
To explain our results in more detail, fix a prime p, and let HA be the cat*
*egory
of graded, bicommutative Hopf algebras A over Fp which are connected in the sen*
*se that
A0 ~=Fp. The objects in HA are the abelian objects in the category CA of graded*
* connected
coalgebras over Fp; hence, we call an object in HA an abelian Hopf algebra. Let*
* sHA be
the category of simplicial objects in HA.
If f : A ! B is a morphism in sHA there are two obvious ways to specify whe*
*n f
is a weak equivalence. On the one hand, if A 2 sHA, it is, among other things, *
*a graded
abelian group and, as such, has graded homotopy groups ss*A ~=H*(A; @), where
@ = (1)idi: An ! An1:
We could demand that f be a weak equivalence if ss*f is an isomorphism. On the *
*other
hand, HA is an abelian category with enough projectives, so sHA acquires a noti*
*on of weak
_________________________
1The first author was supported by the National Science Foundation
1
equivalence from [14, xII.4]; essentially, f : A ! B is a weak equivalence if t*
*he morphism
f, regarded as a map of chain complexes over HA, becomes an isomorphism in the *
*derived
category of HA. Another way to say the same thing is this: the category HA has *
*a set of
small projective generators and, hence, there is an equivalence of categories D*
** : HA ! D
to a category of modules called Dieudonne modules. (See [16]. Dieudonne theor*
*y is
summarized in section 3.) Then the second notion of weak equivalence is equiva*
*lent to
specifying that
ss*D*f : ss*D*A ! ss*D*B
be an isomorphism.
That these two notions of weak equivalence are very different is emphasized*
* by the
following example. Let K 2 HA and let B(K) 2 sHA be the bar construction on K.
Then
ss*B(K) ~=Tor K*(Fp; Fp);
but
ss*D*B(K) ~=D*K
concentrated in degree 1. Nonetheless, our first main result (Theorem 5.12) is*
* that the
two notions of weak equivalence are not so different after all.
Theorem A. Let f : A ! B be a morphism in sHA. Then ss*f : ss*A ! ss*B is an
isomorphism if and only if ss*D*f : ss*D*A ! ss*D*B is an isomorphism.
This would show, for example, that a map H ! K in HA is an isomorphism if a*
*nd
only if
Tor H*(Fp; Fp) ! Tor K*(Fp; Fp)
is an isomorphism. Remember that we are working with graded, connected Hopf alg*
*ebras.
We actually prove much more than Theorem A. We will show that sHA has two c*
*losed
model category structures with respectively, the two specified notions of weak *
*equivalence
and that, furthermore, the two resulting homotopy categories are equivalent. Th*
*e reader
will deduce that our methods are very model theoretic.
Our second main result is a decomposition theorem for Postnikov towers in s*
*HA. If
A 2 sHA, then A has a MoorePostnikov tower {A(n)}n0 . The details are spelled *
*out
in section 5, but one can obtain this tower by taking the MoorePostnikov tower*
* of the
associated Dieudonne module, regarded as a graded simplicial set, and then noti*
*cing that
defines a tower of Hopf algebras. Let F (n) be the fiber at stage n, defined by*
* the pullback
2
diagram of Hopf algebras
F (n) _______wA(n)
 
u u
Fp ______wA(n  1):
It is wellknown (and an observation due to Moore) that if A is a simplicial ab*
*elian group,
then A(n) is noncanonically the product of A(n  1) and F (n). This is essenti*
*ally because
the category of abelian groups has projective dimension 1. The category HA has *
*projective
dimension 2, so no such splitting occurs in HA. However, we have the following*
*. (See
Theorem 7.1.) Note that tensor product is the product in HA.
Theorem B. Let A 2 sHA and suppose ss0A ~=Fp. Then there is a weak equivalence *
*of
simplicial algebras between A(n) and F (n) A(n  1).
In other words, the Postnikov tower for such A splits as algebras, even tho*
*ugh it may
not split as Hopf algebras. The splitting is not canonical and depends on a cho*
*ice of null
homotopy of a kinvariant. These invariants are explained in section 6, and the*
* existence of
the nullhomotopy depends on Theorem A and some remarks on the homological alge*
*bra
of Dieudonne modules from section 4.
We claim that Theorem B has strong implications for the homotopy groups ss**
*A with
A 2 sHA. We now fix p = 2. Because A 2 sHA is a simplicial algebra, ss*A is a D*
*algebra
in the sense of [7] and [19]; in particular there are operations ffii: ssnA ! s*
*sn+iA, 2 i n,
doubling internal degrees and satisfying Cartan and Ademstyle relations. (See*
* section 8.)
Also A is a simplicial coalgebra, so ss*A is an unstable coalgebra over the Ste*
*enrod algebra.
Here the Steenrod algebra is expanded in the sense that Sq0 6= 1; indeed, the a*
*ction of
Sq0 on ss*A is induced by the Verschiebung on A. These two structures are not u*
*nrelated
and both interact with the Hopf algebra structure. In particular, ss*A is an un*
*stable Hopf
algebra over the Steenrod algebra. The details are spelled out in [19] and reca*
*pitulated in
section 8. The resulting object is a Hopf Dalgebra. For the investigation of s*
*s*A, we note
that we may assume that ss0A ~=Fp. For if A+ is the kernel, in sHA, of the natu*
*ral map
A ! ss0A, then ss0A+ ~=Fp and there is a natural isomorphism ss*A ~=ss*A+ ss0A.
The first result concerns the Dalgebra structure of ss*A. If is any Dalg*
*ebra, the
indecomposables Q are a module over the operations ffii above and we write F2 *
*Q for
the quotient of Q by these operations. The augmentation ideal functor on Dalge*
*bras to
bigraded vector spaces has a left adjoint SD ; we say B is free as a Dalgebra *
*if B ~=SD (V )
for some V . Note that in this case F2 QB ~=V . For the following result, see*
* Remark
8.11.3.
3
Theorem C. Let A 2 sHA and suppose ss0A ~=Fp. Then ss*A is a free Dalgebra and
there is a natural isomorphism
F2 Qss*A ~=HQ*A
where HQ*A is the AndreQuillen homology of A.
We do not claim that there is a natural isomorphism ss*A ~= SD (HQ*A). The*
* best
naturality statement we have is in Proposition 8.6. The AndreQuillen homology *
*of A is
that of [15, x4]. One interpretation of Theorem C is that Quillen's fundamental*
* spectral
sequence [15, x7] collapses. As an auxiliary to this result we claim that HQ*A *
*is easy to
compute, especially if one knows ss*D*A. See Section 4. To be fair, the stateme*
*nt about
AndreQuillen homology given in Theorem C is automatic, once one knows that ss**
*A is a
free Dalgebra. We included this statement to emphasize computability.
Theorem C is closely related to one of the main technical results of [12]. *
*(See Lemma
3.1 of that paper.) There the proof is algebraic; here we have understood the h*
*omotopy
theoretic basis for this result. Also, from Theorem C, it is possible to recov*
*er the main
calculational result of [12].
Let HD be the category of Hopf Dalgebras, If H 2 HD one can form the vector
space F2 QH. Although the Steenrod operations act on QH such is the nature of
the interaction with between these operations and the ffii that F2 QH only in*
*herits
an A(0) = F2[Sq0; Sq1]=(Sq1)2 action. Let L be the category of bigraded unstabl*
*e A(0)
modules. The functor
F2 Q(.) : HD ! L
has a right adjoint . Let A 2 sHA be so that ss*A ~= F2. Then Theorem C and t*
*he
arguments of [12] x3 imply that the natural map
ss*A ! (F2 Qss*A)
is an isomorphism. The functor can be easily understood in terms of functors *
*arising
in the theory of unstable algebras over the Steenrod algebra. In fact, the forg*
*etful functor
from unstable coalgebras to modules over Sq0 and Sq1 has a right adjoint . If *
*M is a
module over Sq0 and Sq1, (M) is an unstable Hopf algebra, with diagonal obtaine*
*d by
applying to the diagonal M ! M x M, and there is a natural isomorphism of unst*
*able
Hopf algebras
(M) ~=(M):
4
See [12], x1 for details. Finally, the isomorphism F2 Qss*A ~= HQ*A of Theor*
*em C
endows the AndreQuillen homology of A with the structure of an A(0) module. At*
* the
end of section 8 we give an intrinsic definition of this action.
The reader sensitive to generalization will wonder to what extent our hypot*
*heses are
necessary. First the analogs of Theorems C and D hold at odd primes; however, *
*for
homotopy theoretic applications, we would want to consider topologically commut*
*ative
algebras_that is xy = (1)x yyx. In the absence of an immediately compellin*
*g appli
cation we chose not to venture into the resulting notational swamp. Secondly, *
*one can
work with any perfect field, not just prime fields. However, some care must be *
*taken with
Dieudonne modules. See [16] or [5]. Finally, one could drop the internal gradin*
*g, at the
price of adding hypotheses. Our methods apply explicitly to "connected" Hopf a*
*lgebras
([18] or [5]); other types of Hopf algebras  group rings, for examples  would*
* have to be
treated differently. And Theorem A would not be true as stated here: one would *
*only have
that f : A ! B is a homotopy isomorphism if and only D*f : D*A ! D*B is a homot*
*opy
isomorphism after a suitable completion.
The second author would like to thank Matthew Ando for several useful conve*
*rsations
about Dieudonne modules and the University of Virginia for hospitality during t*
*he period
this work was carried out.
