Comparing Completions of a Space at a Prime
Paul Goerss1
There are two practical ways to complete an abelian group A at a prime p. O*
*ne could
complete A with respect to the neighborhood base of zero given by the subgroups*
* pnA.
This completion, called p-completion, is limA=pnA. Or one could complete A with*
* respect
to the neighborhood base of zero given by subgroups B A of finite p-power inde*
*x. This
completion, called p-pro-finite completion, is limAffwhere Affruns over the fin*
*ite p-group
quotients of A. They agree for finitely generated groups, but not in general: i*
*f V is an Fp
vector space it is p-complete, but its p-pro-finite completion is the double du*
*al V **. Also,
the former, p-completion, is easier to define, but it is neither left nor right*
* exact and the
category of p-complete groups is not abelian - it is not closed under cokernels*
*. The latter,
p-pro-finite completion, is initially less tractable, but it is right exact and*
* the category of
p-pro-finite abelian groups is an abelian category.
There are two corresponding completions for topological spaces. The analog*
* of p-
completion is Bousfield-Kan completion [3] and the analog of p-pro-finite compl*
*etion is
related to Sullivan's pro-finite completion and has recently been given a homot*
*opical def-
inition by Morel [13]. The purpose of this note is to compare these two comple*
*tions; in
addition, we seek to give Morel's completion the same sort of computational foo*
*ting that
the Bousfield-Kan completion enjoys.
To underscore the similarities and differences, let me make some remarks on*
* how these
completions of spaces are constructed.
If X is a space, the mod p homology H*X = H*(X; Fp) is a graded coalgebra o*
*ver
Fp and, as such, there is an isomorphism colimffCff~=H*X, where Cff H*X runs ov*
*er
the filtered system of sub-coalgebras which are finite dimensional in each degr*
*ee. Thus
there is an isomorphism in cohomology, H*X ~=limffC*ff, where C*ffis the algebr*
*a dual to
Cff. In short, H*X is a pro-finite algebra_which is to say the inverse limit eq*
*uips H*X
with a topology_and if X ! Y is a map of spaces, H*Y ! H*X is a continuous map *
*of
algebras.
The Bousfield-Kan completion of X may be obtained by the following program.*
* Define
a tower of fibrations
. .-.! Xs-qs!Xs-1-! . .-.! X1 q1-!X0
_________________________
1The author was partially supported by the National Science Foundation.
1
equipped with maps fs : X ! Xs so that qsfs = fs-1 and Xs is defined recursivel*
*y.
First X0 is a generalized mod p Eilenberg-MacLane space (i.e., with trivial k-i*
*nvariants)
so that ss*X0 is a graded Fp vector space and the continuous map f*0: H*X0 ! H**
*X is
surjective. Having defined Xs-1, choose a generalized Eilenberg-MacLane space K*
*s with
ss*Ks an Fp-vector space and a map ks : Xs-1 ! Ks so that
* f*s-1
H*Ks-ks!H*Xs-1 --! H*X
is an exact sequence of pro-finite algebras. Note that the universal property o*
*f Eilenberg-
MacLane spaces implies that one may specify the continuous map k*sfirst; this t*
*hen specifies
ks up to homotopy. Define qs : Xs ! Xs-1 to be the pull-back, via ks, of the pa*
*th-fibration
over Ks and fs to be any lift of fs-1. In the end, one may identify limsXs with*
* the Bousfield-
Kan completion of X, and the induced map X ! limsXs as the completion map. (See*
* [3,
III x6]).
The p-pro-finite completion of Morel arises when one runs this program afte*
*r neglect-
ing the topology on H*X. To do this one needs a setting where non-topological c*
*ohomology
arises naturally. This is provided by the continuous cohomology of a simplicial*
* pro-finite
set. As this sentence indicates this paper is written simplicially, so "space" *
*means "sim-
plicial set". The details of continuous cohomology and Morel's results are give*
*n in section
1. We linger there a bit to prove that the closed model category on simplicial *
*pro-finite
sets constructed by Morel in [13] is actually part of a simplicial model catego*
*ry structure.
This helps with later constructions. Section 2 gives a similar homotopical foun*
*dation to
the Bousfield-Kan completion. This enlarges on the idea, originally due to Dror*
*-Farjoun
[4], that one might regard the Bousfield-Kan completion as a pro-object.
The later sections provide, among other things, a computational foundation *
*for p-
pro-finite completion. Section 3 gives a description of p-pro-finite completion*
* as the total
space of a cosimplicial space and describes the resulting Bousfield-Kan spectra*
*l sequence.
We prove also that if H*X is finite in each degree, then the Bousfield-Kan comp*
*letion
and the p-pro-finite completion of X are weakly equivalent. This is certainly *
*plausible,
given the constructions above. Section 5 computes the homotopy groups of the p-*
*pro-finite
completion of a connected nilpotent spaces X in terms of the homotopy groups of*
* X. As
a preliminary section 4 discusses the derived functors of p-pro-finite completi*
*on of abelian
groups. Unlike the p-completion functor needed for Bousfield-Kan completion, p-*
*pro-finite
completion is right exact and has straightforward homological algebra. Finally *
*section 6
approaches the problem of computing the homology of the p-pro-finite completion*
* of X
in terms of H*X. This is not simple. Even in the case of the Bousfield-Kan comp*
*letion
2
one must compute H*(limsXs), where the Xs are as above, by no means a straightf*
*orward
operation. See [3, VI x5]. In the case of p-pro-finite completion, one is faced*
* with a spectral
sequence based on the right derived functors of the completion of H*X in the ca*
*tegory of
coalgebras. Some examples are given.
1 Pro-finite spaces and p-pro-finite completion.
This section is devoted to recapitulating and interpreting the result of Mo*
*rel [13].
This includes the definition of p-pro-finite completion.
A pro-finite set X is a filtered inverse limit X = {Xff} of finite sets Xf*
*f, and a
morphism of pro-finite sets X = {Xff} ! Y = {Yff} is a morphism in the pro-cate*
*gory_
that is, an element in
Hom pro(X; Y ) = limficolimffHomsets(Xff; Yfi):
For X a pro-finite set, the inverse limit limffXffacquires a topology by giving*
* each Xffthe
discrete topology and requiring that
Y
limffXff Xff
ff
be a subspace, where the product has the product topology. Thus limffXffis com*
*pact
and totally disconnected. Let bEbe the category of pro-finite sets and F the ca*
*tegory of
compact totally disconnected topological spaces and continuous maps. For the fo*
*llowing
compare [14].
Lemma 1.1. The functor lim: bE! F is an equivalence of categories.
Proof: Let Z 2 F. Define R(Z) to be the set of equivalence relations R Z x Z
so that R is open in Z x Z. Then {Z=R}R(Z) is a pro-finite set and Z ! limZ=R*
* is
R(Z)
an isomorphism. Also if X 2 bE, then the relations defined by the projections *
*X ! Xff
are contained in R(limffXff), so X is pro-isomorphic to {limffXff=R}R(limXff). *
*Finally the
ff
assignment Z ! {Z=R}R(Z) is a functor, for if f : Z ! Z0 is continuous, (f x f)*
*-1 defines
a map R(Z0) ! R(Z) and hence a pro-map {Z=R}R(Z) ! {Z0=R}R(Z0).
Because of this result one often confuses X 2 bEwith limffXff2 F. I hope th*
*e context
makes clear which I'm referring to.
Now let bSbe the category of simplicial pro-finite sets; one can identify t*
*his with the
category of simplicial compact totally disconnected topological spaces. There i*
*s a closed-
model category structure on bS, but before giving the details, let us remark th*
*at bSis a
3
simplicial category in the sense of Quillen [15, II x2]. To see this, first no*
*te that the
"forgetful functor" to simplicial sets
(1:2:1) | . | : bS! S
which sends X to limffXffwithout the topology has a left adjoint given by compl*
*etion:
(1:2:2) bY= {Yff}
where Yffruns over all the quotients of Y that are level-wise finite in the sen*
*se that for all
n the set of n simplices is finite.
To define the simplicial structure on bS, let X 2 bSand K 2 S and define
(1:3:1) X K = X x bK
in bS. For fixed K, the functor X 7! X K from bSto itself has right adjoint X*
* 7!
hom (K; X). If K has finitely many non-degenerate simplices,
(1:3:2) hom (K; X) = {map (K; Xff)}:
For more general K, one must proceed as in [15, II x2]. Finally, there is an e*
*xternal
mapping space functor
map (.; .) : bSopx bS! S
with n-simplices given by the formula
(1:3:3) map (X; Y )n = Hom bS(X n; Y );
where n is the usual n-simplex.
