0
Towards the finiteness of ss* LK(n) S
Ethan S. Devinatz
Abstract
Let G be a closed subgroup of the nthMorava stabilizer group Sn, n 2, and l*
*et
EhGndenote the continuous homotopy fixed point spectrum of Devinatz and Hopkins.
If G = , the subgroup topologically generated by an element z in the p-Sylow
subgroup S0nof Sn, and z is non-torsion in the quotient of S0nby its center, we*
* prove
that the Ehn-homology of any K(n - 2)*-acyclic finite spectrum annihilated b*
*y p is
of essentially finite rank. We also show that the units in En* fixed by z are j*
*ust the
units in the Witt vectors with coefficients in the field of pn elements. If n *
*= 2 and
p 5, we show that, if G is a closed subgroup of S0nnot contained in the cente*
*r, then G
xp)
contains an open subnormal subgroup U such that the mod(p) homotopy of Eh(UxFn
is of essentially finite rank.
Introduction
This paper represents an attempt to make some progress towards the long-standing
open question asking whether ss*LK(n)S0, the homotopy groups of the localizatio*
*n of
S0 with respect to the nthMorava K-theory K(n), is a module of finite type over*
* the
p-adic integers Zp. (If X is any spectrum, LK(n)X is local with respect to the *
*mod(p)
Moore spectrum M, so that ss*LK(n)X is canonically a Zp-module.)
For small values of n_and appropriate values of the prime p_the answer to this
question is known. The computation of ss*LK(1)S0 is by now classical (see for e*
*xample
[14]); these groups are related to the image of the J-homomorphism. This comput*
*ation
also shows that ss*LK(1)S0 is a Zp-module of finite type. ss*LK(2)S0 has also *
*been
computed for p 3, and it too is of finite type; here, however, the computatio*
*ns are
extraordinarily complicated ([19, 20]). (Another way of approaching LK(2)S0 at *
*p = 3
is carried out in [9].) It is easier_but still complicated_to compute ss*LK(2)M*
* and
prove that it's of finite type; these calculations were also done by Shimomura *
*([17, 18])
and used in the computation of ss*LK(2)S0. It is, however, not hard to deduce t*
*hat
ss*LK(2)S0 is of finite type provided that ss*LK(2)M is. Indeed, if ss*LK(2)M *
*is of
finite type, so is ss*LK(2)M(pi), where M(pi) denotes the mod (pi) Moore spectr*
*um.
Therefore,
ss*LK(2)S0- ! limss*LK(2)M(pi).
i
1
Since ss*LK(2)S0
lim___________i0,! limss*LK(2)M(pi),
i p ss*LK(2)S i
it follows that
ss*LK(2)S0
ss*LK(2)S0- ! lim___________i0;
i p ss*LK(2)S
in other words, ss*LK(2)S0 is complete with respect to the p-adic topology. Tog*
*ether
with the fact that ss*LK(2)S0=pss*LK(2)S0 is of finite type, this implies that *
*ss*LK(2)S0
is of finite type over Zp.
If n > 2, explicit determination of ss*LK(n)S0 seems inaccessible. Indeed, t*
*he
chromatic theory approach requires, as a starting point, knowledge of the conti*
*nuous
cohomology with trivial mod (p) coefficients of, say, the p-Sylow subgroup S0no*
*f the
nthMorava stabilizer group Sn. This cohomology has been computed for n = 3 [23],
but is too complicated to do the required further calculations; if n 4, even *
*explicit
calculation of H*c(S0n, Fp) may be impractical.
Of course, a thick subcategory argument shows that ss*LK(n)S0 is of finite ty*
*pe
provided that ss*LK(n)X is of finite type for some finite X which is not ration*
*ally
acyclic. More generally, if ss*LK(n)X is of finite type for some finite X in th*
*e E(n)*-
local category, then ss*LK(n)Y is of finite type for any finite Y in the E(n)*
**-local
category such that
{m n : K(m)*Y 6= 0} {m n : K(m)*X 6= 0}.
