ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE
COMPLEXES
NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Abstract.Relying on the computation of the Andr'e-Quillen homology group*
*s for unstable Hopf
algebras, we prove that the mod p cohomology of the n-connected cover of*
* a finite H-space is
always finitely generated as algebra over the Steenrod algebra.
Introduction
Consider the n-connected cover of a finite complex. Does its (mod p) cohomolo*
*gy satisfy some
finiteness property? Such a question has already been raised by McGibbon and Mo*
*ller in [MM97 ],
but no satisfactory answer has been proposed. We do not ask here for an algorit*
*hm which would
allow to make explicit computations. We rather look for a general structural s*
*tatement which
would tell us to what kind of class such cohomologies belong. The prototypical *
*theorems we have
in mind are the Evens-Venkov result, [Eve61], [Ven59], that the cohomology of a*
* finite group is
Noetherian, the analog for p-compact groups obtained by Dwyer and Wilkerson [DW*
*94 ], and the
fact that the mod p cohomology of an Eilenberg-Mac Lane space K(A, n), with A a*
*belian of finite
type, is finitely generated as an algebra over the Steenrod algebra, which can *
*easily been inferred
from the work of Serre [Ser53] and Cartan [Car55].
This last observation leads us to ask first whether or not the mod p cohomolo*
*gy of a finite
Postnikov piece is also finitely generated as an algebra over the Steenrod alge*
*bra and second, since
a finite complex X and its n-connected cover X only differ in a finite numbe*
*r of homotopy
groups, if H*(X; Fp) satisfies the same property. We offer in this paper a p*
*ositive answer when
X is an H-space, based on the analysis of the fibration P ! X ! X, where P i*
*s a finite
Postnikov piece. In fact we prove a strong closure property for H-fibrations.
Theorem 4.2. Let F ! E ! B be an H-fibration in which both H*(F ; Fp) and H*(*
*B; Fp) are
finitely generated unstable algebras. Then so is H*(E; Fp).
_____________
2000 Mathematics Subject Classification. Primary 55P45; Secondary 13D03, 55S*
*10, 55T20.
All three authors are partially supported by MEC grant MTM2004-06686. The fi*
*rst and third authors were
partially supported by the Mittag-Leffler Institute in Sweden. The third author*
* is supported by the program Ram'on
y Cajal, MEC, Spain.
1
2 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
This applies in particular to highly connected covers of finite H-spaces, se*
*e Theorem 4.3. In our
previous work [CCSa ] we proved that the theorem holds whenever the base space *
*is an Eilenberg-
Mac Lane space. The proof relied mainly on Smith's work [Smi70] on the Eilenber*
*g-Moore spectral
sequence.
Our starting point here is the same and we need to analyze carefully certain*
* Hopf subalgebras
of H*(F ; Fp). Observe that the property for an unstable algebra K to be a fin*
*itely generated
Ap-algebra is equivalent to say that the module of the undecomposable elements *
*QK is finitely
generated as unstable module. It is often more handy to work with this module b*
*ecause it is smaller
than the whole algebra and, above all, the category of unstable modules is loca*
*lly Noetherian,
[LZ86].
The main problem (or interest) with the functor Q(-) is the failure of left *
*exactness. To what
extent this functor is not left exact is precisely measured by Andr'e-Quillen h*
*omology HQ*(-). In
our setting we keep control of the size of these unstable modules.
Proposition 2.3. Let A be a Hopf algebra which is a finitely generated unsta*
*ble Ap-algebra.
Then HQ0(A) = QA and HQ1(A) are both finitely generated unstable modules.
As the higher groups are all trivial (see Proposition 1.3), this gives a qui*
*te accurate description of
Andr'e-Quillen homology in our situation. The relevance of Andr'e-Quillen homol*
*ogy in homotopy
theory is notorious since Miller solved the Sullivan conjecture, [Mil84]. The t*
*ypical result which is
needed in his work, and which has been then extended by Lannes and Schwartz, [L*
*S86], is that the
module of indecomposable elements of an unstable algebra K is locally finite if*
* and only if so are
all Andr'e-Quillen homology groups of K.
Proposition 2.3 yields then our main algebraic structural result about the c*
*ategory of unstable
Hopf algebras.
Theorem 2.4. Let B be a Hopf algebra which is a finitely generated unstable *
*Ap-algebra. Then
so is any unstable Hopf subalgebra.