Remark 1.4: Let * 2 bSbe the terminal object, the one-point space, and let X 2 *
*bS.
Then, as always, one should distinguish between hom (*; X) ~=X and
map(*; X) ~=|X| 2 S:
As above |X| = limffXff, neglecting the topology.
We next turn to the model category structure on bS. For X 2 bS, define the *
*continuous
cohomology with Fp coefficients by
(1:5) H*cont(X; Fp) = colimffH*(Xff; Fp)
4
This has a description as the cohomology of continuous cochains as follows: reg*
*ard X as
a simplicial compact, totally disconnected space and set
Cncont(X) = Hom cont(Xn; Fp):
Then C*cont(X) is a cosimplicial vector space and
H*cont(X) ~=ss*C*cont(X) ~=H*(C*cont(X); (-1)idi):
If X 2 S, then
Cn (X) = Hom sets(Xn; Fp) ~=Hom cont(Xb; Fp);
whence
(1:6) H*X ~=H*cont(Xb):
Now define a morphism f : X ! Y in bSto be
1.7.1)a weak equivalence if H*f : H*contY ! H*contX is an isomorphism;
1.7.2)a cofibration if it is a level-wise injection_that is, a level-wise injec*
*tion of simplicial
topological spaces; and
1.7.3)a fibration if it has the right lifting property with respect to all triv*
*ial cofibrations.
As always, a trivial cofibration is a cofibration which is also a weak equi*
*valence. Also a
morphism f : X ! Y has the right lifting property with respect to a morphism g *
*: A ! B
if any lifting problem
A _____wX
g| ""] |f
|u" |u
B _____wY
can be solved in such a way that both triangles commute.
Morel's result is:
Theorem 1.8. With these definitions of weak equivalence, cofibration, and fibra*
*tion, bS
becomes a simplicial model category.
Proof: The fact that bSis a closed model category is Theoreme 1 of [13]. This l*
*eaves only
Axiom SM7, relating the simplicial structure to the model category structures. *
*But this
follows easily from SM7b [15, II x2], which is a triviality in this case.
We will devote some space below to clarifying what the fibrations and fibra*
*nt objects
of bSare. But let us now describe the homotopical properties of the completion *
*functor.
5
Lemma 1.9. 1) The completion functor (.)b : S ! bSpreserves cofibrations and se*
*nds
H*(.; Fp) isomorphisms to weak equivalences.
2) The forgetful functor | . | : bS! S preserves fibrations and weak equiva*
*lences among
fibrant objects.
Proof: The second statement is a formal consequence of the first. To the first*
*, note
that (.)b preserves injections, since every inclusion of sets is split. Since c*
*ofibrations are
level-wise injections in both categories, completion preserves cofibrations. Th*
*e statement
about H*(.; Fp) isomorphisms follows from 1.6.
An easy consequence of Quillen's Theorem [15, I x4] and Lemma 1.9 is:
Corollary 1.10. The functors (.)band |.| induce an adjoint pair of total derive*
*d functors
L (.)b: Ho (S ) ! Ho (bS) : R | . |:
Remarks 1.11: 1) Since every object of S is cofibrant, L(X)b ' bX. However, not*
* every
object of bSis fibrant. To obtain a model for bS, choose a H*cont(.) isomorphi*
*sm X ! Y
with Y fibrant; then |Y | 2 S is a model for R |X|.
2) By [1] there is a simplicial model category structure on S where f : X !*
* Y is a
cofibration if it is a level-wise inclusion and a weak equivalence if H*f is an*
* isomorphism.
The fibrant objects are the Bousfield-local spaces. Hence Lemma 1.9.1 implies *
*|X| is
Bousfield local if X 2 bSis fibrant.
We now define the p-pro-finite completion of a space X.
Definition 1.12. Let X 2 S . Then the p-pro-finite completion of X is an H*co*
*nt(.)
isomorphism bX! Xp with Xp fibrant in bS.
Of course, Xp is uniquely defined up to homotopy equivalence, |Xp| is a mod*
*el for
R |Xb| and the map
j : X ! |Xp|
in S can be called the completion map. On the level of homotopy categories the *
*morphism
j is a model for the unit of the adjunction X ! R |L (X)b|.
In section 3 we will point out that the choice Xb ! Xp can be made functori*
*ally;
alternatively, one could see this fact by noting that Morel actually proves tha*
*t bS has
functorial factorizations in the sense of [5].
We now turn to a discussion of fibrant objects. We begin by extending the d*
*efinition
of continuous cohomology. Let M be a pro-finite abelian group. Then if X 2 bS*
*, the
continuous cochains C*cont(X; M) are defined by
(1:13:1) Cncont(X; M) = Hom bE(Xn; M)
6
and the continuous cohomology of X with coefficients in M are given by
(1:13:2) H*cont(X; M) = H*C*cont(X; M):
Now, if X = {Xff} and M = {Mfi}, one has
Cncont(X; M)= limficolimffHomE((Xff)n; ; Mfi)
limfiCncont(X; Mfi)
so one gets a spectral sequence
(1:14) limfi(p)Hqcont(X; Mfi) ) Hp+qcont(X; M)
and this has the following consequences:
Lemma 1.15. 1) If X 2 S is a space, there is a spectral sequence
limfi(p)Hq(X; Mfi) ) Hp+qcont(Xb; M)
2) If X 2 S has only finitely many simplices in each degree,
limfiHq(X; Mfi) ~=Hpcont(Xb; M):
Proof: In the first clause, one has Hq(X; Mfi) = Hqcont(Xb; Mfi). In *
* the second,
limfi(p)Hq(X; Mfi) = 0 for p > 0 by [11, 1.1], since Hq(X; Mfi) is a finite abe*
*lian group
for all q and fi.
The next observation is that (normalized) continuous cochains and continuou*
*s coho-
mology are representable functors. The proofs are the usual ones.
Briefly, let sAb be the category of simplicial abelian groups and cAb the c*
*ategory of
chain complexes of abelian abelian groups. The normalization functor N : sAb ! *
*cAb is
an equivalence of categories with inverse K. For an abelian group A, let L(A; n*
*) 2 sAb be
the object with ae
NL(A; n)k = A;0;k =kn;6n=+n1
and @ = 1 : NL(A; n)n+1 ! NL(A; n)n. Then for X 2 S, there is a natural isomorp*
*hism
NCn (X; A) ~=Hom S(X; L(A; n))
7
where NCn (X; A) denotes the normalized cochains. If K(A; n) 2 sAb has NK(A; n)*
* ~=A
concentrated in degree n, then
Zn (X; A) ~=Hom S(X; K(A; n))
where Zn (X; A) NCn (X; A) are the cocycles. Finally, interpreting chain homot*
*opies as
simplicial homotopies yields
[X; K(A; n)] ~=Hn (X; A):
If M is a pro-finite abelian group, L(M; n) and K(M; n) are simplicial pro-fini*
*te abelian
groups and the analogous statements hold.
Lemma 1.16. For X 2 bSand a pro-finite abelian group M, there are natural isomo*
*rphisms
Hom bS(X; L(M; n))~=NCncont(X; M)
Hom bS(X; K(M; n))~=Zncont(X; M)
[X; K(M; n)]bS~=Hncont(X; M):
Proof: Only the statement about homotopy classes requires clarification. Again*
*, one
interprets a (continuous) simplicial homotopy as a continuous chain homotopy.
Strictly speaking, one shouldn't calculate [X; K(M; n)]bSunless K(M; n) is *
*fibrant.
We will see below that K(M; n) is fibrant if M is a pro-finite abelian p-group.*
* For more
general M, a slight modification of the model category structure (so f : X ! Y *
*is a weak
equivalence if H*contf : H*cont(Y; Z=`Z) ! H*cont(X; Z=`Z) is an isomorphism fo*
*r all primes
`) makes K(M; n) fibrant. See [13], after Theoreme 1.
A useful corollary of Lemmas 1.15 and 1.16 is:
Lemma 1.17. Let M be a pro-finite abelian group. Then
ss*|K(M; n)| ~=limffMff
concentrated in degree n.
Proof: Note that |K(M; n)| = limffK(Mff; n) without the topology is a simplicia*
*l group.
Hence for t > 0
sst|K(M; n)| = [St; |K(M; n)|]S~=[St; K(M; n)]bS
~=Hncont(St; M)
since St has only finitely many simplices in each degree. Similarly, for t = 0,*
* sst|K(M; n)| =
Hncont(*; M). Now apply Lemma 1.15.2.