1. Preliminaries on closed model category structures.
Our main structure result for simplicial abelian Hopf algebras uses homotop*
*y theoretic
techniques in an integral way. The purpose of this section is to spell out the *
*necessary closed
model category structures on simplicial algebras and simplicial abelian Hopf al*
*gebras. We
assume familiarity with simplicial model categories as put forth in [14, Part I*
*I], [8], and
many other sources.
Fix a prime field Fp and let A be the category of Fpalgebras. Here and thr*
*oughout
this paper, the algebras (and coalgebras and Hopf algebras) we consider will be*
* graded
and connected  that is, isomorphic to Fp in degree zero.
The category sA of simplicial algebras over Fp is a simplicial category in *
*the sense of
Quillen [14, xII.2]; in particular, there is a mapping space functor to simplic*
*ial sets
map sA (.; .) : sAop x sA ! S:
For fixed A 2 sA, the functor B 7! map sA(A; B) has left adjoint K 7! A K, and*
* for
fixed B 2 sA, the functor A 7! map sA(A; B) from sA to Sop has right adjoint K *
*7! BK .
If A is a simplicial abelian group, let NA be the normalized chain complex *
*on A; thus
NAn = An=Im (s0) + . .+.Im(sn1)
5
n
X nP
and @ = (1)idi: NAn ! NAn1. Then ss*A = H*NA ~=H*(A; (1)idi).
i=1 i=1
Proposition 1.1 [14, xII.4]. The category sA acquires the structure of a simpli*
*cial model
category structure where a morphism f : A ! B is
1)a weak equivalence if ss*f : ss*A ! ss*B is an isomorphism;
2)a fibration if NAn ! NBn is onto for n 1; and
3)a cofibration if f has the left lifting property with respect to all triv*
*ial fibrations.
A morphism is trivial fibration if it is at once a weak equivalence and a f*
*ibration.
There is also a notion of trivial cofibration. A morphism f : A ! B has the le*
*ft lifting
property with respect to q : X ! Y if any diagram
A _____wX
f ""] q
u" u
B _____wY
can be completed so that both triangles commute. Of course, the map q will have*
* the right
lifting property with respect to the map f. The "simplicial" in simplicial mode*
*l category
means Quillen's Axiom SM7 is satisfied: if j : A ! B is a cofibration and q : X*
* ! Y is a
fibration, then the induced map in S
(1:2) map sA(B; X) ! map sA(B; Y ) xmapsA (A;Ym)apsA(A; X)
is a fibration. It is a trivial fibration if j or q is a weak equivalence.
To clarify cofibrations somewhat, we introduce the notion of a saturated cl*
*ass of mor
phisms. This is a class of morphisms in a category C that is closed under isomo*
*rphisms,
coproducts, retracts, cobase change, and sequential colimits. The last two cond*
*itions mean
this: To be closed under cobase change means that if j : A ! B is the class and
A _____wX
j i
u u
B _____wY
is a pushout diagram, then i is in the class. To be closed under sequential c*
*olimits
means that if K is an ordinal number regarded as a category with one morphism s*
* ! t
with s < t < K, and X : K ! C is a functor with colims 0 and E0;*2= ss*(X1 A X2). Thus we have the spectral sequen*
*ce.
It is exactly this method which yields spectral sequence of 1.8, so a spectral *
*sequence
comparison argument implies Y is weakly equivalent to a homotopy pushout.
A specialization of these ideas is the notion of a cofibration sequence. Th*
*is happens
when X1 ~=Fp. Let us write X for X2. Then A ! X ! Y is a cofibration sequenc*
*e if
10
A ! X is a cofibration and Y ~=FpA X. The homotopy cofiber is the homotopy push*
*out
of F2 A ! X. There is a spectral sequence
Tor ss*A(Fp; ss*X) ) ss*Z
if Z is the homotopy cofiber. Again, if A ! X is a morphism of Hopf algebras, *
*it is,
irrelevant, up to weak equivalence, whether we form the homotopy cofiber in sA *
*or sHA.
In any case, if A ! X is a levelwise inclusion of simplicial Hopf algebras, th*
*en Fp A X
is, up to weak equivalence, the homotopy cofiber. This follows from Propositio*
*n 1.9. A
sequence A ! X ! Y will be called a homotopy cofiber sequence if there is a dia*
*gram in
the homotopy category
X _____wZ
=  '
u u
X _____wY
where Z is the homotopy cofiber. A homotopy cofiber sequence A ! X ! Y in sA wi*
*ll
be called split if X ! Y has a section in Ho (sA) and the resulting map A Y ! *
*X is a
weak equivalence.
Finally, there is suspension. If A 2 sA, the suspension A of A is the homo*
*topy
pushout of Fp A ! Fp. There is a spectral sequence
Tor ss*A(Fp; Fp) ) ss*A:
A convenient model for A, at least if A is cofibrant, is given by W A, where W *
*A =
Fp A W A and W A is the contractible simplicial algebra with
(W A)q = Aq Aq1 . . .A1 A0
and face and degeneracy maps given as in [9]. If A is a simplicial abelian Hopf*
* algebra,
then W A is a simplicial abelian Hopf algebra and there is a weak equivalence A*
* ' W A
regardless of whether A is cofibrant. Note that because HA is an abelian categ*
*ory, the
pushout diagram
A _____wW A
 
u u
Fp _____wW A
is also a pullback diagram, so A is a model for W A ' A, at least if W A was f*
*ibrant
and W A ! W A a fibration. We will show how to remove this "fibrancyfibration"*
* clause
in Section 5.
11
2. Dieudonne Theory
The category HA is an abelian category with a set of small projective gener*
*ators
and, as such, is equivalent to a category of modules. Schoeller's Theorem descr*
*ibes that
category. We recapitulate that result here and use it to elaborate the homotopy*
* theory of
sHA.
Let Fp be our fixed field of characteristic p and let Zp be the padic inte*
*gers. We
define the Dieudonne ring on Fp to be the ring R by quotient of the polynomial *
*ring
R = Zp[V; F ]=(V F  p):
Note that R may be graded by requiring deg(F ) = 1, deg(V ) = 1 and Zp = R0.
A graded Dieudonne module M is a positively graded abelian group with an ac*
*tion of
V and F
V : Mn ! Mn=p
F : Mn ! Mpn
so that V F (x) = px, F V (y) = py. Note that M becomes a graded Rmodule if w*
*e ask
that for x 2 Mn and b 2 Rm that
(2:1) deg(bx) = pm n
with the understanding that bx = 0 if deg(bx) is a fraction. This immediately *
*implies
ps+1Mn = 0 if n = psn0 and (n0; p) = 1.
Let D be the category of graded Dieudonne modules. The following is Schoel*
*ler's
result. See [16].
Proposition 2.2. There is an equivalence of categories
D* : HA ! D:
The functor D* is easily described. Write n = psn0 with (n0; p) = 1. Then t*
*he graded
ring Z[x0; x1; : :;:xs], deg(xi) = pin0, has a unique Hopf algebra structure so*
* that the
Witt polynomials
i pi1 i
wi= xp0+ px1 + . .+.p xi
are primitive. Let H(n) = Fp Z Z[x0; x1; : :;:xs] = Fp[x0; x1; : :;:xs] with t*
*his Hopf
algebra structure. Then for H 2 HA, D*H = {DnH} with
DnH = Hom HA (H(n); H):
12
The operators V and F are induced, respectively, by the inclusion H(n) ! H(pn) *
*and the
morphism H(pn) ! H(n) sending xi to xpi1. The Hopf algebras H(n), n 1, form a
system of small, projective generators for HA. Thus Proposition 2.2 follows onc*
*e Schoeller
calculates Dm H(n) for all m and n. Again, the following is in [16]. (Compare [*
*12], Lemma
6.7.)
Lemma 2.3. Let Zp[n], n 1, be the graded Zp module isomorphic to Zp concentrat*
*ed
in degree n. Then there is an isomorphism of Dieudonne modules
D*H(n) ~=R Zp Zp[n]:
The grading must be interpreted as in (2.1). Note that there are natural is*
*omorphisms
(2:4) Hom D(D*H(n); M) ~=Hom Zp(Zp[n]; M) ~=Mn
Thus D*H(n) is "free" projective in D, by which we mean this module represents *
*the exact
functor M 7! Mn.
There are also "free" injectives. The functor on D given by
M 7! Hom Zp(Mn; Z=p1 )
*
* h i
is exact and representable. To see this, fix n, and let Hom Zp(R; Z=p1 ) denote*
* the Z 1_p
graded Rmodule with
Hom Zp(R; Z=p1 )pin= Hom Zp(Ri; Z=p1 )
and if OE 2 Hom Zp(Ri; Zp1 ), then
(V OE)(x) = OE(F x) (F OE)(x) = OE(V x):
Let J(n) Hom Zp(R; Zp1 ) be the largest subR module which is a Dieudonne modu*
*le.
Thus J(n) is the submodule of homomorphisms OE : Ri! Z=p1 so that
V kOE = 0
if deg(V kOE) = pikn is a fraction. Standard adjoint functor arguments imply
(2:5) Hom D (M; J(n)) ~=Hom Zp(Mn; Z=p1 );
so J(n) is injective. If (n) is the (unique up to isomorphism) Hopf algebra so*
* that
D*(n) ~=J(n), then (n) is injective in HA.