Acknowledgments. We would like to thank Jean Lannes for his guided reading of [*
*GLM92 ] at
the Mittag-Leffler Institute and Carles Broto for many discussions on this mate*
*rial.
1.Andr'e-Quillen homology of Hopf algebras
In this section, we compute Andr'e-Quillen homology for Hopf algebras, and i*
*ntroduce the action
of the Steenrod algebra in the next one. A clear and short introduction to Andr*
*'e-Quillen homology
can be found in Bousfield's [Bou75, Appendix], see also Goerss' book [Goe90].
Let us briefly recall from Schwartz's book [Sch94] how one computes Andr'e-Q*
*uillen homology
in our setting. The symmetric algebra comonad S(-) yields a simplicial resoluti*
*on So(A) for any
commutative algebra A. The Andr'e-Quillen homology group HQi(A) is the i-th hom*
*ology group
ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE COMPLEXES *
* 3
of the complex obtained from So(A) by taking the module of indecomposable eleme*
*nts (and the
differential is the usual alternating sum). This is a graded Fp-vector space. L*
*ong exact sequences
arise from certain extensions, just like in the dual situation for the primitiv*
*e functor, [Bou75,
Theorem 3.6].
Lemma 1.1. Let A be a Hopf subalgebra of a Hopf algebra B of finite type. Then *
*there is a long
exact sequence
. .!.HQ2(B==A) ! HQ1(A) ! HQ1(B) ! HQ1(B==A) ! QA ! QB ! Q(B==A)
in Andr'e-Quillen homology.
Proof.Long exact sequences in Andr'e-Quillen homology are induced by cofibratio*
*ns of simplicial
algebras. However, the inclusion A B of a sub-Hopf algebra is not a cofibrati*
*on in general (seen
as a constant simplicial object). To get around this difficulty we use Goerss' *
*argument from [Goe90,
Section 10] and we reproduce it here in our framework.
For any morphism f : A ! B of simplicial algebras, there is a spectral sequen*
*ce, [Goe90, Propo-
sition 4.7], Torss*Ap(Fp, ss*B)q converging to the homotopy groups ssp+qCof(f) *
*of the homotopy
cofiber. Now, because B is of finite type, it is always a free A-module by the *
*Milnor-Moore result
[MM65 , Theorem 4.4]. Thus the E2-term is isomorphic to TorFp0(Fp, B==A)* ~=B=*
*=A. The spec-
tral sequence collapses and hence Cof(f) is weakly equivalent to B==A. In parti*
*cular, we have the
desired long exact sequence. *
* |
Following the terminology used in [Smi70, Section 6], we introduce the follow*
*ing definition.
Definition 1.2. A sequence of (Hopf) algebras
Fp -! A -! B -! C -! Fp
is coexact if the morphism A ! B is a monomorphism and its cokernel B==A is iso*
*morphic to C
as a (Hopf) algebra.
We can thus restate the previous lemma by saying that coexact sequences of Ho*
*pf algebras
induce long exact sequences in Andr'e-Quillen homology.
By the Borel-Hopf decomposition theorem [MM65 , Theorem 7.11], any Hopf alge*
*bra of finite
type is isomorphic, as an algebra, to a tensor product of monogenic Hopf algebr*
*as, i.e. either a
ki
truncated polynomial algebra of the form Fp[xi]=(xpi ), where pkiis the height *
*of the generator
xi, or a polynomial algebra of the form Fp[yj], or, when p is odd, an exterior *
*algebra (zi). Let us
denote by , the Frobenius map, sending an element x to its p-th power xp.
Proposition 1.3. Let A be a Hopf algebra of finite type. Then HQ0(A) = QA and *
*HQ1(A) is
isomorphic to the Fp-vector space generated by the elements ,kixi of degree pki*
*. |xi| where xi2 A
is a generator of height pki, 0 < ki< 1 . Moreover HQn(A) = 0 if n 2.
4 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Proof. Consider the symmetric algebra S(QA) and construct an algebra map S(QA) *
*i A by
choosing representatives in A of the indecomposable elements. We have then a co*
*exact sequence
of algebras Fp[,kixi] ,! S(QA) i A and A, as Hopf algebra, can be seen as the q*
*uotient
S(QA)==Fp[,kixi]. Since S(QA) is a free commutative algebra, HQn S(QA) = 0 fo*
*r all n 1.