8
That said, we can use these objects to give some feel for the fibrations in*
* bS. Note that
the coboundary
@ : NCncont(X; M) ! Zn+1cont(X; M)
defines a map of pro-finite simplicial abelian groups
L(M; n) ! K(M; n + 1):
The fiber is K(M; n).
Lemma 1.18. Let M be a pro-finite abelian p-group. The for all n 0, K(M; n) is*
* fibrant
and L(M; n) ! K(M; n + 1) is a fibration.
Proof: This is a variant of [13], x1.4, Lemma 2. The proof is the same, since n*
*ormalization
is exact.
As mentioned above, the restriction to p-groups is necessary since we defin*
*ed weak
equivalences using only mod p cohomology.
All fibrations may be characterized as follows. Define a class of morphism*
*s to be
saturated if it is closed under isomorphisms, products, pull-backs, retracts, a*
*nd countable
inverse limits. This last means that given a tower of morphisms
. .!.Xs ! Xs-1 ! . .!.X1 ! X0
in the class, then limXs ! X0 is in the class. The class of fibrations in a clo*
*sed model
category is saturated. The saturation of a class of morphisms is the smallest *
*saturated
class containing those morphisms.
Lemma 1.19. The fibrations in bSare at once
1)the saturation of the class of morphisms L(M; n) ! K(M; n + 1) n 0 and, *
*with
M a pro-finite abelian p-group;
2)the saturation of the set of morphisms, L(Z=p; n) ! K(Z=p; n + 1), n 0.
Proof: This is implicit in the construction of fibrations given in [13] x1.4, P*
*roposition 1.
We close with a discussion of how p-pro-finite completion behaves with resp*
*ect to
coproducts. This requires a very short discussion of connectedness in bS.
`
A pro-finite space X 2 bSis connected if any isomorphism X1 X2 ~=X in bSi*
*mplies
X1 = ; or X2 = ;. A connected component X Y is a maximal connected sub-object *
*in
9
bS. Let X 2 bSbe arbitrary and I = {Xi} be the set of connected components of X*
*. Then
a
the usual arguments imply Xi~= X and there is an isomorphism
I
I 7! Hom Fp-alg(H*contX; Fp)
sending Xi to
H0contX ! H0contXi~= Fp:
Thus if f : X ! Y is any trivial cofibration in bS, f induces an isomorphism fr*
*om the set
of connected components of X to the connected components of Y and we have:
Lemma 1.20. A morphism in bSis a fibration if and only if it has the right lift*
*ing property
with respect to all trivial cofibrations X ! Y with X connected.
We know that Bousfield-Kan completion commutes with all disjoint unions. T*
*he
following is the best we can do for p-pro-finite completion.
Proposition 1.21. Let Xi, 1 i n, be a finite set of spaces. Then there is a *
*weak
equivalence
na an
(Xi)p ! ( Xi)p:
i=1 i=1
Proof: Let bXi! Yi be a set of weak equivalences with Yi2 bSfibrant. Then the n*
*atural
map
an an
H*( Xi) H*cont( Yi)
i=1 i=1
*
*an
is an isomorphism, since the disjoint union is finite. Thus we need only show *
* Yi is
*
*i=1
fibrant. But this follows from Lemma 1.20.
2 The homotopical background of Bousfield-Kan completion.
As an exercise in using new thoughts to understand old ones, we show how th*
*e ideas
of the previous section can be specialized to show how the Bousfield-Kan p-comp*
*letion of a
space can be obtained as the inverse limit of a fibrant object in a suitable mo*
*del category
structure on towers of simplicial sets. This has the aesthetic advantage of ma*
*king the
important tower lemmas of [3, xIII.6] a straightforward consequence of standard*
* model
category facts. For the sake of brevity, we will only sketch certain proofs.
10
One way to interpret this section is as follows: it is a useful idea of Dro*
*r-Farjoun's
[4] that one can regard the Bousfield-Kan compeletion as a pro-object. What we *
*do here
provides a homotopical foundation for this observation.
Let Tow be the category of towers X = {Xn} of simplicial sets with morphis*
*ms the
pro-maps
Hom (X; Y ) = limncolimkS(Xk; Yn):
Thus a morphism X ! Y is an equivalence class of tower maps, meaning a "commuta*
*tive
ladder" of the form
. . .____wXks+1_____wXks _____w.X.k.2____wXk1
| | | |
|u |u |u |u
. . .____wYns+1 _____wYns _____w.Y.n.2____wYn1
where the horizontal maps are induced from the tower projections of X and Y and*
* limks =
1 = limns.
The first thing to notice is that Tow is a simplicial category. Indeed if*
* K 2 S and
X 2 Tow , then the functors
(2:1:1) X 7! X K = {Xn x K}
and
(2:1:2) X 7! XK = {map S(K; Xn)}
are adjoint and the mapping space functor map : Tow x Tow op! S with N simpli*
*ces
(2:1:3) map (X; Y )n = Hom Tow (X n; Y )
completes the structure.
There is also a notion of continuous cohomology with Fp coefficients. If X*
* 2 Tow ,
define
(2:2) H*contX = H*cont(X; Fp) = colimnH*(Xn; Fp):
The notation is justified by the observation that if limXn is topologized with *
*the inverse
Q
limit topology (i.e., limXn is a subspace of Xn, where each Xn has the discre*
*te topology),
thus H*contX can be computed using continuous cochains on limXn.
Now define a morphism f : X ! Y in Tow to be
2.3.1)a weak equivalence if H*contf : H*contY ! H*contX is an isomorphism;
2.3.2)a cofibration if limXn ! limYn is an inclusion;
2.3.3)a fibration if it has the right lifting with respect to all trivial fibra*
*tions.
11
Theorem 2.4. With the simplicial structure of (2.1) and the notion of weak equi*
*valence,
cofibrations, and fibration of (2.3), the category Tow becomes a simplicial mo*
*del category.
The proof is essentially the same as that given by Morel for bS. The releva*
*nt obser-
vations are, first, that Hkcont(.) is a representable homotopy functor and, sec*
*ond, that if
V = {Vn} is any tower of Fp vector spaces, then the map L(V; k) ! K(V; k) in T *
*ow is a
fibration. The representing object for Hkcont(.) is K(Z=pZ; k) regarded as a co*
*nstant tower.
Note that if X is a constant tower H*contX = H*X.
Definition 2.5. Let X 2 S be a space. Then we may regard X as a constant tower *
*in
Tow . Define the Bousfield-Kan p-completion (Fp)1 X of X to be a fibrant repla*
*cement
for X in Tow .
Thus (Fp)1 X is well-defined up to weak equivalence in Tow . As with the p-*
*pro-finite
completion one somtimes confuses the difference between (Fp)1 X 2 Tow and its *
*inverse
limit; in fact, I've already done so, as Bousfield and Kan define (Fp)1 X to as*
* an inverse
limit. I hope that, in context, this confusion will cause no difficulty.
We now reprove one of Bousfield and Kan's tower lemmas, using Theorem 2.4.
Proposition 2.6. (Fp-nilpotent tower lemma, [3], p. 88). Let X 2 S be regarded*
* as a
constant tower and X ! Y a map of towers so that
1)H*X ! H*Y = {H*Yn} is a pro-isomorphism;
2)for all n, Yn is an Fp-nilpotent space.
Then Y is weakly equivalent to the Bousfield-Kan p-completion of X.
Proof: The first statement is equivalent to the assertion that
H*X ~=H*contX ~=H*contY;
so we need only show that the second statement implies Y is fibrant. Fix n and *
*let Y
be the tower with ae
Y k = Yn* kk n< n:
Then, by Proposition 5.3iii of [3, p. 83] (a refined Postnikov tower for Fp nil*
*potent spaces
given at the beginning of section 5) and the fact that L(V; k) ! K(V; k) is a f*
*ibration in
Tow for all towers of vector spaces V , the tower Y is fibrant in Tow . No*
*w Y = limnY ,
and since a directed inverse limit of fibrant objects is fibrant, Y is fibrant *
*in Tow .
12
3 A cosimplicial description of p-pro-finite completion.
In this section, we interpret a construction of Morel's to describe Xp as t*
*he total
space of a cosimplicial p-pro-finite space. We describe the E2 terms of the res*
*ulting homo-
topy spectral sequence and use this to compare Bousfield-Kan p-completion to p-*
*pro-finite
completion.
Let X 2 bSbe a simplicial pro-finite set. Define FpX by the equation
(3:1) FpX = {FpXff}
where FpXffis the simplicial vector space on Xff.