(See [2] for a discussion of the E(n)*-local thick subcategory theorem.) If X *
*is a
K(n - 1)*-acyclic finite spectrum in the E(n)*-local category, then it is stand*
*ard that
ss*LK(n)X is of finite type ([5, Proposition 4.1]). The next case to consider i*
*s therefore
when X is K(n-2)*-acyclic. If p n, the Smith-Toda complex V (n-2) exists E(n)*
**-
locally, and we may ask whether ss*LK(n)V (n - 2)is of finite type. Still, how*
*ever,
the inaccessibility of the cohomology of S0nprevents explicit calculation. A di*
*fferent
approach is needed.
Let En denote the Landweber exact spectrum with coefficient ring W Fpn[[u1, .*
* .,.
un-1]][u, u-1], where W Fpn denotes the ring of Witt vectors with coefficients *
*in the
field Fpn of pn elements, and whose BP*-algebra structure map r : BP* ! En* is *
*given
by 8
i
< uiui-p i < n
r(vi) = u1-pn i = n
:
0 i > n,
where viis the ithHazewinkel generator. The nthextended Morava stabilizer group
Gn Sn o Gal(Fpn=Fp) Sn o Gal
acts on En, and in [7], the continuous homotopy fixed point spectrum EhGnis con-
structed for G a closed subgroup of Gn. These spectra have the expected functor*
*ial
properties, and there is a "continuous homotopy fixed point" spectral sequence
H*c(G, E*nX) ) (EhGn)*X.
2
If H / G, there is also a spectral sequence
i j
H*c G=H, (EhHn)*X ) (EhGn)*X
([3]). These spectral sequences are strongly convergent and have a horizontal v*
*anishing
line at E1 independent of X. Since EhGnn~=LK(n)S0, it follows that if G is a cl*
*osed
subnormal subgroup of Gn_that is, there exists a chain of closed subgroups
G = H0/ H1/ H2/ . ./.Ht= Gn
_and if X is a finite (in the E(n)*-local category) spectrum for which ss*(EhGn*
*^ X) is
of finite type, then so is
0 hG
ss*LK(n)X = ss* (LK(n)S ) ^ X = ss*(En n ^ X).
(See [5] for a discussion of this.) For subgroups G whose continuous cohomology*
* with
trivial coefficients Fpis easily understood, one might hope to compute ss* EhGn*
*^ V (n - 2)
and see that it is of finite type.
In [5], we carried out this computation in the case where p 3 and G is the
"diagonal" subgroup H2 of G2. Unfortunately, ss*(EhH22^ M) is not of finite typ*
*e. In
fact, subsequent (unpublished) calculations of mine seemed to indicate that ss**
*(EhG2^
M) is almost never of finite type, yet almost always "almost" of finite type in*
* the sense
of [5], the meaning of which we now recall.
Let M be a graded Fp[ ]-module, where is even if p 6= 2. We will say that M*
* is
good if
i.M is -adically complete; i.e. M = limM= iM
i
ii.M= M is an Fp-vector space of finite type
iii.ker( : M ! M) is an Fp-vector space of finite type.
If M is good, then
Y Y
(0.1) M nffFp[ ] x mfiFp[ ]=( ifi)
ff fi
(see [22]). Condition iii is not needed to obtain this decomposition; it will, *
*however,
prove useful later. M is said to be of essentially finite rank if there are onl*
*y_a_finite
number of ff's in the_decomposition. This is equivalent to the statement that M*
*=T has
finite rank, where T is the closure of the torsion module T . Note, however, th*
*at M=T
may be uncountable even if M is of essentially finite rank.
Now suppose that X is a K(n - 2)*-acyclic finite spectrum in the E(n)*-local
category with vn-1 self-map and such that p : X ! X is trivial. Also write *
* :
ss*(EhGn^ X) ! ss*(EhGn^ X) for the map on homotopy groups induced by EhGn^ .
Then ss*(EhGn^ X) is an Fp[ ]-module and, as in [5, Proposition 4.5], ss*(EhGn^*
* X) is
good. Moreover, since vn-1 self-maps are essentially unique (cf. [10]), it is e*
*asy to see
that whether or not ss*(EhGn^ X) is of essentially finite rank does not depend *
*upon the
choice of .