*
* Q
Likewise HQn Fp[,kixi] = 0 for all n 1. Now, Lemma 1.1 allows us to identif*
*y H1 (A) ~=
HQ0(Fp[,kixi]) ~= kFp<,kixi>, as a graded vector space. *
* |
The vanishing of the higher Andr'e-Quillen homology groups, or in other word*
*s the fact that the
functor Q(-) has homological dimension 1 for Hopf algebras, has been analyzed*
* by Bousfield in
the dual situation [Bou75, Theorem 4.1]. The next lemma is now a reformulation *
*of the preceding
proposition.
Lemma 1.4. Let A be a Hopf algebra of finite type and denote by xi the truncate*
*d polynomial
*
* ki
generators. Then HQ1(A) is isomorphic to the Fp-vector space generated by the e*
*lements xip 2
S(A), where pkiis the height of xi.
Proof. We have to compute the first homology group of the complex
. .!.S2(A) d-!S(A) m-!A .
The morphisms are given by the alternating sums of the face maps. Let us use th*
*e symbols for
the tensor product in S(A) and for the next level in S(S(A)). If jA : S(A) ! *
*A is the counit
defined by jA(a b) = ab, the two face maps S2(A) ! S1(A) are then S(jA) and j*
*S(A).
Thus m(a) = S(jA)(a) - jS(A)= a - a = 0 and m(a b) = ab, since jS(A)(a b*
*) = a b is
decomposable. Likewise d(w) = jA(w) on elements w 2 S(A) and d(v w) = v w -*
* jA(v)
ki
jA(w) for v, w 2 S(A). The elements xip clearly belong to the kernel of m. To *
*compare them
to the generators {,kixi} of HQ0(Fp[,kixi]) ~=HQ1(A), we apply So to the coexac*
*t sequence of
algebras Fp[,kixi] ,! S(QA) i A. The snake Lemma yields a connecting morphism K*
*er(m) !
ki
HQ0(Fp[,kixi]), which sends precisely xip to ,kixi. *
* |
Remark 1.5. Alternatively, one could use the identification of the first Andr'e*
*-Quillen homology
group HQ1(A) with the indecomposable elements of degree 2 in TorA(Fp, Fp), [Goe*
*90, Section 10].
ki-1
As an Fp-vector space it is generated by the transpotence elements [xpi |xi]. *
*When p = 2 and
ki= 0, this is not technically speaking a transpotence element, but still it is*
* [xi|xi] which appears
in degree 2, see for example [Kan88 , Section 29-2].
2. Bringing in the action of the Steenrod algebra
The results of the previous section apply to Hopf algebras which are finitel*
*y generated as algebras
over the Steenrod algebra: they are of finite type. Our aim in this section is *
*to identify the action of
the Steenrod algebra on the unstable module HQ1(A). Good references on Andr'e-Q*
*uillen homology
ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE COMPLEXES *
* 5
for unstable algebras are [Sch94, Chapter 7] and of course [Mil84], which showe*
*d the importance
of Andr'e-Quillen homology in a topological context.
The Fp-vector space HQ1(A) is equipped with an action of Ap because the Steen*
*rod algebra acts
on the symmetric algebra via the Cartan formula. This yields the same unstable *
*module HQ1(A)
as the derived functor computed with a resolution in the category of unstable a*
*lgebras, [LS86] and
[Sch94, Proposition 7.2.2].
As expected with this type of questions, the case when p = 2 is slightly simp*
*ler than the case
when p is odd. To write a unified proof, we use the well-known trick [LZ86] to *
*consider, in the
odd-primary case, the subalgebra of Ap concentrated in even degrees. If M is a *
*module over Ap,
the module M0 concentrated in even degree is defined by (M0)2n = M2n and (M0)2n*
*+1= 0. This
is not an Ap-submodule of M, but it is a module over Ap on which the Bockstein *
*fi acts trivially.
Hence it can be seen as a module over the algebra A0p, the subalgebra of Ap gen*
*erated by the
operations Pi. When p = 2 we adopt the convention that A02= A2, U0= U, and writ*
*e Pi for Sqi.
Like in [Sch94, 1.2], for a sequence I = ("0, i1, "1, . .,.in, "n) where the "k*
*'s are 0 or 1, we write PI
for the operation fi"0Pi1fi"1. .P.infi"n.
In [LZ86, Appendice B], Lannes and Zarati prove that the category U is locall*
*y noetherian,
which they do by reducing the proof to the case of U0. We use their computation*
*s in the following
lemma, in fact the explicit version from [Sch94, 1.8].