Note that FpX is a simplicial pro-finite vector space and the functor X ! F*
*pX is left
adjoint to the forgetful functor from bSto the category of simplicial pro-finit*
*e Fp vector
spaces. We can also forget the vector space structure and obtain a functor
__
Fp(.) : bS! bS:
Thus, by the usual properties of adjoint functors, is the functor of a triple o*
*n bS; hence for
each X 2 bSwe obtain an augmented cosimplicial object in bS
__o
(3:2) X ! FpX
__ __ __
where (FopX)s = F pO . .O.FpX, the composition taken s + 1 times. The cosimpli*
*cial
__
object FopX is a resolution of X in the sense that, applying functor Fp degree-*
*wise yields
an augmented cosimplicial pro-finite vector space
__o
FpX ! Fp(FpX)
which comes equipped with a canonical contraction.
__
Morel's next observation is that FopX can be used to build the p-pro-finite*
* completion
on X. To state the result, note that bSis a simplicial model category, it has *
*homotopy
colimits in the style of Bousfield and Kan. In particular, if Zo is a cosimpli*
*cial object
in bS, the coequalizer diagram in bSone can form the total space Tot (Zo) defin*
*ed by the
coequalizer diagram in bS
Y Y
(3:3) Tot (Zo) ! hom (n; Zn ) !! hom (n; Zm )
[n] [n]![m]
where n 2 S is the canonical n-simplex and hom (n; Zn ) is the internal mapping*
* space
functor
hom (n; Zn ) = {hom (n; Znff)}:
13
The second product is over all morphisms in the ordinal number category .
To further analyze the structure of Tot (Zo) as an object of bS, let o be t*
*he standard
cosimplicial space which is k in cosimplicial degree k and let skn be the nth s*
*keleton
functor. If Zo is a cosimplicial object in bS, then Zo = {Zo=R}R2R(Zo), using t*
*he notation
of Lemma 1.1. For all n < 1, let
Tot n Zo = map cbS(skno; Zo) = {map cS(skno; Zo=R)}R2R(X) :
Since skno is generated, as a cosimplicial space, by the canonical n-simplex n *
*2 (n)n,
Tot n(Zo) is an object in bSand there is an isomorphism in bS
(3:4) Tot (Zo) ~=limnTotn(Zo):
To obtain the p-pro-finite completion, we have
Proposition 3.5. Let X 2 bS be a simplicial pro-finite set. Then the induced *
*map
__ __
X ! Tot (FopX) is weak equivalence and Tot (FopX) is fibrant. If X = bY, where *
*Y is a
space, then
__o
Y 7! Tot (FpYb)
is a model for the p-pro-finite completion of Y .
Proof: The second clause follows from the first and the definition p-pro-finite*
* completion.
__
To see X ! Tot (FopX) is a cohomology isomorphism note that
__o
H*contX ~=colimffH*Xff~=colimffcolimnH*Tot n FpXff
~=H*contTot(__FopX):
__ *
* __
Thus one need only show Tot (FopX) is fibrant. For this it is sufficient to sh*
*ow FopX
is fibrant in the Reedy model category structure on cosimplicial objects in bS.*
* But the
standard argument [3, p. 276] shows that the canonical map to the nth matching *
*object
__n __o
s : FpX ! MnF pX
is a split surjection of simplicial pro-finite vector spaces. The result now f*
*ollows from
Lemma 1.18.
This resolution is a key tool in the investigation of p-pro-finite completi*
*on. We begin
with the following preliminary result. Let | . | : bS! S be the forgetful funct*
*or.
14
Lemma 3.5. Let Zo be a cosimplicial object in bS. Then there is a natural isomo*
*rphism
|Tot (Zo)| ~=Tot (|Zo|):
Proof: If K 2 S has only finitely many simplices, then for all X 2 bS,
| hom(K; X)|~= limmap (K; X=R)
R(X)
~=map (K; |X|):
Since | . | is a right adjoint it preserves products and equalizers. The result*
* now follows by
applying | . | to the diagram of (3.3).
An object X 2 bS is pointed if it comes equipped with a map * ! X in bSfrom
the one-point space; this amounts to choosing a basepoint for |X|. A cosimplici*
*al object
Zo is pointed if there is a map * ! Zo from the constant one-point cosimplicial*
* object
to Zo. This amounts to choosing a basepoint for |aZo| where aZo is the equaliz*
*er of
d0; d1 : Z0 ! Z1. If Zo is pointed, there is a homotopy spectral sequence
(3:7) ssssst|Zo| ) sst-sTot (|Zo|) ~=sst-s|Tot (Zo)|:
For the spectral sequence to make homotopical sense, |Zo| must be fibrant in th*
*e sense of
[3], X x4. Thus we have the extremely technical:
Lemma 3.8. Let Zo be a cosimplicial object in bS. If Zo is fibrant in the Reed*
*y model
category structure on cosimplicial objects in bS, then |Zo| is fibrant as a cos*
*implicial space.
Proof: One needs
|s| : |Zn | ! Mn|Zo|
to be a fibration. By hypothesis s : Zn ! MnZo is a fibration in bS, and Lemma*
* 1.9.2
says | . | preserves fibrations. Now one need only observe that Mn|Zo| ~=|MnZo|*
* since the
matching space is an inverse limit.
__
This lemma applies, in particular, to FopX, X 2 bS. See the proof of Propos*
*ition 3.5.
__
We next interpret the E2 term ssssstFopX.
Lemma 3.9. Let W = {Wff} be a simplicial pro-finite vector space. Then
ss*|W | ~=limffss*|Wff|:
Proof: Every short exact sequence of pro-finite vector spaces is split. Hence *
*there are
pro-finite vector space Un, n 1, Vn, n 0 and a (non-natural) isomorphism
Y Y
W ~= L(Un; n) x K(Vn; n):
n n
15
The result now follows from Lemma 1.17.
In this result, we chose 0 2 |W | as the basepoint; however, because |W | i*
*s simplicial
group, the result is equally true for any choice of basepoint.
__
We now use Lemma 3.9 to compute sstFpX, X 2 bS. The result is similar to t*
*he
non-pro-finite case (see, for example [13] x1 for a lucid exposition); therefor*
*e, I will be
brief.
Let K be the category of unstable algebras over the Steenrod algebra. If X *
*2 bS, then
H*contX 2 K. Let K* be the category of augmented objects in K, i.e., objects H*
* 2 K
equipped with a morphism ffl : H ! Fp. If X 2 bSis pointed, then H*contX 2 K*.*
* The
augmentation ideal functor from K* to graded vector spaces has a left adjoint G*
*, and we
__ __
let G : K* ! K* be the composite functor. Then G is the functor of a cotriple o*
*n K*. If
__ __
X 2 bSis pointed, then the unit X ! FpX gives FpX a basepoint.
__
Lemma 3.10. Let X 2 bSbe pointed. Then with the induced basepoint on FpX, there*
* is
a natural isomorphism
__ __ * * t
sstFpX ~=Hom K* (G HcontX; H S ):
Proof: Applying Lemma 3.9 yields
__
sstFpX ~=sstFpX ~= lim sstFp(X=R):
R(X)
Then we have natural isomorphisms
__ * * t
lim sstFp(X=R)~= lim Hom K*(G H (X=R); H S )
R(X) R(X)
~= Hom K*(__GH*contX; H*St)
__
since G commutes with filtered colimits.
__ __
Now, if H 2 K*, let G oH ! H be the cotriple resolution H induced by G and *
*define
__ * t
Ext sK*(H; H*St) = sssHom K* (G oH; H S ):
If t = 0, this is defined only for s = 0.
Proposition 3.11. Let X 2 bSbe pointed. Then the homotopy spectral sequence of *
*the
__
cosimplicial space |FopX| is
__o
Ext sK*(H*contX; H*St) ) sst-s|Tot (FpX)|:
Proof: Combine the spectral sequence of 3.7 with Zo = FopX with Lemma 3.10.
16
Corollary 3.12. Let Y 2 S be a pointed space. Then there is a homotopy spectr*
*al
sequence
Ext sK*(H*Y; H*St) ) sst-s|Yp|:
__
Proof: One has H*Y = H*contbYand |Tot (FopbY)| = |Yp| by Proposition 3.5.
One can use these tools to give a comparison between the Bousfield-Kan and *
*p-pro-
finite completions of a space Y . For Y 2 S, let FpY be the simplicial vecto*
*r space on
__
Y , and let Fp : S ! S be the resulting triple. Then the augmented cosimplicia*
*l space
__ __
Y ! FopY is fibrant and the induced map Y ! Tot (FopY ) = (Fp)1 Y is the Bousfi*
*eld-Kan
p-completion of Y by [3], p. 88. By the same reasoning that arrived at Proposit*
*ion 3.11,
__
the homotopy spectral sequence of FopY is the Bousfield Bousfield-Kan spectral *
*sequence
__o
(3:13) Ext sCA*(H*St; H*Y ) ) sst-sTot (FpY ):
Here CA* is the category of augmented unstable coalgebras over the Steenrod*
* algebra.