Note that (2.4) and (2.5) supply explicit recipes for showing D (and hence *
*HA) has
enough projectives and enough injectives. Schoeller adds:
13
Proposition 2.6. The categories D and HA have projective and injective dimensio*
*n 2.
We actually give a proof of this fact in Corollary 3.6.
We now turn to the study of the homotopy theory of sD, which, by Schoeller'*
*s The
orem, is the equivalent to studying sHA. There are several possible ways to put*
* a closed
model category structure on sD_one supplied, for example, by combining Proposit*
*ion 1.5
with Schoeller's result. We shall use another also. Recall that the normalizati*
*on functor
N : sD ! ch*D
is an equivalence of categories between sD and nonnegatively graded chain comp*
*lexes over
D.
Proposition 2.7. The category sD, with its standard simplicial structure, becom*
*es a
simplicial model category where f : A ! B is
1)a weak equivalence if ss*f = H*Nf is an isomorphism;
2)a fibration if NAn ! NBn is surjective for n 1; and
3)a cofibration if An ! Bn is injective for all n and the cokernel of An ! *
*Bn is
projective in D for all n.
Proof: See [14, II.4]
Note that 3) could easily be rephrased to say NAn ! NBn is injective for al*
*l n, and
coker{NAn ! NBn} is projective in D for all n.
We note that a morphism f : A ! B in sD is a fibration if and only if the m*
*aps
induced in internal (i.e. not simplicial) degree n is a fibration of simplicial*
* sets; that is,
f : A ! B is a fibration if and only if
f* : map sD(D*H(n); A) ! map sD(D*H(n); B)
is a fibration of simplicial sets. See [14, II.4].
The category sD is also cofibrantly generated. The forgetful functor from D*
* to graded
abelian groups has a left adjoint
M 7! R Zp M
with grading interpreted as in 2.1. The cofibrations in sD are generated by mo*
*rphisms
1 f : R Zp A ! R Zp B where f : A ! B is a levelwise inclusion of simplicial *
*abelian
groups with NB free and finitely generated over Z. Similarly the trivial cofibr*
*ations are
generated by morphisms 1 g : R A ! R B with g : A ! B a levelwise inclusion*
* of
simplicial abelian groups, NB free and finitely generated over Z, and ss*g an i*
*somorphism.
We close this section with a result intended to help characterize cofibrati*
*ons in sHA.
14
Lemma 2.8. Let f : A ! B be a cofibration in sHA. Then D*f : D*A ! D*B is injec*
*tive
in sD and F operates injectively on the cokernel. In particular if B 2 sHA is c*
*ofibrant, F
operates injectively on D*B.
Proof: Notice that the class of injective morphisms M ! N in sD so that F opera*
*tes
injectively on the cokernel is saturated. Also notice that if g : S(C) ! S(D) i*
*s any of the
generators of the class of cofibrations in sHA, then D*g is injective and F act*
*s injectively
on the cokernel. This last is because the cokernel is isomorphic to S(C=D). T*
*hus since
the class of cofibrations is saturated, the result follows.
We conjecture that the converse of Lemma 2.8 is true, but we won't need tha*
*t result
here.
3. Homological algebra of Dieudonne modules
The other piece of information necessary for our splitting result is a vani*
*shing criterion
for certain Ext groups of Hopf algebras. This will be accomplished through the*
* medium
of Dieudonne modules.
We begin with indecomposables and primitives for Dieudonne modules.
Definition 3.1. Let M 2 D be a Dieudonne module. Then the module of indecomposa*
*bles
is defined by
QM = coker{F : M ! M}
and the module of primitives is defined by
P M = ker{V : M ! M}:
Note that QM inherits an action of V and P M inherits an action of F . Thes*
*e modules
are so named because if H 2 HA, then there is a natural isomorphism of Fp[V ] m*
*odules
(3:2:1) QD*H ~=QH;
where QH inherits an action by V from the Verschiebung : H ! H, and there is a
natural isomorphism of Fp[F ] modules
(3:2:2) P D*H ~=P H
where P H inherits an F action from the pth power map. To see the isomorphism (*
*3.2.1)
for example, let (x) be the divided power algebra (dual to a polynomial algebra*
*) on a
primitive generator of degree n. Then
(QH)*n~=Hom HA (H; (x)) ~=Hom D (D*H; D*(x)) ~=(QD*H)*n;
15
where (.)* denotes the Fpdual. Equation (3.2.2) is proved similarly.
The functor Q on D is right exact and has left derived functors; the functo*
*r P is left
exact and has right derived functors. Let
: D ! D
be the "doubling" functor; thus
ae
(M)m = Mn0 mm=6pn 0 mod p
with inherited F and V action.
Proposition 3.3. 1) LsQ = 0 for s > 1 and there is an exact sequence in D
0 ! L1QM ! M F!M ! QM ! 0:
2) RsP = 0 for s > 1 and there is a short exact sequence in D
0 ! P M ! M V!M ! R1P M ! 0:
Proof: The argument is standard, once one observes F : M ! M is onetoone for M
projective, and V : M ! M is onto for M injective.
We can use the functors Q and P to characterize projective and injective ob*
*jects in D
and HA. Call a Dieudonne module M F projective if F : M ! M is onetoone; the*
*re is
an obvious dual notion of V injective: we require that V : M ! M be onto. The *
*following
is inspired by [11]; the argument is originally due to F. Morel.
Proposition 3.4. 1) A module M 2 D is projective if and only if M is F project*
*ive and
QM is a projective graded Fp[V ] module.
2) A module M 2 D is injective if and only if M is V injective and P M is an i*
*njective as
an Fp[F ] module.
Proof: We prove part 2. Part 1 is dual. If M is injective it is a retract of a *
*product of the
injectives J(n) of section 2, hence has the stipulated property. This leaves th*
*e converse.
Suppose j : M ! J is an onetoone map in D to an injective J. To show M is
injective we need only show it splits. Since P j : P M ! P J is an injection an*
*d P M is an
injective Fp[F ] module, P j splits. Thus if K is the cokernel of P j, we may c*
*hoose a map
of Fp[F ] modules K ! P J so that the composite K ! P J ! K is the identity. Re*
*gard
K ! P J ! J as a morphism of Dieudonne modules, and let J1 be the cokernel. Th*
*en
16
if P M ! P J was a bijection in degrees k m, the induced map P M ! P J1 is sti*
*ll an
injection and a bijection in degrees k m + 1. Repeat the process and recursive*
*ly define
Dieudonne modules Jn and quotients Jn1 ! Jn so that P M ! P Jn is an injection*
* and
a bijection in degrees k m + n. Finally, let J1 = colimJn. Then P M ! P J1 *
* is an
isomorphism. Since M and J1 are both V injective, M ! J1 is an isomorphism.
From this result one can get detailed information about projective and inje*
*ctive res
olutions.
Lemma 3.5. 1) Let M 2 D be an F projective Dieudonne module. Then M has a pro
jective resolution of length 1.
2) Let M 2 D be a V injective Dieudonne module. Then M has an injective resolu*
*tion of
length 1.
Proof: For part 1, let 0 ! K ! P ! M ! 0 be the beginning of a projective resol*
*ution.
Then K is F projective since P is. Also
0 ! QK ! QP ! QM ! 0
is exact, since L1QM = 0, by Proposition 3.3.1. Since QP is a projective Fp[V ]*
* module,
so is QK. Hence K is projective.
Part 2 is similar.
Corollary 3.6. Every object M of D has a projective resolution of length 2 and *
*an
injective resolution of length 2.
Proof: Let 0 ! K ! F0 ! M ! 0 be the beginning of a projective resolution. Then*
* K
is F projective, so has a projective resolution
0 ! F2 ! F1 ! K ! 0:
Splicing the resolutions together yields the result. The claim about injective *
*resolutions is
similar.
Corollary 3.7. Let M; N 2 D. Then Ext sD(M; N) = 0 for s > 2 and if either M is
F projective or N is V injective, then
Ext 2D(M; N) = 0:
Remark 3.8: We make several assertions here without proof. First, if H 2 HA, i*
*s an
abelian Hopf algebra, then the following statements are equivalent:
1)D*H is F injective;
17
2)L1QH = 0;
3)the Frobenius (.)p : H ! H is injective; and
4)there is a vector space V H and an isomorphism of algebras S(V ) ~=H.
These facts, and Proposition 3.4.1 can be used to characterize projectives in H*
*A. For
injectives, one has that the following statements are equivalent.
10)D*H is V injective;
20)R1P H = 0;
30)the Verschiebung : H ! H is injective; and
40)there is a quotient vector space H ! V and automorphism of coalgebras H ~*
*=S*V .
Thus one can also characterize injectives in HA. Statements of this type ar*
*e addressed
in [1] and [11, x1]. One line of proof might go, 1 , 2, by Proposition 4.3, 2 *
*, 4 as in
[1x3], and 3 , 4.
4. The total derived functors of indecomposables and primitives
This section extends the homological algebra of the previous section to the*
* category
sD. The total derived functors of indecomposables can be identified with Andre*
*Quillen
homology. This will be done in the next section.
Define the total left derived functor LQ of the functor Q on sD be the form*
*ula
(4:1) LQM = QN
where N ! M is a weak equivalence in sD with N cofibrant. The simplicial vect*
*or
space LQM is welldefined up to weak equivalence. Since N is cofibrant, it is l*
*evelwise
projective, so, by Proposition 4.4, there is a short exact sequence
0 ! N F!N ! QN ! 0;
hence a long exact sequence
(4:2) . .!.ssqM F!ssqM ! ssqL QM ! ssq1M ! . .s.s0L Q ! 0:
Here we used that is exact and ss*N ~=ss*M.