Lemma 2.1. Let M be an unstable module which is finitely generated over Ap. The*
*n so is the
module M0, over A0p.
Proof.The statement is a tautology when p = 2. Let us assume p is an odd prime.*
* In the category
U0 of unstable modules concentrated in even degrees, F 0(2n) is the free object*
* on one generator '2n
in degree 2n. We must show that M0 is a quotient of a finite direct sum of such*
* modules. As we
know that M is a quotient of a finite direct sum of F (n)'s, it is enough to pr*
*ove the lemma for a
free module F (n).
A basis over Fp for the module F (n) is given by the elements PI'n where I is*
* admissible with
excess e(I) n. Define F (n)0kto be the span over Fp of the elements PI'n with*
* e(I) k. Then
F (n)0k=F (n)0k+1is zero when k + n is odd and it is generated by the images of*
* the elements PI'n
where the k Bocksteins appear in the first k + 1 possible slots. In particular,*
* F (n)0is generated by
these elements as an A0p-module. *
* |
The generators for HQ1(A) will be related to certain elements in QA we descri*
*be next.
Lemma 2.2. Let A be a Hopf algebra which is a finitely generated unstable Ap-al*
*gebra and let N
be the submodule of QA generated by the truncated polynomial generators xi. The*
*n N0 is finitely
generated in U0. There exists an integer d and a finite set {xk,i| 1 k d, 1*
* i nk} of generators
6 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
P
such that any element in N of height pk can be written i`k,ixk,ifor some (adm*
*issible) operations
`k,i2 A0p.
Proof. Since U is locally noetherian, [Sch94, Theorem 1.8.1], the unstable modu*
*le N is finitely
generated, being a submodule of QA. Thus, by Lemma 2.1, N0 is finitely generate*
*d over A0p. This
implies in particular that the height of the truncated generators is bounded by*
* some integer pd
(the action of the Steenrod algebra on xican only lower the height by the formu*
*las [Sch94, 1.7.1]).
For 1 k d, write N0(k) for the submodule of N0 generated by the xi's of *
*height pk and
choose representatives in A of generators xk,ifor the module N0(k), with 1 i *
* nk. Hence the
finite set {xk,i| 1 k d, 1 i nk} generates N0. *
* |
P
The relation x = i`k,ixk,iholds in the module of indecomposable elements (*
*in fact in N0).
Beware that the same relation holds also in the algebra A, but only up to decom*
*posable elements.
Proposition 2.3. Let A be a Hopf algebra which is a finitely generated unstable*
* Ap-algebra. Then
HQ0(A) = QA and HQ1(A) are both finitely generated unstable modules.
ki
Proof. Lemma 1.4 allows us to identify HQ1(A) ~= kFp, as a graded vector*
* space. We must
now identify the action of the Steenrod algebra.
ki Q
We claim that the finite set of elements xkp,i generates H1 (A) as unstable *
*module. More
P *
* ki Q
precisely we show that the relation x = i`k,ixk,iin QA yields a relation for *
*x p in H1 (A). To
simplify the notation, let us assume that the height of x is pk and that the re*
*lation is of the form
P
x = j`jxj for generators xj of the same height. The relation for x holds in A*
* up to decomposable
elements which must have lower height. But if apk= 0 = bpk, then
k pk pk pk pk pk pk pk pk*
* pk
d[a p b - (a b) ] = a b - a b - (a b) + (ab) *
* = (ab)
P *
* pk
and hence the decomposable elements disappear in HQ1(A). Therefore x pk = i*
*`jxj in
HQ1(A). The operations `j live in A0p, so that we have basically to perform the*
* following compu-
tation in the symmetric algebra: (Pnx) pk = Ppkn(x pk). There exist thus operat*
*ions j 2 A0p
such that
k X pk X pk
x p = (`jxj) = j(xj )
j
and the claim is proven. *
* |
Theorem 2.4. Let B be a Hopf algebra which is a finitely generated unstable Ap-*
*algebra. Then so
is any unstable Hopf subalgebra.