The claim is that one can compare this spectral sequence to the one of Corollar*
*y 3.12.
Dualization induces a map
Hom CA*(H*St; H*Y ) ! Hom K*(H*Y; H*St):
This extends to a map of derived functors
(3:14) Ext sCA*(H*St; H*Y ) ! Ext sK*(H*Y; H*St):
This can be realized homotopically as follows. Let bY = {Yff} be the completio*
*n of Y .
__ __
Then the induced maps FpY ! FpYffdefine a map of cosimplicial spaces FopY ! |Fo*
*pbY|
and one gets:
__ __
Proposition 3.15. Let Y 2 S be pointed. Then the map FopY ! |FopbY| induces a m*
*ap
of homotopy spectral sequences
Ext sCA*(H*St;?H*Y )==) sst-s(Fp)1 Y
y ?y
Ext sK*(H*Y; H*St) ==) sst-s|Yp|:
A space Y is of finite type if HkY is finite for all k. If Y is of finite t*
*ype, so is FpY
and this implies that
Ext sCA*(H*St; H*Y ) ~=Ext sK*(H*Y; H*St):
Corollary 3.16. Let Y 2 S be of finite type. Then the natural map of completions
(Fp)1 Y ! |Yp|
is a weak equivalence.
17
Proof: If Y is connected, this follows from Proposition 3.15 and the mapping l*
*emma
of [3], p. 285. In general, Y will have only finitely many components, so one*
* can apply
Proposition 1.21 and [3] xI.7.1.
Remark 3.17: One of the main points of this section was to discuss the comparis*
*on of
spectral sequences of Proposition 3.15, and Corollary 3.16 was a by-product. A*
* shorter
non-functorial route to this last result is as follows: for Y of finite type, d*
*efine a tower of
fibrations
. .!.Ys!qsYs-1 . .!.Y2 q2!Y1 q1!Y0
and a map fs : Y ! Ys so that
1)qsfs = fs-1,
2)each Ys is of finite type, Y0 is a product of Eilenberg-MacLane spaces of*
* type
K(Z=p; n), and there is a homotopy pull-back square for s 0,
Ys+1 ______w*
| |
|u |u
Ys ______wsKs
where Ks is also a product of Eilenberg-MacLane space of type K(Z=p; n) a*
*nd of
finite type and
3)H*Ks ! H*Ys ! H*Y is exact in the sense that
Fp H*Ks H*Ys ~=H*Y:
Then colimH*Ys ~=H*Y and Lemma 1.19 implies limYs is weakly equivalent to |*
*Yp|.
On the other hand, the tower lemma Proposition 2.6 implies limYs ' (Fp)1 Y .
4 The p-pro-finite completion of abelian groups.
In this section we discuss the derived functors of p-pro-finite completion.
Let Ab be the category of abelian groups and Ab pthe category of pro-finite*
* abelian
p-groups. This category is equivalent, via inverse limit, to the category of co*
*mpact, totally
disconnected abelian groups all of whose open subgroups have p-power index. Th*
*ere is
also an extremely useful Pontrjagin duality. Let T Abp be the category of p-tor*
*sion abelian
groups. Then if A 2 T Abp, the dual
(4:1:1) A* = Hom Ab(A; Z=p1 )
18
is naturally an object in Abp, since every torsion group is the filtered colimi*
*t of its finite
subgroups. Conversely, if B = {Bff} 2 Abp, then the continuous dual
(4:1:2) B# = colimffHomAb(Bff; Z=p1 )
is naturally object in T Abp and the functors (.)* and (.)# induce an equivalen*
*ce of cate-
gories.
The forgetful functor | . | : Abp ! Ab given by |B| = limffBff(without topo*
*logy) has
left adjoint (A) = {Aff} where the filtered system has objects all surjections *
*A ! Aff
with Affa finite abelian p-group. We call this functor p-pro-finite completion.*
* If we regard
(A) as a compact totally disconnected topological group_that is, take (A) = lim*
*ffAff
with the topology_then the unit of the adjunction A ! |(A)| really is a complet*
*ion map
A ! limffAff.
It is possible to give a simple description of the Pontrjagin dual of (A). *
*If A is an
abelian group, let
T Hom (A; Z=p1 ) = colimnHom(A; Z=pnZ):
This is the p-torsion in Hom (A; Z=p1 ).
Lemma 4.2. Let A be an abelian group. Then there is a natural isomorphism
(A)# ~=T Hom (A; Z=p1 ):
Proof: For if we write (A) = {Aff} where Affis a finite abelian p-group
T Hom (A; Zp1 )~=colimnHom(A; Z=pnZ)
~=colimHom Ab ((A); Z=pnZ)
n p
~=colimcolim Hom (Aff; Z=pnZ)
n ff
~=colimHom (Aff; Z=p1 Z) = (A)# :
ff
It follows that, in contrast to p-completion, p-pro-finite completion is right *
*exact: indeed,
since Zp1 is an injective abelian group, T Hom (A; Z=p1 ) is left exact.
Now let Ls : Ab ! Ab pbe the left derived functors of . Since is right ex*
*act,
L0 ~= and, of course, Ls = 0 for s > 1. If we let T : Ab ! Ab be the functor wh*
*ich
assigns to an abelian group its torsion sub-group, T is left exact and has righ*
*t derived
functors RsT ; again, R0T ~=T and RsT = 0 for s > 1.
19
Proposition 4.3. For all A 2 Ab, there is a natural isomorphism
(Ls(A))# ~=(RsT )Hom (A; Z=p1 ):
Proof: The case s = 0 is Lemma 4.2. Also, if A is free abelian, Hom (A; Z1p) *
*is an
injective abelian group, so one has
Ls(A) = 0 = (RsT )Hom (A; Zp1 )
for s > 0. The result follows from general homological algebra.
Example 4.4: If A = Z=p1 , then Hom (A; Z=p1 ) ~=Zp, the p-adic integers. The s*
*hort
exact sequence
0 ! Zp ! Qp ! Z=p1 ! 0
show (R1T )Hom (A; Z=p1 ) = Z=p1 and T Hom (A; Z=p1 ) = 0. Hence (Z=p1 ) = 0 *
*and
L1(Z=p1 ) ~=Zp.
We would now like to compare p-pro-finite completion to the p-completion fu*
*nctor
(A) = limnA=pnA ~=limn(Z=pnZ A):
Because lim is only left exact and tensor product is only right exact is neith*
*er left nor
right exact; nonetheless, still has left derived functors Ls. While Ls = 0 for*
* s > 1,
one can only claim there is a natural surjection L0 ! . In fact, one has
Lemma 4.5. There is a natural exact sequence
0 ! lim1Tor (Z=pnZ; A) ! L0(A) ! lim(Z=pnZ A) ! 0
and a natural isomorphism
lim Tor (Z=pnZ; A) ~=L1(A):
Proof: See [8]. If A is free abelian group, lim1(Z=pnZ A) = 0. This is a conse*
*quence
of the exact sequence in limiand the short exact sequence of towers
n
{0 ! A p-!A ! Z=pnZ A ! 0}:
Let F* ! A be a projective resolution. Then there is a short exact sequence of*
* chain
complexes
Y @ Y
0 ! lim(Z=pn F*) ! (Z=pnZ F*) ! (Z=pnZ F*) ! 0:
n
Analyzing the long exact sequence in homology completes the proof.
As a consequence one has the description of Ls familiar to topologists (see*
* [3], p.
166).
20
Corollary 4.6. There are natural isomorphisms
L0(A) ~=Ext (Z=p1 Z; A)
L1(A) ~=Hom (Z=p1 Z; A):
Proof: There are natural isomorphisms Tor (Z=pnZ; A) ~= Hom (Z=pnZ; A), so, *
* by
Lemma 4.5
L1(A) ~=limHom (Z=pnZ; A) ~=Hom (Z=p1 Z; A):
Since L0(A) = 0 if A is injective, the result follows.
Note that Corollary 4.6 also implies Lemma 4.5 using the composite functor *
*spectral
sequence
lim(p)Extq(Z=pnZ; A) ) Ext p+q(Z=p1 Z; A)
and the fact that Ext q(Z=pnZ; A) ~=Tor 1-q(Z=pnZ; A).
To compare p-completion (.) to p-pro-finite completion , note that any map *
*A !