Lemma 4.3. 1) If M 2 sD is levelwise F projective, then
ss*L QM ~=ss*QM:
2) For all M 2 sD, there is a short exact sequence of Dieudonne modules
0 ! QssqM ! ssqL QM ! L1Qssq1M ! 0:
18
Proof: For part 1, choose N ! M with N cofibrant. Then there is a diagram of si*
*mplicial
abelian groups
0 _____wN _____wFN _____wQN _____w0
  
u u u
0 _____wM _____wFM _____wQM _____w0:
The result follows from the induced map on long exact sequences and the five le*
*mma. Part
2 follows from 4.2 and the definition of Q and LiQ and Proposition 3.3.1.
There is a corresponding notion of total right derived functors of primitiv*
*es, muddled
by the fact that the functor P on sD does not preserve weak equivalences among *
*fibrant
objects_or, put another way, the model category structure we have chosen on sD *
*is built
from projectives and thus is not well adapted to studying right derived functor*
*s. There
are several ways out of this difficulty, including a closed model category stru*
*cture on sD
built from injectives, but the following is quick.
Lemma 4.4. Let M 2 sD. Then there is a natural morphism j : M ! J(M) in sD so t*
*hat
j is a levelwise injection and a weak equivalence and the normalized Dieudonne*
* modules
NJ(M)q are V injective for q > 0.
Proof: The forgetful functor from D to graded pointed sets has a right adjoint *
*I: if X
is a graded pointed set Y Y
I(X) = J(n)
n Xn*
where * is the basepoint. If K 2 D, we get a short exact sequence
(4:4:1) 0 ! K ! I(K) ! I1(K) ! 0
with I(K) and I1(K) both V injective. If K 2 ch*D is a chain complex of Dieud*
*onne
modules, one gets an augmented bicomplex
K ! I*(K)
from 4.4.1. The total complex of I*(K) is given by
8
< I(Kq) x I1(Kq+1) q 1
tI*(K)q = : :
Ker{I(K0) x I1(K1) ! I1(K0)} q = 0
The augmentation K ! I*(K) induces a levelwise injection K ! tI*(K) that is a *
*homol
ogy isomorphism. Note that tI*(K)q is V injective if q > 0. If M 2 sD, define *
*J(M) by
the formula
NJ(M) = tI*(NM):
19
We now simply define the total derived functor as primitives by the equatio*
*n, for
M 2 sD,
(4:5) RP (M) = P J(M):
An immediate consequence is the existence of a long exact sequence
(4:6) . .!.ssqR P (M) ! ssqM V!ssqM ! ssq1R P (M) ! . .!.ss0M:
And we have the analog of Lemma 4.3.
Lemma 4.7. 1) If M 2 sD and NMq is V injective for q > 0, then
ssqR P (M) ~=ssqP M:
2) For all M 2 sD there is a short exact sequence of Dieudonne modules
0 ! R1P ssq+1M ! ssqR P (M) ! P ssqM ! 0:
We can now prove the following technical lemma, which is crucial for our sp*
*litting
result. Let S* : A ! HA be right adjoint to the forgetful functor.
Lemma 4.8. Let A 2 sA. Then for all q > 0 the Dieudonne module ssqD*S*A is V 
injective.
Proof: Fix an integer n and write n = psn0 where (n0; p) = 1. Then we have the
projective Hopf algebra H(n) = Fp[x0; : :;:xs] with Witt vector diagonal and if*
* H 2 HA,
DnH ~=Hom HA (H(n); H):
The homomorphism V : DnH ! Dn=pH is induced by the inclusion H(n=p) ! H(n). The
claim is that this map is a cofibration of constant Hopf algebras in sHA. We pr*
*ove this
after completing our proof.
Assuming this fact, one has a cofibration sequence in sHA
Fp ! H(n=p) ! H(n) ! Fp[xs] ! Fp
where xs 2 Fp[xs] is a primitive generator of degree n. Thus, for H 2 sHA fibra*
*nt, one
has a fiber sequence of simplicial sets
(4:9) map sHA (F[xs]; H) ! map sHA(H(n); H) ! map sHA(H(n=p); H):
20
Since Hom sHA(Fp[xs]; H) = (P H)n (internal degree, not simplicial degree), th*
*is fiber
sequence is
(P H)n ! DnH V!Dn=pH:
In particular, ND*Hq is V injective for q > 0, since this is what it means for*
* a morphism
of simplicial abelian groups to be a fibration. [14, xII.4]. So, applying homot*
*opy groups
to 4.1 yields the long exact sequence 4.6.
Now let H = S*A. Then H is fibrant in sCA, hence in sHA. (See Proposition
1.5). Therefore, the sequence of 4.9 is a fibration sequence. By adjointness, t*
*his fibration
sequence is isomorphic to
mapsA (Fp[xs]; A) ! map sA(Fp[x0; : :;:xs]; A) ! map sA(Fp[x0; : :;:xs1]*
*; A):
Since Fp[x0; : :;:xs1] ! Fp[x0; : :;:xs] is split as algebras, this fibration *
*sequence is split
as simplicial sets (but not as simplicial groups). Hence
ssqmap sHA (H(n); S*A) ! ssqmap sHA (H(n=p); S*A)
is surjective_in fact, split surjective as groups for q 1. Thus 4.6 breaks up*
* into a
sequence of short exact sequences.
0 ! ssqA ! ssqD*S*A V!ssqD*S*A ! 0
and the result follows.
To see H(n=p) ! H(n) is a cofibration in sHA, consider the diagram
S(H(n=p)) _____wS(H(n))
 
u u
H(n=p) ________wH(n):
The vertical maps are onto, and the top is a cofibration. We show the bottom ma*
*p is a
retract of the top. Since H(n) is a projective, we choose a splitting oe : H(n)*
* ! S(H(n))
of S(H(n)) ! H(n). Since
S(H(n=p))k ~=S(H(n))k
in degrees k n=p, and all the generators of H(n=p) are in degrees k n=p, the *
*splitting
oe restricts to a splitting of S(H(n=p)) ! H(n=p).
Example 4.10: One cannot make the dual assertion that ssqD*S(C) is F projectiv*
*e for
C 2 sCA. For example, let p = 2 and E(x) an exterior algebra on a generator x o*
*f degree
k > 0. Let C = W E(x) regarded as a coalgebra. Then a simple calculation shows
ss1D*S(C) ~=F2
concentrated in degree k.
21
5. Postnikov Towers and the equivalence of homotopy theories
Before stating and proving the main result on the decomposition of Postniko*
*v towers
of simplicial Hopf algebras, we give a description of what the Postnikov tower *
*is for a
general abelian category with enough projectives. We also prove one of our main*
* results,
namely, Ho (sHA) ~=Ho (sD).
Let C be an abelian category with enough projectives and let X 2 sC. Define*
* the nth
Postnikov section of X as follows: for fixed k, let In;k! Xk be the kernel of t*
*he map
Y
d : Xk ! Xn
OE:[n]![k]
where OE runs over all injections in the ordinal number category with m n and *
*d is
induced by the maps OE* : Xk ! Xn. Define
(5:1) X(n)k = Xk=In;k:
Notice that there is a quotient map in sC from X ! X(n) and that if k n, X(n)k*
* = Xk.
There are also quotient maps
(5:2) qn : X(n) ! X(n  1)
and X ~=limX(n). Let F (n) be the fiber of qn, defined by the pullback diagram
F (n) _______wX(n)
(5:3) u u
0 _______wX(n  1):
Note that in an abelian category, this diagram is also a pushout diagram since*
* X(n) !
X(n  1) is surjective. The reader familiar with the MoorePostnikov tower of a*
* simplicial
set [13] will see that this new Postnikov tower agrees with the MoorePostnikov*
* tower if
C = Ab , the category of abelian groups. For if x; y 2 Xk are ksimplices, then*
* x = y in
X(n)k if and only if OE*x = OE*y for OE : [m] ! [k] as above, which occurs if a*
*nd only if the
nth faces of x and y agree.
We now describe the normalization NX(n). Let NX be the normalization of X,
ZnX = Ker{@ : NXn ! NXn1}
BnX = Im{@ : NXn+1 ! NXn} ~=NXn+1=Zn+1X:
Lemma 5.4. The normalization NX(n) of X(n) is the quotient chain complex of NX
. .!.0 ! BnX ! NXn ! . .!.NX1 ! NX0:
22
Proof: Use the description of NX as the kernel of
Y
Xk ! Xm
OE:[m]![k]
where OE runs over all injections in so that OE(0) = 0 and m < k. If k > n *
*+ 1, then
any injection OE : [m] ! [k] with m < n can be written as a composition of inje*
*ction
OE = OE0 where OE0 : [k  1] ! [k] and OE(0) = 0. Thus NXk In;k. If k = n +*
* 1, then
NXn+1 \ In;n+1= Zn+1X, and the result follows.