Proof. Consider an unstable Hopf subalgebra A B and the quotient B==A. By Lem*
*ma 1.1, we
have an associated exact sequence in Andr'e-Quillen homology
HQ1(B==A) ! QA ! QB,
ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE COMPLEXES *
* 7
in which the unstable modules QB and HQ1(B==A) are finitely generated by Propos*
*ition 2.3. Thus
so is QA. *
* |
Example 2.5. Let us consider the Hopf algebra B = H*(K(Z=p, 2)). When p is odd *
*it is the tensor
product of a polynomial algebra Fp['2, fiP1fi'2, fiPpP1fi'2, . .]., concentrate*
*d in even degrees, with
an exterior algebra (fi'2, P1fi'2, ...).
We consider the Hopf subalgebra A given by the image of the Frobenius ,. This*
* is the polynomial
subalgebra
Fp[('2)p, (fiP1fi'2)p, (fiPpP1fi'2)p, . .].
The quotient B==A has an exterior part and a truncated polynomial part where al*
*l generators have
height p. The module of indecomposable elements Q(B==A) is isomorphic to QB. It*
* is a quotient
of F (2), and thus generated, as an unstable module, by a single generator '2 i*
*n degree 2. The
submodule concentrated in even degree is a module over A0p. It is finitely gene*
*rated as well, by
Lemma 2.1, but one needs two generators '2 and fiP1fi'2. Explicit computations *
*of the action of
the Steenrod algebra can be found in [Cre01].
Therefore HQ1(B==A) is an unstable module, which is generated by the elements*
* '2p and
(fiP1fi'2) p.
Remark 2.6. For plain unstable algebras, Theorem 2.4 is false, as pointed out t*
*o us by Hans-
Werner Henn. Consider indeed the unstable algebra
H*(CP 1 x S2; Fp) ~=Fp[x] E(y)
where both x and y have degree 2. Take the ideal generated by y, and add 1 to t*
*urn it into an
unstable subalgebra. Since y2 = 0, this is isomorphic, as an unstable algebra,*
* to Fp 2Fp
2He*(CP 1; Fp), which is not finitely generated.
3.H-fibrations over Eilenberg-Mac Lane spaces
In the second part of this paper we turn now our attention to topological app*
*lications of the
Andr'e-Quillen homology computation we have done previously. More precisely, we*
* concentrate on
H-fibrations.
Definition 3.1. An H-space B satisfies the (strong) fg closure property if, for*
* any H-fibration
F ! E ! B, the cohomology H*(E) is a finitely generated unstable algebra if (an*
*d only if) so is
H*(F ).
We prove in this section that Eilenberg-Mac Lane spaces enjoy the strong fg c*
*losure property.
In [CCSa ] we established the fg closure property for K(A, n) with n 2, which*
* was sufficient to
our purposes there.
8 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Given n 2, consider a non-trivial H-fibration F -i!E -ss!K(A, n) where A i*
*s either Z=p or
a Pr"ufer group Zp1. This situation has been extensively and carefully studied*
* by L. Smith in
[Smi70].
The E2-term of the Eilenberg-Moore spectral sequence is given by TorH*(K(A,n*
*))(H*(E), Fp)
and converges to H*(F ). Since we deal with an H-fibration, [Smi70, Theorem 2.*
*4] applies and
E2 ~=H*(E)==ss* TorH*(K(A,n))\\ss*(Fp, Fp) as algebras, where H*(K(A, n))\\ss*
** is the Hopf
subalgebra kernel of ss*. The first differential is dp-1 [Smi70, Theorem 4.7]. *
*The same argument as
in [Smi70, Section 5] (done for stable Postnikov pieces) is valid in our situat*
*ion as well, and has been
in fact already used in this setting, see [Smi70, Proposition 7.3*]: on algebra*
* generators the next
differentials must be zero, so that the spectral sequence collapses at Ep. This*
* term is generated
by H*(E)==ss* = E0,*p, s-1,0CokerfiP0 E-1,*p, and s-1,0QH*(K(A, n))\\ssodd E*
*-1,*p, where
fiP0:QHodd(K(A, n)) ! QHeven(K(A, n)) is defined by fiP0(x) = fiPt(x) with 2t +*
* 1 = |x|.
The algebra structure is described in [Smi70, Proposition 7.3*] by means of *
*coexact sequences,
see Definition 1.2.