Afffrom A to a finite abelian p-group factors through Z=pnZ A ! Afffor some n.*
* This
yields a map of groups
(4:7) : (A) = lim(Z=pnZ A) ! |(A)| = limffAff:
One can include the topology: (A), with the inverse limit topology, is a Hausdo*
*rff abelian
group and the natural map of topological abelian groups in (4.7) is continuous.
This can be made more explicit: by Lemma 4.2,
(A) = (T Hom (A; Z=p1 ))*
= limnHom(A; Z=pnZ)*
where * is Z=p1 duality. Evaluation defines a homomorphism
(4:8:1) ffl : Z=pnZ A ! Hom (A; Z=pnZ)*
and the morphism of (4.7) is isomorphic to
(4:8:2) ffl : limn(Z=pnZ A) ! limnHom(A; Z=pnZ)*:
This also makes it possible to describe behavior on derived functors. Since (.)*
** is exact,
evaluation (4.8.1) defines morphism
(4:8:3) Tor (Z=pnZ; A) ! Ext (A; Z=pnZ)*
and one has
21
Lemma 4.9. The natural maps of derived functors Ls(A) ! Ls(A) are isomorphic to
(s = 1) limnTor(Z=pnZ; A) ! limnExt(A; Z=pnZ)*
and
(s = 0) L0(A) ! limn(Z=pnZ A) ! limnHom(A; Z=pnZ)*:
Proof: This follows from the formulas of 4.8, the proof of Lemma 4.5, and the f*
*act that
lim1Hom (A; Z=pnZ)* = 0 since Hom (A; Z=pnZ)* is a pro-finite group.
Proposition 4.10. Let A be an abelian group with the property that Z=pZ A and
Tor (Z=pZ; A) are finitely generated. Then L0(A) ~=limn(Z=pnZ A) and the natu*
*ral
maps
Ls(A) ! Ls(A)
are isomorphisms.
Proof: By induction on n, Tor (Z=pnZ; A) is finite, hence lim1Tor (Z=pnZ; A) = *
*0. So
L0(A) ~=limn(Z=pnZ A) by Lemma 4.5. Also by induction on n, the maps
ffl : Tor s(Z=pnZ; A) ! Ext s(Z=pnZ; A)*
are isomorphisms for all s. Hence the result follows from Lemma 4.9.
Examples of groups satisfying the hypothesis of Proposition 4.10 include fi*
*nitely
generated groups, their p-completions, and finite direct sums of the standard i*
*njectives
Q; Q=Z; Zp1 ; Qp, etc.
The p-complete abelian groups do not form an abelian sub-category of the ca*
*tegory
of all abelian groups. The smallest abelian sub-category containing the p-compl*
*ete groups
is the category of Ext -p-completehgroupsi(or p-cotorsionhgroups);ithat is, tho*
*se abelian
groups A so that Ext (Z 1_p; A) = 0 = Hom (Z 1_p; A). A clear explanation o*
*f this
difficulty is in [10, x4].
As a final note we record
Lemma 4.11. Let A be an abelian group.
1)The p-pro-finite completion A of A is Ext -p-complete.
2)The group L1A is torsion free.
22
Proof: For 1) write A = limffAffwhere Affare finite abelian p-groups. Then ther*
*e is a
spectral sequence
limpffExtq(Z 1_p; Aff) ) Ext p+q(Z 1_p; A):
But the E2 term is zero.
For 2), Proposition 4.3 implies L1(A)# is divisible. Hence L1(A) is torsion*
* free.
5 The homotopy groups of the p-pro-finite completion of a nilpotent space.
Let X be a pointed, connected nilpotent space. Then there is a tower of pr*
*incipal
fibrations
. ._.___wXsu_____wXs-1 . . .__wX2u___wX1 = K(A1; n1)
| |
| |
K(As; ns) K(A2; n2)
equipped with a weak equivalence X ! limsXs and so that slim!1ns = 1. Using thi*
*s fact
and the results of the previous section, we give a formula for calculating ss*|*
*Xp|.
We begin with the case of an Eilenberg-MacLane space. Let Ls denote the der*
*ived
functors of p-pro-finite completion of abelian groups.
Lemma 5.1. Let A be an abelian group and n 1. then there are natural isomorphi*
*sms
ssn|K(A; n)p|~=(A)
ssn+1|K(A; n)p|~=L1(A)
and ssk|K(A; n)p| = 0 if k 6= n or n + 1.
Proof: Recall that K(A; n) is the simplicial abelian group whose normalization *
*NK(A; n)
~=A in degree n. Define a simplicial abelian group Kc(A; n) (c for cofibrant) a*
*nd a weak
equivalence Kc(A; n) ! K(A; n) as follows. Choose a free resolution
0 ! F1 ! F0 ! A ! 0
of A as an abelian group and let Kc(A; n) be the simplicial abelian group with *
*normaliza-
tion on the chain complex F1 ! F0 with F0 in degree n. Then there is a weak equ*
*ivalence
Kc(A; n) ! K(A; n). Hence Kc(A; n)p ! K(A; n)p is a weak equivalence. Let Kc(A;*
* n)
be the simplicial p-pro-finite obtained by applying the p-pro-finite completion*
* functor level-
wise. Since normalization is exact, this result follows from the next.
23
Lemma 5.2. The simplicial p-pro-finite abelian group Kc(A; n) is a model for th*
*e p-pro-
finite completion of Kc(A; n).
Proof: By Lemmas 1.18 and 1.19, Kc(A; n) is fibrant. Thus we need only show tha*
*t the
map Kc(A; n)^ ! Kc(A; n) adjoint to Kc(A; n) ! |Kc(A; n)| induces an isomorphism
H*Kc(A; n) ~=H*contKc(A; n). To do this we introduce a spectral sequence.
If V is a cosimplicial vector space and (.) is the exterior algebra functor*
*, there is a
functor G0(.) on graded vector spaces so that
G0(ss*V ) ~=ss*(V ):
This is a result of Dold. Since commutes with filtered colimits so does G0. *
*If B is a
simplicial abelian group which is free in each degree, there is a spectral sequ*
*ence
___
(5:3) G0(ss*Hom (B; Z=pZ)) ) H*W B:
___
Here W B is the "suspension functor" on simplicial abelian groups; on normaliza*
*tions
___ ___
(NW B)k = Bk-1. In particular W Kc(A; n - 1) ~= Kc(A; n). We will construct *
*the
spectral sequence below. Assuming this to be the case, consider the map B^ ! B
adjoint to B ! |B|. Then, since Z=pZ is a finite and, hence, p-pro-finite abeli*
*an group
Hom (B; Z=pZ) ~=Hom cont(B; Z=pZ) ~=colimffHom(Bff; Z=pZ):
Hence one has a diagram of spectral sequences
___
G0(ss*Hom? (B; Z=pZ)) ==) H*W B
y ?y
___
colimffG0(ss*Hom (Bff; Z=pZ))==)colimffH*W Bff:
Since G0 commutes with filtered colimits, one has an isomorphism on E2 terms. *
*Since
___ ___
normalization is exact, W B ~=W B and we conclude
___ * ___
H*W B ~=HcontW B:
Setting B = Kc(A; n - 1) proves the result.
___
To produce the spectral sequence (5.3), form the bisimplicial group W oB *
* with
___ ___ ___ ___
(W oB)s;t= (W Bs)t. Then there is a weak equivalence diag W oB ' W B. Applying
___
the cochains functor Hom = Hom sets(.; Z=pZ) to W oB and filtering degree in s*
* yields a
spectral sequence
___ s+t___
ssssstHom (W oB) ) H W B:
___
For fixed s, ss*Hom (W Bs) ~=H*(Bs) ~=(Hom (Bs; Z=pZ)) since Bs is free. The *
*spectral
sequence follows.
We now come to the analog of Bousfield and Kan's nilpotent fiber lemma.
24
q
Proposition 5.4. Let F ! E ! B be a nilpotent fiber sequence of connected space*
*s.
Then there is a homotopy fiber sequence
Fp ! Ep ! Bp:
Proof: One can argue exactly as in [3], Chap. II. Here is another way. By [3] I*
*I.4.7, the
fibration q : E ! B has a refined Postnikov tower
. . .____wEsu_____wqsEs-1.___w._._wE2u________wq2E1u_____wq1E0 = B
| | |
| | |
K(As; ns) K(A2; n2) K(A1; n1)
so that E ' limEs and q ' limqs, each qs is a principal fibration and slim!1ns *
*= 1. If Fs
is the fiber of Es ! Bs, one gets an induced tower
. . .__wFsu_____wqsFs-1 ___w._._.wF2u_____wF1 = K(A1; n1)
| |
| |
K(As; ns) K(A2; n2)
with limFs ~=F .