As an immediate consequence we have
Lemma 5.5. The normalization NF (n) of the fiber of X(n) ! X(n  1) is
. .!.0 ! BnX ! ZnX ! 0 ! . .!.0
concentrated in degrees n + 1 and n. Furthermore
ae
sskF (n) ~= ssnX0 kk=6n= n
Note that the projection map ZnX ! ssnX defines a canonical weak equivalence
(5:6) F (n) '!K(ssnX; n) ~=W nssnX:
A word on notation is in order here. If C is any abelian category, then th*
*ere is a
functor W : sC ! sC characterized by the equation
(NW X)n = (NX)n1
where NX1 = 0. It is this definition of W which is used in equation (5.6). Th*
*e functor
W is a model for suspension in sC. This functor W is different than the funct*
*or W defined
on simplicial algebras at the end of section 1. But it is not that different: t*
*hey are both
models for suspension, and if A is a simplicial abelian Hopf algebra, the two n*
*otions of W
agree. Since both meanings are ingrained in the literature we will persist.
We now prove that D* : sHA ! sD induces an equivalence of homotopy category.
We will, in fact prove more_we will show A ! B in sHA is a weak equivalence in *
*sHA
if and only if D*A ! D*B is a weak equivalence in sD. We will also relate Andre*
*Quillen
homology of A 2 sHA to the total derived object LQD*A. The proofs go by a seque*
*nce
of lemmas.
Lemma 5.7. The functor D* : sHA ! sD preserves fibrations and trivial fibration*
*s.
23
Proof: Let H(n) 2 HA be the projective so that DnK = Hom HA (H(n); K). Then the
constant simplicial object H(n) 2 sHA is cofibrant, by Lemma 2.8, so if q : X !*
* Y is a
fibration in sHA
q* : map sHA(H(n); X) ! map sHA(H(n); Y )
is a fibration of simplicial sets. This map is isomorphic to the map of simplic*
*ial abelian
groups DnX ! DnY . Since this morphism of simplicial abelian groups is a fibrat*
*ion of
simplicial sets if and only if (NDnX)q ! (NDnY )q is onto for q > 0,
D*q : D*X ! D*Y
is a fibration. If q is a trivial fibration, q* is a trivial fibration, hence s*
*sqDnX ! ssqDnY is
an isomorphism for q 0. Hence D*q is a trivial fibration.
If A 2 sA is a simplicial algebra, the AndreQuillen homology HQ*A is defin*
*ed as
follows. Choose a trivial fibration X ! A in sA with X cofibrant. Then HQ*A = s*
*s*QX.
Of course, HQ* is a functor on the homotopy category of simplicial algebras. I*
*f A is a
simplicial Hopf algebra, we may take X ! A to be a trivial fibration in sHA wit*
*h X
cofibrant in sHA. Then X is cofibrant in sA, so HQ*A ~=ss*QX.
Lemma 5.8. Let A 2 sHA. Then there is a natural isomorphism
HQ*A ~=ss*L QD*A:
Proof: Choose a trivial fibration X ! A in sHA with X cofibrant. By Lemma 2.8 a*
*nd
Lemma 5.7, D*X ! D*A is a weak equivalence and D*X is F projective. Thus Lemma
4.3.1 implies
ss*L QD*A ~=ss*QD*X:
But we proved in 3.2.1 that QD*X ~=QX.
The following result, which is crucial to our enterprise, uses the fact tha*
*t we are
working with graded connected objects in a crucial way  the last sentence of t*
*he proof is
false without this assumption.
Lemma 5.9. Let X 2 sHA be a simplicial abelian Hopf algebra. Then ss*D*X = 0 if*
* and
only if ss*X ~=Fp.
Proof: First assume ss*D*X = 0. Then, for all n 0, the inclusion
BnD*X ! ZnD*X
24
is an isomorphism. Thus D*F (n) has a simplicial contraction and, then, F (n) *
*has a
simplicial contraction. Hence ss*F (n) ~=Fp. Since (5.2) is a pushout diagram,*
* Proposition
1.9 shows X(n) ! X(n  1) is a weak equivalence. Since X(0) = F (0) we have ss**
*X(n) ~=
Fp. Now, for any k, sskX ~=sskX(n) for n sufficiently large, so ss*X ~=Fp.
Conversely, suppose ss*X ~=Fp. Thus the unit map X ! Fp is a weak equivalen*
*ce,
and HQ*X ~= HQ*Fp = 0. Then Lemma 5.8 and the long exact sequence of 4.2 impli*
*es
F : ssqD*X ! ssqD*X is an isomorphism. This can only happen if ssqD*X = 0.
Lemma 5.10. The functor D* : sHA ! sD sends trivial cofibrations to weak equiva*
*lences
and, therefore, preserves all weak equivalences.
Proof: Let A ! B be a trivial cofibration. Then it is an inclusion. Let C = Fp *
*A B.
Then
0 ! D*A ! D*B ! D*C ! 0
is an exact sequence of Dieudonne modules and ss*C ~= Fp, by Proposition 1.9. *
*Hence
ss*D*C = 0 by Lemma 5.9 and D*A ! D*B is a weak equivalence.
Since every weak equivalence in sHA can be factored as a trivial cofibratio*
*n followed
by a trivial fibration, the result follows from Lemma 5.7.
Lemma 5.11. The adjoint D1*: sD ! sHA of D* preserves cofibrations and all weak
equivalences.
Proof: By Lemma 5.7 and adjointness, D1*preserves cofibrations and trivial cof*
*ibra
tions. To show D1*preserves weak equivalences, it is equivalent to show that i*
*f f : A ! B
is a morphism in sHA so that ss*D*f is an isomorphism, then so was ss*f. Facto*
*r f as
A j!X q!B with j is a trivial cofibration and q is a fibration. Then Lemma 5.10*
* shows
ss*D*j is an isomorphism, so we have that ss*D*q is an isomorphism. We use this*
* data to
show ss*q is an isomorphism. Since D*q is a fibration, by Lemma 5.7, and ss0D**
*q is an
isomorphism, D*q is surjective. Hence q is surjective. Consider the pullback d*
*iagram
F _____wX
 q
u u
Fp _____wB:
Since q is surjective, this is also a pushout diagram. Since
0 ! D*F ! D*X ! D*B ! 0
is exact, ss*D*F = 0. Lemma 5.9 implies ss*F = 0, so Proposition 1.9 implies ss*
**q is an
isomorphism.
25
Theorem 5.12. The functor D* : sHA ! sD and its adjoint D1*both preserve all w*
*eak
equivalences and define an adjoint equivalence of categories
D* : Ho (sHA) ! Ho (sD)
Proof: Combine Lemmas 5.10 and 5.11 and Quillen's result [14, xI.3].
We can use these results to make a simple application to homotopy pullback*
*s in sHA.
If X1 ! A X2 is a diagram in sHA, the homotopy pullback is defined in the sa*
*me
manner as the homotopy pushout: factor X2 ! A as X2 j!Z q!A where j is a weak
equivalence and q is a fibration. Then the homotopy pullback is the pullback *
*X1A Z. It
is welldefined up to weak equivalence. A pullback diagram
Y _____wX2
 
u u
X1 ______wA
will be called a homotopy pullback diagram if Y is weakly equivalent to the h*
*omotopy
pullback. By analogy with Proposition 1.9 we have:
Proposition 5.13. Consider a pullback diagram in sHA
Y _____wX2
 q
u u
X1 _____wA:
If the map q is surjective, it is a homotopy pullback diagram.
Proof: There is a diagram of MeyerVietoris sequences
0 ________wD*Y ________wD*X1 D*X2 _____w D*A _____w0
  
u u u
0 _____wD*(X1*Z) ______wD*X1 D*Z ______w D*A _____w0:
Since ss*X2 ~=ss*Z, we have ss*D*X2 ~=ss*D*Z, whence ss*D*Y ~=ss*D*(X1A Z).
6. On kinvariants and looped kinvariants
In the last section we introduced the Postnikov tower of a simplicial Hopf *
*algebra. In
this section, we give a short description of how kinvariants behave. Although *
*we will use
the looped kinvariant for our splitting result, we begin with the kinvariant,*
* as it is more
familiar and because we can use it to emphasize some of the subtleties.
26
Let C be an abelian category, X 2 sC, and {X(n)}n0 the Postnikov tower of *
*X. Let
F (n) be the fiber of X(n) ! X(n  1).
For all Z 2 sC, choose a natural inclusion Z ! CZ so that the projection ma*
*p CZ ! 0
induces a chain equivalence NCZ ! 0. Define the object BF (n) and the kinvaria*
*nt by
the pushout diagram
X(n) _______wCX(n)
(6:1) qnu uq
X(n  1) _____wknBF (n):
This is also a pullback diagram, since q is surjective and a homotopy pullbac*
*k diagram.
The connection with what is normally known as a kinvariant is supplied by the *
*following.
Lemma 6.2. There are natural weak equivalences in sC
BF (n) W F (n) ! K(ssnX; n + 1):
Proof: There is a pushout diagram
F (n) _____wCF (n)
 
u u
0 _______wF (n)
and a map of MeyerVietoris sequences induced by the diagram of short exact seq*
*uences
in sC
0 _____wF (n) _________w0 CF (n) _________wF (n) _____w0
  
u u u
0 _____wX(n) _____wX(n  1) CX(n) _____wBF (n) _____w0:
An examination of these MeyerVietoris sequences shows F (n) ! BF (n) is a weak
equivalence. But W F (n) is a model for F (n) and there is a weak equivalence W*
* F (n) !
W K(ssnX; n) = K(ssnX; n + 1), by 5.6.