Proposition 3.2. [Smi70] Let n 2 and consider an H-fibration F -i!E ss-!K(A, *
*n) where A is
either Z=p or a Pr"ufer group Zp1. Then there is a coexact sequence of Hopf alg*
*ebras
*
Fp -! H*(E)==ss* i-!H*(F ) -! R -! Fp,
and R is described in turn by a coexact sequence of Hopf algebras
Fp -! -! R -! S -! Fp,
where is an exterior algebra generated by s-1,0CokerfiP0, and S H*(K(A, n -*
* 1)) is a Hopf
subalgebra. *
* |
Example 3.3. Let us see how the well-known cohomology of S3<3> can be identifie*
*d with these
tools. Consider the fibration
S3<3> i!S3 ss!K(Z, 3)
In this situation H*(S3)==ss* = 0 and H*(S3<3>) ~= S by Proposition 3.2. R*
*ecall that
H*(K(Z, 3)) = Fp[fiPku3 : k 1] E(Pku3 : k 0) where Pk = Ppk-1. .P.1and P0*
*u3 = u3.
Here the Hopf algebra kernel H*(K(Z, 3))\\ss* is Fp[fiPku3 : k 1] E(Pku3 : *
*k 1). One sees
next that the cokernel of fiPp is {fiP1u3} and S H*(K(Z, 2)) is generated by *
*P1'2 = 'p2. This
implies that H*(S3<3>) ~=Fp[x2p] E(fix2p).
Proposition 3.2 will allow us to improve [CCSa , Theorem 6.1]. We rely on th*
*e following obvious
lemma, which we will use again in the next section.
Lemma 3.4. Consider an H-space B and assume that there exists an H-fibration B0*
*! B ! B00
such that both B0 and B00satisfy the (strong) fg closure property. Then so does*
* B.
ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE COMPLEXES *
* 9
Proof.Consider an H-fibration F ! E ! B and construct the pull-back diagram of *
*fibrations
E0 ____//_E____//B00
p0|| p|| ||||
fflffl| fflffl| ||
B0 ____//_B____//B00
The homotopy fiber of p0is F , which allows to conclude. *
* |
Theorem 3.5. Let A be a finite direct sum of copies of cyclic groups Z=pr and P*
*r"ufer groups
Zp1, and n 2. Consider an H-fibration F -i!E ss-!K(A, n). Then H*(F ) is a fi*
*nitely generated
Ap-algebra if and only if so is H*(E).
Proof.If we consider the fibration of Eilenberg-Mac Lane spaces induced by a gr*
*oup extension
A0! A ! A, we see from Lemma 3.4 that we can assume that A = Z=p or Zp1.
Since H*(K(A, n)) is finitely generated as algebra over Ap, so is its image I*
*m(ss*) H*(E).
Hence, to prove the theorem, it is enough to show that the module of indecompos*
*able elements
Q(H*(E)==ss*) is a finitely generated Ap-module if and only if so is QH*(F ).
Let us now apply Lemma 1.1 to the coexact sequences from Proposition 3.2. The*
* unstable Hopf
algebra S is an unstable Hopf subalgebra of H*(K(A, n)). Thus Theorem 2.4 impl*
*ies that S is
finitely generated over Ap. Moreover, the exterior algebra is identified with*
* E(s-1,0CokerfiP0),
where the cohomological operation fiP0 is to be understood as an operation from*
* the odd degree part
of QH*(K(A, n)) to the even degree part. The latter module is finitely generate*
*d by Lemma 2.1.
Hence the cokernel is finitely generated as well, as a module over the Steenrod*
* algebra. The exact
sequence in Andr'e-Quillen homology for the coexact sequence involving R and Pr*
*oposition 2.3 show
that both QR and HQ1(R) are finitely generated unstable modules. Finally, sinc*
*e U is a locally
noetherian category, [Sch94, Theorem 1.8.1], the exact sequence
HQ1(R) ! Q(H*(E)==ss*) ! QH*(F ) ! QR
implies that QH*F is a finitely generated Ap-module if and only if so is Q(H*(E*
*)==ss*). |
Remark 3.6. Another approach to Theorem 3.5 is to dualize the work of Goerss, L*
*annes, and
Morel in [GLM92 , Section 2]. They analyze the homology sequence
H*( 2K(A, n)) ! H*( F ) ! H*( E) ! H*( K(A, n))
for a fibration of spaces F ! E ! K(A, n) and measure its failure to be exact. *
* This can be
dualized and actually works for H-fibrations, not only loop fibrations. Let us *
*quickly sketch the
key ideas. Consider now an H-fibration F ! E ! K(A, n) and the complex
* i*
H*(K(A, n)) ss-!H*E -! H*F -! H*K(A, n - 1).