Choose a weak equivalence B^ ! Bp with Bp fibrant and factor bE ! bB ! Bp as
Eb!j Ep f!Bp where j is a weak equivalence and f is a fibration. As the notatio*
*n indicates
Ep and Bp are models for the p-pro-finite completions. Let F 0be the fiber of *
*f. Then
F 0is automatically fibrant, so we need only show the induced map bF! F 0induce*
*s an
isomorphism H*F ~= H*contF 0. Suppose we knew this to be true in the case wher*
*e q is
principal with fiber K(A; n) = F . Then, we can show inductively that there is *
*a fibration
sequence (Fs)p ! (Es)p ! Bp. Indeed, with proper choices of fibrant models, one*
* has a
diagram with columns and top and bottom rows fibration sequences.
K(As; ns)p _____w'K(As; ns)p ______w*
| | |
|u |u |u
(Fs)p __________w(Es)p _______wBp
| | |=
|u |u |u
(Fs-1)p ________w(Es-1)p ______wBp
It follows that the middle row is also a homotopy fibration sequence. To see th*
*is, let (Fs)0
be the fiber of (Es)p ! Bp. Then there is a map of fibration sequences
K(As; ns)p _____w(Fs)p _____w(Fs-1)p
|= | |=
|u |u |u
K(As; ns)p _____w(Fs)0_____w(Fs-1)p
25
whence, by the Serre Spectral Sequence in bS(cf. [13], the final remark), H*co*
*nt(Fs)p ~=
H*cont(Fs0). Since both are fibrant (Fs)p ' (Fs)0.
Finally, note that lim(Es)p is a model for Ep. This is because lims(Es)p is*
* fibrant and
(because slim!1ns = 1)
H*E ~=colimsH*Es ~=colimsH*cont(Es)p ~=H*cont(lim(Es))p:
Now the fiber of lims(Es)p ! Bp is lim(Fs)p. But lim(Fs)p is a model for Fp, by*
* the same
reasoning and we are left only with the following lemma:
Lemma 5.5. Let K(A; n) ! E q!B be a principal fibration. Then
K(A; n)p ! Ep ! Bp
is a fibration sequence in bS.
Proof: Consider the principal fibration
___
K(A; n) ! W K(A; n) ! W K(A; n):
Since W K(A; n) is contractible Lemma 5.2 implies that the result holds in this*
* case; indeed,
___
Kc(A; n) ! W Kc(A; n) ! W Kc(A; n) is an exact sequence of simplicial p-pro-fin*
*ite
___
abelian groups. Let B ! W K(A; n) classify q, and define E0 by the homotopy pul*
*l-back
diagram in bS.
E0 _____wW K(A; n)p
| |
|u ___ |u
Bp _____wWK(A; n)p:
We need only show the induced map bE! E0 induces an isomorphism H*Eb ~=H*contE0.
Then E0 is a model for Ep and since the fiber of E0 ! Bp is K(A; n)p, we will b*
*e done.
To finish the argument, consider the diagram
K(A; n)^ _____wbE_____wbB
| | |
|u |u |u
K(A; n)p _____wE0 _____wBp:
This induces a map of Serre Spectral Sequences
H*(B; H*K(A;?n)) ==) H*E
y ?y
H*cont(Bp; H*contK(A; n)p)==) HcontE0:
Since the E2 terms are isomorphic, the result follows.
With this result in hand, we can make use of the following definition.
26
Definition 5.6. Let G be a group. Define the derived functors of p-pro-finite c*
*ompletion
by the formula
|Ls(G)| = sss+1|K(G; 1)p|:
By Lemma 5.1, this agrees with the usual definition if G is abelian.
Here is a model for K(G; 1)p. Choose a free simplicial resolution Xo ! G i*
*n the
category of groups and let Xo denote the level-wise p-pro-finite completion of *
*X. Now
___
there is a weak equivalence W Xo ' K(G; 1). As a consequence of Proposition 2, *
*x1.5 of
___
[13], W Xo is a fibrant. See Corollaire 1, x1.5 of [13]. Hence it is only a mat*
*ter of showing
___ ___
that H*W Xo ~=H*contWXo. Now, one has, by Kan [12], that
___ n
Hn+1 W Xo ~=ss Hom (Xo; Z=pZ):
Hence ___
Hn+1 W Xo ~=ssnHom cont(Xo; Z=pZ)
~=colimssnHom ((Xo)ff; Z=pZ))
ff
~=H*cont_WX:
Note that because is a left adjoint and limis exact on pro-finite groups,
___
(5:7) L0(G) = ss1|W Xo| ~=|ss0Xo| ~=|G|:
Lemma 5.8. 1) If G is a nilpotent group, Ls(G) = 0 for s > 1.
2) If 1 ! K ! G ! H ! 1 is a short exact sequence of nilpotent groups, there is*
* a long
exact sequence
0 ! L1K ! L1G ! L1H ! K ! G ! H ! 1:
Proof: First consider the short exact sequence of the second statement. Then
K(K; 1) ! K(G; 1) ! K(H; 1)
is a nilpotent fibration sequence. So there is a long exact sequence in L*(.).*
* If G is
nilpotent, one can use this long exact sequence to induct over the lower centra*
*l series to
show Ls(G) = 0 for s > 2.
Proposition 5.9. Let X be a connected nilpotent space. Then there is a splittab*
*le short
exact sequence for all n 1
0 ! (ssnX) ! ssn|Xp| ! L1(ssn-1X) ! 0:
27
Proof: The existence of the short exact sequence follows from the refined Postn*
*ikov
tower for X, and Lemmas 5.1 and 5.8, and the nilpotent fiber lemma Proposition *
*5.4.
The fact that sequence splits follows from [9, p. 370] because (ssnX) is Ext -p*
*-complete
and L1(ssn-1X) is torsion free. See Lemma 3.11. Note that if G is nilpotent L1(*
*G) is
torsion free by Lemma 5.8.2 and Lemma 3.11.
If X is connected and nilpotent, and (Fp)1 X is its Bousfield-Kan completio*
*n, there
is a splittable short exact sequence ([3], p. 183)
(5:10) 0 ! L0(ssnX) ! ssn(Fp)1 X ! L1(ssn-1X) ! 0
where Ls are the derived functors of p-completion. It is an easy exercise to sh*
*ow that
the induced map ss*(Fp)1 X ! ssn|Xp| fits into a map of short exact sequence 5.*
*10 to 5.9
with the natural maps Ls(ssnX) ! Ls(ssnX) on the ends.
6 The homology of the p-pro-finite completion.
Let X be a space. One would like the mod p homology H*|Xp| as a functor of *
*H*X,
at least if X is connected and nilpotent. Since H*contXp ~=H*X, one can subsum*
*e this
question into the larger problem of computing H*|Y | as a functor of H*contY . *
*This seems to
be a difficult problem. This section is devoted to what I know, and is mostly a*
*n outgrowth
of previous work on the homology of homotopy inverse limits [7].
We begin with some algebra. A pro-finite graded (cocommutative) coalgebra C*
* = {Cff}
is a filtered system of graded coalgebras Cffwhich are finite in each degree. T*
*he category
CApf of pro-finite graded coalgebras has these coalgebras as objects and pro-mo*
*rphisms
(as in 1.1) as morphisms. If X = {Xff} 2 bS, then H*X = {H*Xff} 2 CApf.
The category CApf is equivalent, via a Pontrjagin style duality, to the cat*
*egory A of
graded commutative algebras over Fp. Indeed, if C = {Cff} 2 CApf, then
(6:1:1) C# = Hom cont(C; Fp) = colimffC*ff
is in A, and if A 2 A, then
(6:1:2) A* = {A*ff} 2 CApf
where Aff A runs over the sub-algebras of finite type. These two dualization fu*
*nctors
define the equivalence of categories. Note that this implies that if X 2 bS, th*
*en H*X 2 CApf
is determined, up to isomorphism in CApf, by the algebra
H*contX ~=(H*X)#
28
in A. Thus we may recast the question that opened this section as follows: if *
*Y 2 bSis
fibrant, can one compute H*|Y | as a functor of H*Y 2 CApf? The answer we give *
*is an
attenuated affirmative one.
Incidentally, the fibrancy condition of Y is to guarantee that the weak eq*
*uivalence
class of |Y |, and hence the coalgebra H*|Y |, depends only on the weak equival*
*ence class
of Y in bS.