Remark 6.3: Because (6.1) is a homotopy pullback diagram, the weak homotopy ty*
*pe of
X(n) depends only on the class of kn in [X(n1); BF (n)]sC. This follows from a*
* standard
argument, recapitulated in some detail (in a dual situation) in the proof of on*
*e of our
results. See 7.3. Now there is a spectral sequence (cf. 7.4 below)
Ext pC(ssqX(n  1); ssnX) ) [(n+1)(p+q)X(n  1); BF (n)]sC:
27
So if C has projective dimension 1, [X(n  1); BF (n)]sC = 0 and there is a wea*
*k homotopy
equivalence (noncanonical)
X(n) ' X(n  1) x F (n):
This happens when C is the category of abelian groups. If C = HA, then
[X(n  1); BF (n)] ~=Ext 2sHA(ssn1X; ssnX)
and this group need not be zero. So X(n  1) ! X(n) need not split as Hopf alge*
*bras.
We now define the looped kinvariant. For all Z 2 sCA, choose an object Z a*
*nd a
natural map Z ! Z so that 0 ! Z is a weak equivalence. Thus Z is a model for the
path space on Z. If Z is simplicially connected in the sense that Z0 = 0 or, mo*
*re generally,
if ss0Z = 0, we may take Z ! Z to be a surjection. Define Y (n  1) and the lo*
*oped
kinvariant kn by the pullback diagram
Y (n  1) _____wX(n)
(6:4) kn u u
F (n) _______wX(n):
Note that Y (n  1) ! X(n) is an injection, since F (n) ! X(n) is an injection.*
* If X(n)
is simplicially connected, X(n) ! X(n) is a surjection, so this is a pushout d*
*iagram and
a homotopy pushout diagram.
The following result says that Y (n  1) is a model for the loops on X(n  *
*1).
Lemma 6.5. Suppose X is simplicially connected. Then each X(n) is connected and*
* there
is a natural weak equivalence W Y (n  1) ' X(n  1).
Proof: That each X(n) is simplicially connected is clear. Also, 6.4 is now also*
* a pushout
diagram, so the cokernel of Y (n  1) ! X(n) is isomorphic to the cokernel of F*
* (n) !
X(n), which is X(n  1). But X(n) ' 0 and Y (n  1) ! X(n) is an inclusion, so *
*the
cokernel is a model for Y (n  1) ' W Y (n  1).
Remark 6.6: Under the isomorphism
[Y (n  1); F (n)]sCA ~=[X(n  1); BF (n)]sCA
supplied by Lemmas 6.2 and 6.5, the class of kn is identified with the class of*
* kn. We
won't need this fact so we won't supply more details. But it explains our notat*
*ion.
28
7. The splitting results
We will say a simplicial Hopf algebra X is simplicially connected if X0 ~=F*
*p. Let X
be a simplicial abelian Hopf algebra and let {X(n)} be the Postnikov tower for *
*X. Then
each of the sequences, n 1,
F (n) ! X(n) qn!X(n  1)
is a homotopy cofibration sequence. The point of this section is to prove
Theorem 7.1. Suppose X is simplicially connected. Then this homotopy cofibrati*
*on
sequence is split in the category of simplicial algebras; that is, for all n 1*
*, there is an
isomorphism in the homotopy category of simplicial algebras X(n1)F (n) ! X(n) *
*that
fits into a homotopy commutative diagram of cofibration sequences of simplicial*
* algebras.
There is a further decomposition of the fiber F (n) which we will present a*
*t the end
of the section. We also discuss how to weaken the simplicially connected hypot*
*hesis in
Lemma 8.3.
Theorem 7.1 is proved by calculating with the looped kinvariant kn : Y (n *
* 1) !
F (n) as defined in section 3.
Proposition 7.2. Let X be any simplicial abelian Hopf algebra. In the homotopy *
*cate
gory Ho (sA), the looped kinvariant kn : Y (n  1) ! F (n) is trivial for all *
*n 2.
7.3. Proof of Theorem 7.1: We assume Proposition 7.2. Recall that Y (n  1) is
defined by a pullback diagram in sHA
Y (n  1) _____wX(n)
(7:4:1) kn u u
F (n) _______wX(n)
where X(n) is a model for a contractible cover for X(n). Since X is simpliciall*
* connected,
this is also a pushout diagram. Note that, for the same reason, we have that
Fp Y(n1)X(n) ~=Fp F(n)X(n) ~=X(n  1):
Now if kn was actually the constant map, the universal property of pushouts wo*
*uld
demonstrate that there was one isomorphism
X(n) ~=F (n) X(n  1):
29
The idea, then, is to deform kn to the constant map and claim that the resultin*
*g push
out weakly equivalent to X(n). Since Y (n  1) may not be cofibrant, this esse*
*ntially
straightforward argument requires a little care.
Choose a cofibrant object Y 2 sHA equipped with a weak equivalence Y ! Y (n*
*  1)
and factor the composite Y ! Y (n1) ! X(n) and Y !jC ! X(n) where the first map
is a cofibration and the second a weak equivalence. Define X1 by the pushout d*
*iagram
Y _______wC
(7:4:2) knOj u u
F (n) _____wX1:
There is a map from (7.4.2) to (7.4.1) and a comparison of the homotopy spectra*
*l se
quences implies X1 ! X(n) is a weak equivalence. For the same reason Fp Y C !
Fp Y (n1)X(n) ~=X(n  1) is a weak equivalence.
Since Y is cofibrant, Proposition 7.2 implies there is a homotopy in h : Y *
* 1 ! F (n)
in the category sA from kn O j to the constant map. Define X2 by the pushout d*
*iagram
Y 1 _____wC 1
(7:4:3) uh u
F (n) ________wX2
The inclusion of the 0simplex {0} ! 1defines a map from 7.4.2 to 7.4.3. Sinc*
*e Y 1 !
C 1 is a morphism in sHA, a comparison of the homotopy spectral sequences imp*
*lies
X1 ! X2 is a weak equivalence. Finally, the inclusion of the 0simplex {1} ! *
*1defines,
in a similar manner, a weak equivalence F (n) FpY C ! X2. Finally one has a di*
*agram
F (n) _____wF (n) Fp Y C ______wFp Y C
=   =
u u u
F (n)u___________wX2u____________wFpuY C
=   =
  
F (n) ___________wX1 ____________wFp Y C
=   
u u u
F (n) __________wX(n) __________wX(n  1):
All the vertical maps are weak equivalences and all the horizontal sequences ar*
*e homotopy
cofibration sequences.
We now start to work on Proposition 7.2. We begin with a standard lemma.
30
Lemma 7.4. Let X 2 sHA be any simplicial abelian Hopf algebra so that ss*D*X ~=M
concentrated in degree n. Then for any A 2 sHA there is a first quadrant cohom*
*ology
spectral sequence
Ext pD(ssqD*A; M) ) [n(p+q)A; X]sHA :
This spectral sequence is equipped with an edge homomorphism
Ext 2D(ssqD*A; M) ! [n(q+2)A; X]sHA :
Proof: See [4, Chap. XVII]. The edge homomorphism arises because Ext psD(N; M) *
*= 0
for p > 2.
Lemma 7.5. Let X 2 sHA have the property that ssqD*X = 0 for q n  1 and suppo*
*se
Y 2 sHA has the property that Y (q) ~=Fp for q < n. Then
[X; Y ]sHA ~= Ext 2D(ssn2D*X; ssnD*Y ):
Proof: Let {Y (q)} be the Postnikov tower for Y and let F (q) be the fiber of *
*Y (q) !
Y (q  1). We can assume D*X is fibrant in sD. Then there is a tower of fibra*
*tions of
simplicial sets
{map sD(D*X; D*Y (q))}:
The claim is that
(7:6:1) ss0map sD(D*X; D*Y (q)) = Ext 2D(ssn2D*X; ssnD*Y )
if q n and that
(7:6:2) lim1ss1map sD(D*X; D*Y (q)) = 0:
The result will follow since
[X; Y ]sHA ~= [D*X; D*Y ]sD ~=ss0map sD(D*X; D*Y (q)):
For (7.6.1), we argue inductively. If q = n, then Y (q) = F (n) and (7.6.1) fo*
*llows from
Lemma 7.4. For larger q, the homotopy pullback square
Y (q + 1) _____wCY (q + 1)
 
u u
Y (q) _________wBF (q)
31
of 6.1 yields a homotopy fibration sequence
mapsD (D*X; D*Y (q + 1)) ! map sD(D*X; D*Y (q)) ! map sD(D*X; D*BF (q)):
Thus, Lemma 7.4 implies
sstmap sD(D*X; D*Y (q + 1)) ! sstmap sD(D*X; D*Y (q))
is an isomorphism for t q  n. Thus at once supplies the inductive step for 7.*
*6.1 and
proves 7.6.2.
Proof of Proposition 7.2: We need to show that the class of kn in [Y (n1); F (*
*n)]sHA
goes to zero in [Y (n  1); F (n)]sA. We show, in fact, that
[Y (n  1); F (n)]sA = 0:
The forgetful functor sHA ! sA and its right adjoint S* satisfy the hypotheses *
*of Quillen's
adjoint functor theorem [14, xI.3]; hence there is an isomorphism
[Y (n  1); F (n)]sA ~=[Y (n  1); S*F (n)]sHA :
Since F (n)q ~=Fp for q < n, S*F (n)q ~=Fp for q < n; therefore, Lemma 5.5 impl*
*ies
[Y (n  1); S*F (n)]sHA~=Ext2sD(ssn2D*Y (n  1); ssnD*S*F (n))
~=Ext 2sD(ssn1D*X(n  1); ssnD*S*F (n)):
However, the Ext groups are zero by Lemma 4.8 and Corollary 3.7.