10 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Define K to be the Hopf cokernel of the morphism i*: H*E ! H*F and M to be the *
*kernel
of the morphism ss* on primitive elements P H*K(A, n) ! P H*E. The cohomology s*
*uspension
morphism oe : H*(K(A, n)) ! H*(K(A, n - 1)) restricted to M defines a morphism*
* M ! K.
The adjoint M ! K induces an isomorphism of Ap-Hopf algebras U M ! K (compare *
*with
[Smi67, Proposition 5.7]), where U is Steenrod-Epstein's functor, left adjoint *
*to the forgetful functor
K ! U.
Denote by N the cokernel of M ,! P H*K(A, n). The above complex is then exac*
*t at H*E,
[Smi67, Proposition 5.5], and its homology at H*F is isomorphic to U 1N, where *
* 1 is the first
left derived functor of .
Theorem 3.5 now follows from the fact that 1N is a finitely generated unsta*
*ble module. This
simply reflects the fact that the functor (which is the "doubling" functor wh*
*en p = 2), [Sch94,
1.7.2] takes finitely generated unstable modules to finitely generated ones.
We have thus shown that Eilenberg-MacLane spaces K(A, n) satisfy the strong *
*fg closure prop-
erty. But in fact, it can be easily generalized to p-torsion Postnikov pieces.
Proposition 3.7. Consider an H-fibration F -i!E -ss!B, where B is an p-torsion *
*H-Postnikov
piece whose homotopy groups are finite direct sums of cyclic groups and Pr"ufer*
* groups. Then H*(E)
is a finitely generated Ap-algebra, if and only if so is H*(F ).
Proof. An induction on the number of homotopy groups of B with Lemma 3.4 reduce*
*s to the case
when B is an Eilenberg-Mac Lane space K(A, n). We know from Theorem 3.5 that th*
*e statement
holds in this case. *
* |
Our first corollary has already been proved in [CCSa ].
Corollary 3.8. Let F be an H-Postnikov piece of finite type. Then H*(F ) is fin*
*itely generated as
unstable algebra.
Proof. The result is true for an Eilenberg-Mac Lane space K(A, n) where A is an*
* abelian group of
finite type. The proof then follows by induction on the number of homotopy grou*
*ps. |
4.Closure properties of H-fibrations
The aim of this section is to extend the results of the preceding section to*
* arbitrary base spaces.
We will prove that any H-space B such that H*(B) is a finitely generated algebr*
*a over Ap satisfies
the fg closure property. We need here some input from the theory of localizati*
*on. Recall (cf.
[Far96]) that, given a pointed connected space A, a space X is A-local if the e*
*valuation at the base
point in A induces a weak equivalence of mapping spaces map(A, X) ' X. When X i*
*s an H-space,
it is sufficient to require that the pointed mapping space map*(A, X) be contra*
*ctible.
ON THE COHOMOLOGY OF HIGHLY CONNECTED COVERS OF FINITE COMPLEXES *
* 11
Dror-Farjoun and Bousfield have constructed a localization functor PA from sp*
*aces to spaces
together with a natural transformation l : X ! PAX which is an initial map amon*
*g those having
an A-local space as target (see [Far96] and [Bou77]). This functor is known as *
*the A-nullification.
It preserves H-space structures since it commutes with finite products. Moreove*
*r, when X is an
H-space, the map l is an H-map and its fiber is an H-space.
Recall that, for any elementary abelian group V , tensoring with H*V has a le*
*ft adjoint, Lannes'
T -functor TV , [Lan92]. When V = Z=p, the notation T is usually used instead o*
*f TZ=p and the
reduced T -functor is left adjoint to tensoring with the reduced cohomology of *
*Z=p. This allows to
characterize the Krull filtration of the category U of unstable modules as foll*
*ows: M 2 Un if and
__n+1
only if T M = 0, [Sch94, Theorem 6.2.4].
Lemma 4.1. Let X be an H-space such that TV H*(X) is of finite type for any ele*
*mentary abelian
p-group V . Then H*(PBZ=pX) is finite if and only if, for any n, H*(P nBZ=pX) *
*is a finitely
generated Ap-algebra.
Proof.By [Bou94], for any n there are fibrations P nBZ=pX ! P n-1BZ=pX ! K(An, *
*n+1) where
An is a p-torsion abelian group, which is a finite direct sum of copies of cycl*
*ic groups Z=pr and
Pr"ufer groups Zp1 (the technical hypothesis on the T functor allows to apply [*
*CCSa , Theorem 5.4]).