To begin with, let C = {Cff} 2 CApf. Then we can form the limit limffCffin*
* the
category CA of graded commutative coalgebras. This limit is only a sub-object o*
*f the limit
limffFpCffof graded vector spaces; the inverse limit functor need not be exact,*
* only left
exact. See Example 6.12 below. If Y 2 bS, then there is a natural map
(6:2) j : H*|Y | = H* limYff! limH*Yff
and one might hope that this is an edge homomorphism in a spectral sequence who*
*se E2
term depends on right derived functors of the limit functor. This is what I wis*
*h to explain.
The first point to be made is that the limit functor lim : CApf ! CA has su*
*itably
defined right derived functors. Since CApf is not an abelian category, one woul*
*d expect
non-abelian derived functors defined with the aid of a cosimplicial resolution.*
* To facilitate
this one has the next result. Let cCApf be the category of cosimplicial objects*
* in CApf.
Lemma 6.3. The category cCApf has the structure of a simplicial model category *
*where
a morphism f : Co ! Do is
1)a weak equivalence if ss*f : ss*Co ! ss*Do is an isomorphism, and
2)a cofibration if the normalized cochain complex Nf : NCo ! NDo is an inje*
*ction
above cochain degree 0.
The simplicial structure is determined by
Y
hom (K; Co)n ~= Cn
x2Kn
where K 2 S and Co 2 CApf, and the induced coface and codegeneracy maps.
Proof: The opposite category of CApf is equivalent to the category A of graded *
*com-
mutative algebras over Fp. The result follows by interpreting Quillen's standar*
*d simplicial
model category structure on the category sA of simplicial objects in A. See [15*
*, II x2 and
x4].
29
Definition 6.4. Let C 2 CApf. The total right derived functor R lim is defined *
*by the
formula
R limC = limDo
where C ! Do is a weak equivalence in cCApf to a fibrant object, and lim is app*
*lied in
each cosimplicial degree. Thus R limC is a cosimplicial coalgebra and the deriv*
*ed functors
are defined by
Rslim C = sssR limC:
As usual R* limC is well-defined up to isomorphism. Also, since limDo is a *
*cosimpli-
cial coalgebra, R* limC is a bigraded, cocommutative coalgebra.
To interpret this object in homotopy theory one needs:
Lemma 6.5. Let V = {Vff} be a simplicial pro-finite vector space. Then the natu*
*ral map
j : H*|V | ! limH*Vff
is an isomorphism.
Proof: This is a consequence of Lemma 3.9 and the fact that if W is any simplic*
*ial vector
space, then the coalgebra H*W is a functor of ss*W .
The main result of this section is the following. No assertion is made her*
*e about
convergence. This question will be addressed in the remark following.
Proposition 6.6. Let Y 2 bSbe fibrant. Then there is a second quadrant homolo*
*gy
spectral sequence
R* limH*Y ) H*|Y |:
__
Proof: Let Y ! FopY be the cosimplicial resolution of section 2. See 2.2. Then *
*there is
a second-quadrant homology spectral sequence
__o __o
sssHt|FpY | ) Ht-sTot |FpY |:
We need only interpret the E2-term and the abutment. For the latter, we have, b*
*y Propo-
__ __
sition 3.5, that |Y | ! Tot |FopY | ~=|Tot (FopY )| is a weak equivalence. For *
*the former, we
have that
__o __o
H*|FpY | ~=limH*FpY;
__
by the previous result; hence, we need only assert that H*FopY is fibrant in cC*
*Apf. If (.)#
is the duality functor of 6.1.1, then one need only assert that
__o # * __o
(H*FpY ) ~=HcontFpY
30
is cofibrant in the category of simplicial commutative algebras. This is well-*
*known, and
essentially follows from the fact that the cohomology of an Eilenberg-MacLane s*
*pace is a
free commutative algebra. (See [6], among many sources.)
Remark 6.7: Convergence is always a problem with this type of spectral sequence*
*. Since
we are especially interested in the case of non-finite type spaces, the best so*
*urce is probably
[2, x3]. In particular, each of the spaces in the cosimplicial space |FopY | is*
* p-nilpotent, so if
[Rslim H*X]t = 0
for t - s 1, and for each s there are only finitely many k so that [Rslim H*X]*
*s+k = 0,
the spectral sequence will converge strongly.
Example 6.8: There are times when Rslim C = 0 for s > 0. For example suppose
C = {Cff} 2 CApf is a diagram of coalgebras obtained from a diagram of connecte*
*d,
bicommutative Hopf algebras by forgetting the algebra structure. Then the techn*
*iques of
[7] and the fact that limit is exact on pro-finite groups implies Rslim C = 0 s*
* 1. It is
always true that R0 limC ~=limC.
This applies to the spectral sequence of Proposition 6.6 as follows. Let Y *
*= {Yff} 2
bS be pointed and fibrant and suppose each Yffis 2-connected. Then map *(S2; Y*
* ) =
{map *(S2; Yff)} is also fibrant and H*map *(S2; Yff) 2 CApf is a diagram under*
*lying a
diagram of connected bicommutative Hopf algebras. The spectral sequence will co*
*nverge
if [limH*map *(S2; Y )]1 = 0, by Bousfield's result. Actually, it should conve*
*rge without
the last assumption, but I've not looked very hard for a proof.
If X 2 S is 2-connected, then there exists a model for Xp = {Xff} 2 bSso th*
*at each
Xffis 2-connected. See the construction of fibrations given by Morel in [13], f*
*or example.
We now interpret Proposition 6.6 in the case where Y = Xp is the p-pro-fini*
*te comple-
tion of a space X. This amounts to providing an interpretation of the pro-finit*
*e coalgebra
(H*X)*, where (.)* is as in (6.1.2). If C is any graded coalgebra, we may defin*
*e C 2 CApf
to be the diagram {Cff} obtained from taking finite type coalgebras under C. Th*
*us one
considers the category with objects surjective coalgebra maps C ! Cffand morphi*
*sms
commutative triangles and obtains C by sending C ! Cffto Cff. There is a natur*
*al
map C ! limC; this may be regarded as a pro-finite completion of C in the categ*
*ory of
coalgebras.
The description of (H*X)* given in (6.1.2) implies
(H*X)* ~=H*X
31
and Proposition 6.6 implies there is a spectral sequence
(6:9) R* limH*X ) H*|Xp|:
Example 6.10: Suppose Z 2 S is pointed, 2-connected, and fibrant. Then map *(S2*
*; Zp)
is a model for map *(S2; Z)p. We know map *(S2; Zp) is fibrant and
map *(S2; Z) ! map *(S2; Zp)
induces an isomorphism H*contmap*(S2; Zp) ~=H*map *(S2; Z), by the Serre Spectr*
*al Se-
quence. By Example 6.7,
Rslim H*map *(S2; Zp) = 0
for s > 0. Also limH*map *(S2; Zp) = limH*map *(S2; Zp). Thus, if Z is 3-conn*
*ected
(see Example 6.7 for this hypothesis),
H*map *(S2; Zp) ~=H*map *(S2; Zp):
Example 6.11: Some sort of hypothesis is needed to guarantee convergence. If X *
*is finite
type, then H*X has a final object_namely, H*X itself and one easily checks that*
* this
implies Rslim H*X = 0 for s > 0 and limH*X = H*X. But one cannot conclude from
(6.9) that H*|Xp| ~=H*X, without some further hypothesis.. By the results of se*
*ction 2,
|Xp| is weakly equivalent to Bousfield-Kan completion (Fp)1 X and there are spa*
*ces X for
which H*X 6~=H*(Fp)1 X.
Example 6.12: We show, by example, that lim: CApf ! CA is not right exact. Let *
*Cn,
n 1, be the connected coalgebra so (Cn)k = 0, (Cn)2 ~=Fp with generator z, (Cn*
*)1 ~=F2np
with generators x1; x2; : :;:xn; y1; y2 : :;:yn, and the only non-trivial diago*
*nal given by
Xn
z = z 1 + 1 z + xi yi+ yi xi:
i=1
Thus Cn is the homology of an n-holed torus. There are quotient maps Cn ! Cn-1 *
*sending
z to z and xi, yito the like-named object, except xn, yn go to zero. Let C = {C*
*n} 2 CApf.
It is a simple calculation to show the inverse limit (limCn)k = 0 for k 2. In *
*particular if
D = H*S2 and Cn ! H*S2 is the non-trivial map, then C ! D is surjective in CApf*
*, but
limC ! limD ~=H*S2
is not.
32
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*logy 14
(1975), pp. 133-150.
2. A.K. Bousfield, "On the homology spectral sequence of a cosimplicial space",*
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3. A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations,*
* Lec-
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33