Remark 7.8: A consequence of Theorem 7.1 is that for all n 0, there is an isom*
*orphism
in Ho(sA), for X 2 sHA with the property that ss0X ~=Fp, there is a splitting
~=On
OEn : X(n) ! F (q)
q=1
and diagrams in Ho(sA)
On
X(n) _______wOEnF (q)
 q=1

 
 u
u n1O
X(n  1) _____wOEn1F (q):
q=1
32
nN
Since, in any given simplicial degree k, the towers {X(n)k} and { F (q)k} sta*
*bilize in
q=1
the sense that X(n)k ~=X(n  1)k for n sufficiently large, we may write an isom*
*orphism
in Ho(sA)
~=O1
OE : X ! F (q):
q=1
Note that the tensor product makes sense since F (q)k ~=Fp if k < q. Also, by (*
*6.5), there
is a weak equivalence in sHA
F (q) ~=W qH(q)
where D*H(q) ~=ssqD*H.
The next proposition further decomposes F (q). This splitting result is a m*
*inor varia
tion on a result noticed by the second author [19].
Proposition 7.9. Let H 2 HA. Then there is a homotopy cofibration sequence of
simplicial algebras
W S(QH) ! W H ! W 2S(L1QH)
and this sequence is split. Furthermore, the sequence is natural up to homotopy*
* in maps
H ! K in HA, but not the splitting.
Proof: We use the fact that [W kS(V ); A]sA ~=Hom Fp(V; sskA). Choose a weak e*
*quiva
lence A ! W H in sA with A almost free in the sense of [14, xII.4]. Then A is c*
*ofibrant in
sA. Filter A by powers of the augmentation ideal and get a spectral sequence
Es;t1~=sssSt(QA) ) sssA:
Here St(W ) ~=W t =tis the tthhomogeneous piece of the symmetric algebra functo*
*r. This
is Quillen's fundamental spectral sequence. Since Es;11~=sssQA ~=HQs(W H) ~=Ls*
*1QH
we have
E1 ~=E(QH) (L1QH)
where E(.) denotes the exterior algebra and (.) denotes the divided power algeb*
*ra. Since
Quillen's spectral sequence is a spectral sequence of algebras, it must collaps*
*e. Hence
ss1A ~=sssW H ~=QH. Choose a map in sA
j : W S(QH) ! W H
inducing an isomorphism on ss1. This map is unique and natural up to homotopy. *
*Factor
j as
W S(QH) i!B q!W H
33
where i is a cofibration in sA and q is a weak equivalence, and let C = Fp W S(*
*QH) B.
This is the homotopy cofiber of the morphism j. The spectral sequence
Tor ss*W S(QH)(F2; ss*W H) ) ss*C
shows that ss*C ~=(L1QH). Thus there is a weak equivalence W 2S(L1QH) ! C, uniq*
*ue
and natural up to homotopy. Thus it only remains to show that the sequence is *
*split.
However ss2B ~=ss2W H ! ss2C ~=L1QH is onto, so there is a lift
B
[ [] 
[ u
W 2S(L1QH) _____w'C
and the map
W S(QH) W 2S(QH) ! B
is a weak equivalence by direct calculation.
Corollary 7.10. Let H 2 HA. Then for all n 1 there is a homotopy cofibration
sequence of simplicial algebras
W nS(QH) ! W nH ! W n+1S(L1QH)
and this sequence is split. Furthermore, the sequence is natural up to homotopy*
* in mor
phisms H ! K in HA.
Proof: Apply W n1 to the sequence of Proposition 7.9.
8. The homotopy of simplicial abelian Hopf algebras:
the Dalgebra structure
In this section and the next, we give our major application: the calculatio*
*n of ss*A
as a functor of HQ*A, for A 2 sHA. We work only at the prime 2, as this simplif*
*ies the
presentation considerably.
Let A be a simplicial abelian Hopf algebra over F2. Then ss*A comes equippe*
*d with
a great deal of structure. The totality of that structure makes ss*A into what *
*is known as
a Hopf Dalgebra1 [19]. We now give a brief review.
_________________________
1The authors apologize for using the letter D both here and for Dieudonne modul*
*es. These
usages are both in the literature. We hope this causes no difficulty.
34
To begin with, ss*A is a bigraded abelian Hopf algebra. Also, since A is a *
*simplicial
cocommutative coalgebra, ss*A is an unstable coalgebra over the Steenrod algebr*
*a. Thus
there are operations on the right
(.)Sqi : ssnA ! ssniA
which halve the internal degree and are subject to the usual instability, Carta*
*n, and Adem
formulas. In particular, Sqi = 0 if 2i > n, Sqn=2 is the Verschiebung of the Ho*
*pf algebra
ss*A and Sq0 is induced from the Verschiebung of the Hopf algebra An. Note that*
* Sq0 6= 1.
Also, since the simplicial algebra multiplication A A ! A is a morphism of sim*
*plicial
cocommutative coalgebras, the algebra multiplication ss*A ss*A ! ss*A is a mor*
*phism
of unstable coalgebras over the Steenrod operations; that is, the multiplicativ*
*e Cartan
formula holds. We say ss*A is an unstable Hopf algebra over the Steenrod algebr*
*a.
Next, since A is a simplicial commutative algebra, ss*A supports higher div*
*ided power
operations  operations that go back to work of Cartan and are studied in detai*
*l in [3] and
[7]:
ffii(.) : ssnA ! ssn+iA; 2 i n
doubling internal degree and defined only for 2 i n. The following formulas h*
*old:
8.1.1): ffiiis a homomorphism for 2 i < n and ffin = fl2, the divided square, *
*so ffin(x+y) =
ffinx + ffiny + xy;
8.1.2): ffii(xy) = 0 unless degree of x or the degree of y is zero, when ffii(x*
*y) = x2ffiy (if the
degree of x is zero) or ffii(xy) = ffii(x)y2 (if the degree of y is zero);
X j  i + s  1
8.1.3): If i < 2j, ffiiffij(x) = ffii+jsffis(x). Note *
*that 8.1.1 forces
i+1_2si+j_3 j  s
x2 = 0 if x 2 ssnA, n 2. In fact:
8.1.4): if x 2 ssnA, n 1, then x2 = 0.
A bigraded algebra equipped with operations ffii satisfying 8.1.1)_4) is kn*
*own as a D
algebra. Since the diagonal A ! A A is a morphism of simplicial commutative al*
*gebras,
ss*A ! ss*A ss*A commutes with the action of the ffii, where ffii(x y) can be*
* computed
using 8.1.2. One might call such an object a DHopf algebra except that this wo*
*uld cause
some confusion with Hopf Dalgebras, so we avoid this terminology.
Hopf Dalgebras arise from the observation the ffii and the Sqj are not ind*
*ependent,
but satisfy some Nishidastyle relations. These relations are the subject of [*
*19], and we
refer the reader to that reference for details. We record here only the followi*
*ng relation,
because it makes clear that if ss*A is free as a Dalgebra, then many Steenrod *
*operations
35
must be nonzero. Let be the Verschiebung on ss*A and fl2 the second divided *
*power
operation; then if x 2 ssnA and 2 j n
X
(8:2) (ffijx)Sqj = fl2(x) + ffis(xSqs):
2s>j
Since fl2(x) is often nonzero, the implications of the formula are very strong*
*. Note,
however, that j 2, so this formula says nothing about Sq0 or Sq1.
The rest of this section is devoted to studying the Dalgebra structure of *
*ss*A, with
A 2 sHA.
We begin our calculations by reducing to the case where A is simplicially c*
*onnected.
For any A 2 sHA, regard ss0A as a constant object in sHA and define A+ 2 sHA by*
* the
short exact sequence in sHA
F2 ! A+ ! A ! ss0A ! F2:
Lemma 8.3. The object A+ 2 sHA is weakly equivalent to a simplicially connected*
* sim
plicial Hopf algebra and there is a natural isomorphism of Hopf Dalgebras
ss*A ~=ss0A ss*A+ :
Proof: To see A+ is weakly equivalent to a simplicially connected object, note*
* that
D*ss0A ~= ss0D*A concentrated in degree 0. Hence ss0D*A+ = 0. There is a simp*
*licial
Dieudonne module M and a weak equivalence (in sD) M ! D*A+ so that M0 = 0. Let
B 2 sHA be so that D*B ~=M and B ! A+ the induced map. Then B is simplicially
connected and weakly equivalent to A+ . The statement about ss*A follows from t*
*he short
exact sequence of Hopf Dalgebras
Fp ! ss*A+ ! ss*A ! ss0A ! Fp:
Remark 8.4: Let A 2 sHA. Then D*ss0A ~= ss0D*A, so to recover ss*A from ss*D*A,
we need only worry about the case where A is simplicially connected. This will *
*be in the
hypothesis of all our results in the rest of the section.
We now calculate ss*A with its action by the operations ffii. If B is any *
*simplicial
commutative algebra, then the bigraded algebra ss*B equipped with the action by*
* the ffii,
the associated Cartan and Adem formulas and the fact that x2 = 0 if x 2 ssnB, n*
* 1,
is an example of a Dalgebra [19, x2]. There is a category of such and the augm*
*entation
36
ideal functor I from Dalgebras to bigraded vector spaces has a left adjoint SD*
* ; in fact, if
V is a simplicial graded vector space
(8:5) SD (ss*V ) ~=ss*S(V ):
By [6], this formula determines SD .