In this situation, we can apply Theorem 3.5 to show that H*(P nBZ=pX) is a fini*
*tely generated
algebra if and only if H*(P n-1BZ=pX) is so. The statement follows by induction*
* since H*(PBZ=pX)
is always locally finite. *
* |
Theorem 4.2. Consider an H-fibration F -i!E ss-!B. If H*(F ) and H*(B) are fini*
*tely generated
Ap-algebras, then so is H*(E).
Proof.Since both H*(F ) and H*(B) are finitely generated Ap-algebras, the modul*
*es of indecom-
posable elements QH*(F ) and QH*(B) are finitely generated Ap-modules. Therefo*
*re, [CCSa ,
Lemma 7.1], they belong to some stage Un-1 of the Krull filtration. By [CCSa , *
*Theorem 5.3], we
know that both F and B are nBZ=p-local spaces. Since nullification preserves f*
*ibrations whose
base space is local (see [Far96, Corollary 3.D.3]), it follows that E is also *
*nBZ=p-local.
Let us consider the fibration ~PBZ=pB ! B ! PBZ=pB. We know from Lemma 4.1 t*
*hat the
nullification PBZ=pB has finite mod p cohomology. The homotopy fiber ~PBZ=pB is*
* an H-Postnikov
piece whose homotopy groups are finite direct sums of cyclic groups Z=pr and Pr*
*"ufer groups Zp1
by [CCSa , Theorem 5.4]. Lemma 3.4 and Proposition 3.7 show then that it is eno*
*ugh to prove the
theorem when H*(B) is finite.
In that case, B is a BZ=p-local space, and we have thus a diagram of fibratio*
*ns
F _________//_E______//B
| | ||
| | ||
fflffl| fflffl| ||
PBZ=pF _____//PBZ=pE____//_B
12 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
The mod p cohomology H*(PBZ=pF ) is finite and hence so is H*(PBZ=pE) by an eas*
*y Serre spectral
sequence argument. Finally, we can apply Lemma 4.1 to conclude that H*(E) ~=H*(*
*P nBZ=pE)
is a finitely generated Ap-algebra. *
* |
Corollary 4.3. Consider an H-space X with finite mod p cohomology. Then the mod*
* p cohomology
of its n-connected cover X is a finitely generated Ap-algebra.
Proof. Consider the H-fibration (X[n]) ! X ! X. The fiber is an H-Postnikov*
* piece of finite
type and the cohomology of the base is finite. The result follows. *
* |
This can be seen as the mirror result of [CCSb ], where we proved that any H*
*-space with finitely
generated cohomology as algebra over the Steenrod algebra is an n-connected cov*
*er of an H-space
with finite mod p cohomology, up to a finite number of homotopy groups.
Remark 4.4. We have already seen in the proof of Theorem 4.2 that the module of*
* indecomposable
elements of a finitely generated cohomology, as algebra over the Steenrod algeb*
*ra, belongs to some
stage of the Krull filtration. In fact, for any H-space X with finite mod p coh*
*omology, QH*X
belongs to Un-2 by [CCSa , Theorem 5.3], because n-1(X) is BZ=p-local.
Remark 4.5. Theorem 4.2 cannot be improved to an "if and only if" statement. C*
*onsider for
example the path-fibration for the 3-dimensional sphere S3 ! P S3 ! S3. It is *
*well-known that
H*( S3) is a divided power algebra, which is not finitely generated over Ap.
We conclude with a characterization of the H-spaces which satisfy the strong*
* fg closure property.
Proposition 4.6. Let X be an H-space X which satisfies the strong fg closure pr*
*operty. Then X
is, up to p-completion, a p-torsion Postnikov piece.
Proof. If X satisfies the strong fg closure property, then H*( X) is a finitely*
* generated Ap-algebra.
But in this case, by [CCSa , Corollary 7.4], X is, up to p-completion, a p-tor*
*sion Postnikov piece. |
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Nat`alia Castellana and J'er^ome SchererJuan A. Crespo
Departament de Matem`atiques, Departamento de Econom'ia,
Universitat Aut`onoma de Barcelona,Universidad Carlos III de Madrid,
E-08193 Bellaterra, Spain E-28903 Getafe, Spain
E-mail: natalia@mat.uab.es, E-mail: jacrespo@eco.uc3m.es,
jscherer@mat.uab.es