Now let A 2 sHA. Then ssqD*A is a Dieudonne module and we can form QssqD*A
and L1QssqD*A as in 3.0. Let Qss*D*A be the resulting bigraded vector space an*
*d let
L1Qss*D*A be the bigraded vector space with
(L1Qss*D*A)p;q= (L1Qssq1D*A)p:
Thus we have shifted degrees in homotopy.
Proposition 8.6. Let A 2 sHA be a simplicially connected abelian Hopf algebra. *
*Then
there is a natural exact sequence of Dalgebras
Fp ! SD (Qss*D*A) iA!ss*A ! SD (L1Qss*DA) ! Fp:
The sequence is split, but not naturally.
Before proving this, we will construct the map SD (Qss*D*A) ! ss*A. To begi*
*n with
we produce a map ss*D*A ! ss*A. Since
ssqDnA ~=ssqmap sHA (H(n); A);
an element x 2 ssqDnA determines a unique homotopy class of maps
fx : W qH(n) ! A
and, hence, a unique map, ss*fx : ss*W qH(n) ! ss*A. Since H(n) is a polynomial*
* algebra,
(8:7) ss*W qH(n) ~=SD (qQH(n))
by (8.5). Here qQH(n) means QH(n) ~= ssqW qH(n). Let in 2 QH(n)n be the top
class. Then the assignment x 7! ss*fx(in) defines a homomorphism ssqDnA ! (ssq*
*A)n.
The operator F : ssqDnA ! ssqDpnA is defined by the Hopf algebra map H(pn) ! H(*
*n)
sending xi2 H(pn) to xpi12 H(n). Hence if x = F y 2 ssqDpnA, then 8.7 implies
ss*fx(ipn) = 0 2 (ssqA)pn
37
and we have defined a natural
QssqD*A ! ssqA:
This extends to a map of Dalgebra
(8:8) iA : SD (Qss*D*A) ! ss*A:
This is the map of Proposition 8.6.
8.9: Proof of Proposition 8.6: By combining Theorem 7.1 and Corollary 7.10, we
immediately have that iA is injective. Let
C(A) ~=Fp SD (Qss*D*A)ss*A:
We need only show there is a natural isomorphism
(8:9:1) C(A) ~=SD (L1Qss*D*A):
Then the sequence of 8.6 is split, because C(A) is a free Dalgebra.
Suppose, for a moment, that we can construct such a natural isomorphism for*
* A
simplicially connected and ss*D*A concentrated in a single degree. Then we indu*
*ct over
the Postnikov tower to get a natural isomorphism
C(A(n)) ~=SD (L1Qss*D*A(n)):
Indeed, if one has such an isomorphism for n  1, one need only complete the di*
*agram
F2 ________wC(F (n)) ______________wC(A(n)) ______________wC(A(n  1)) _______*
*_wF2
~=  ~=
u u u
F2 ___wSD (L1Qss*D*F (n)) ___wSD (L1Qss*D*A(n)) ___wSD (L1Qss*D*A(n  1)) ___w*
*F2:
But both rows are uniquely split as Dalgebras, as C(F (n)) ~=F2 in degrees les*
*s than n+1
and the generators on C(A(n  1)) as a Dalgebra all lie in degrees less than n*
* + 1. Thus
the result follows from the next lemma.
Lemma 8.10. Let A 2 sHA be simplicially connected and suppose ss*D*A ~=M concen
trated in degree m. Then
C(A) ~=SD (L1QM):
Furthermore this isomorphism is natural with respect to maps in sHA among simpl*
*icially
connected objects with one nonvanishing group in ss*D*(.).
38
Proof: The existence of the isomorphism follows from Corollary 7.10. We prove n*
*atural
ity. Suppose A is as stated and B is simplicially connected with ss*D*B ~=N con*
*centrated
in degree n. By Lemma 7.4 [A; B]sHA = 0 unless 0 n  m 2, whence
[A; B]sHA ~= Ext nmD(M; N):
If m  n = 1 or 2, then C(A) ! C(B) is the trivial map, since C(B) ~=F2 in degr*
*ee less
than n + 1 and the generators C(A) as a Dalgebra lie in degree m + 1. If m = *
*n, let
H; K 2 HA be so that D*H ~=M and D*K ~=N. Then a model for f : A ! B, up to
homotopy is given by
W ng : W nH ! W nK
where g is the image of f under the isomorphism
[A; B]sHA ~= Hom D(M; N) ~=Hom HA (H; K):
The result now follows from the naturality clause of Corollary 7.10.
Remark 8.11: 1) It is easy to construct examples where the splitting of Propos*
*ition
8.6 is not natural. For example, if A is a cofibrant model for W E(x) where E(x*
*) is the
exterior algebra on a primitive generator of degree n and Fp[y] is the polynomi*
*al algebra
on a primitive generator of degree 2n, then there is a map A ! W 2F2[y] which i*
*s surjective
on homotopy. However, both the end terms of 8.6 must map trivially.
2) One consequence of Proposition 8.6 is that the Quillen spectral sequenc*
*e for ss*A
collapses. Assuming A is cofibrant, one filters A by powers of the augmentation*
* ideal (cf.
the proof of 7.9) and gets a spectral sequence
ss*S(QA) ) ss*A:
Assuming A is simplicially connected, one uses 8.5 to get
(8:12) ss*S(QA) ~=SD (HQ*A)
and Proposition 8.6 to get that the spectral sequence collapses. Here one uses *
*the exact
sequence
0 ! QssqD*A ! HQqA ! L1Qssq1D*A = 0:
Note that the nonnaturality of the splitting in Proposition 8.6 is absorbed in*
* the filtration
of the spectral sequence.
39
3) The previous remark and Proposition 8.6 immediately imply Theorem C of *
*the
introduction.
Fix A 2 sHA and suppose that A is simplicially connected. Then formula 8.1.*
*2) im
plies that the vector space of indecomposables Qss*A supports an action by the *
*operations
ffii (now all homomorphisms by 8.1.1) subject to the relations of 8.1.3. Let F2*
* Qss*A
be the quotient of Qss*A by this action. In light of Remark 8.11.2, there is a*
* natural
isomorphism:
F2 Qss*A ~=HQ*A:
The module Qss*A is also an unstable module over the Steenrod operations; h*
*owever,
because of formula 8.2, F2 Qss*A does not get an induced unstable structure. N*
*onethe
less, F2 Qss*A is a module over the subalgebra A(0) = F2[Sq0; Sq1]=(Sq1)2 A *
*of the
Steenrod algebra generated by Sq0 and Sq1 and the quotient map
(8:12) Qss*A ! F2 Qss*A ~=HQ*A
is a module over A(0). As mentioned in the introduction, the Hopf Dalgebra ss*
**A is
determined by this A(0) module. The reader may prefer, on aesthetic grounds, t*
*o have
a description of the A(0) action internal to HQ*A, rather than have it imposed *
*from the
outside form ss*A. We will sketch such a description.
Choose a weak equivalence of simplicial Hopf algebras B ! A so that F acts *
*injectively
on D*B. Then by Lemmas 4.3 and 5.8, there is a natural isomorphism
HQ*A ~=ss*QD*B
where QD*B fits into the short exact sequence
(8:13) 0 ! D*B F!D*B ! QD*B ! 0:
Define the action of Sq0 on HQ*A to be the action induced by V on ss*QD*B. Defi*
*ne the
action of Sq1 to be the "Bockstein" of the short exact sequence of (8.13); thus*
* Sq1 is the
composite
ssnQD*B ! ssn1D*B ~=ssn1D*B ! ssn1QD*B:
Note that Sq1Sq1 = 0 and Sq0Sq1 = Sq1Sq0. Both Sq1 and Sq0 divide internal degr*
*ee by
2.
The claim is that this A(0) action on HQ*A is the same as that considered a*
*bove in
8.12. The proof is by the method of universal examples. Let (xk) be the divided*
* power
algebra on a single primitive generator of degree k.
40
Lemma 8.14. Let A 2 sHA. Then there is a natural isomorphism
[A; W n(xk)]sHA ~= [HQnA]*k:
Proof: In the category of Dieudonne modules
Hom D(D*A; D*(xk)) = [QD*A]*k:
Choose a weak equivalence B ! A in sHA so that F acts injectively on D*B. Then
[A; W n(xk)]sHA ~= [D*B; W nD*(xk)]sD
and the result follows.
Now to show that the two definitions of the A(0) structure on HQ*A agree fo*
*r all A,
we need only show they agree on HQ*W n(xk) for all n and k 1. This is a calcul*
*ation,
and we omit it. It is similar, but far easier, than the similar calculation giv*
*en in [12]. x6.
References
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2. A.K. Bousfield, "On the homology spectral sequence of a cosimplicial space",*
* Amer. J.
Math. 109 (1987), 361394.
3. A. K. Bousfield, "Operations on Derived Functors of Nonadditive Functors," *
*preprint,
Brandeis University, 1967.
4. H.E. Cartan and S. Eilenberg, Homological Algebra, Princeton University Pres*
*s, Prince
ton, 1956.
5. M. Demazure, Lectures on pdivisible groups, Lecture Notes in Mathematics 3*
*02,
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