3
hG ^X) depends very much on the size of H*(G,*
* F ),
Since the computability of ss*(En c *
* p
the easiest case to consider ought then to be G = Zp, the closed subgroup *
*of
S0ntopologically generated by a single non-torsion element z. Here we have comp*
*lete
qualitative results.
Recall that Sn S0no Fxpnand that the center C of Sn is isomorphic to Zxp.
Moreover, C0 S0n\ C Z0p, the ring of p-adic units congruent to 1 mod (p). T*
*he
following is our first main result.
Theorem 1 Let n 2, and suppose that X is a K(n - 2)*-acyclic finite spectrum
in the E(n)*-local category with p1X ' *. If z 2 S0nis non-torsion in S0n=C0, *
*then
ss*(Ehn^ X) is of essentially finite rank.
Remark. If g 2 C0, then g acts trivially on En*=(p) (Proposition AI.1); thus
H*c(, En*X) H*c(, En*X)
if, for instance, z and zg are non-torsion. Since the continuous homotopy fixed*
* point
spectral sequences converging to ss*(Ehn^X) and ss*(Ehn^X) both collapse*
* in this
case, it follows that ss*(Ehn^ X) ss*(Ehn^ X), at least if En*X is con*
*centrated
in even degrees.
Remark. If p > n+1, then S0n=C0 is torsion-free (Remark AI.4), so Theorem 1 app*
*lies
to any z 2 S0n, z 62 C0.
Theorem 1 also gives information about the action of Gn on En*. The first part
of the next result is a consequence of Theorem 1_provided V (n - 2) exists E(n)*
**-
locally_but will be proven in the course of proving this theorem. The second pa*
*rt is
a consequence of the first.
Theorem 2 Let z be as in Theorem 1. Then
a. (Fpn[[un-1]][u, u-1])= Fpn[vn-1]
b.The ring of units in (En*)is (W Fpn)x.
Remark. The ideal (p, u1, . .,.un-2) in En* is invariant, so that Fpn[[un-1]][u*
*, u-1] =
En*=(p, u1, . .,.un-2) is a Gn-module. If V (n - 2) exists, this module is the *
*same as
En*V (n - 2).
This result, in turn has the following immediate consequence.
Corollary. Let G be a closed subgroup of S0ncontaining an element z which is no*
*n-
torsion in S0n=C0. Then (ss*EhGn)x = W Fxpn, so that ss*EhGnis not periodic.
Proof. Any element c in (ss*EhGn)x must be detected by a unit in H0c(G, En*) = *
*EGn*
En*in the continuous homotopy fixed point spectral sequence.
Remark. The condition on z is necessary. For example, if p 3, there is a subg*
*roup
Z=(p) in S0p-1for which ss*Eh(Z=(p))p-1contains a unit in degree 2p2(p - 1)2 (s*
*ee [13]).
4
One would, of course, like to extend Theorem 1 to subgroups more general than
. Here our results are unfortunately much less complete.
Theorem 3 Let p 5 and G a closed subgroup of S02not contained in C0. Then G
xp)
contains an open subnormal subgroup U such that ss*(Eh(UxF2 ^ M) is of essentia*
*lly
finite rank.
Remark. We believe that the strategy used in proving this result ought to gener*
*alize
to n > 2. That is, if G is a closed subgroup of S0nnot contained in C0, and if *
*p n,
then our construction should also apply to give an open subnormal subgroup U su*
*ch
xp)
that ss*(Eh(UxFn ^ V (n - 2)) is of essentially finite rank.
The reader may at this point wonder why we must use U x Fxpinstead of U and
why M is not replaced by an arbitrary rationally acyclic finite spectrum. The e*
*ssential
problem is that modules of essentially finite rank do not behave well with resp*
*ect to
exact sequences. In particular, the kernel of an Fp[ ]-module map between modul*
*es of
essentially finite rank need not have essentially finite rank. This prevents us*
* from imme-
*
*xp)
diately making use of the thick subcategory theorem to conclude that ss*(Eh(UxF*
*2 ^X)
is of essentially finite rank whenever X is rationally acyclic. It also forces *
*us to require
collapsing of the continuous homotopy fixed point spectral sequence in order to*
* obtain
information about homotopy groups from information about group cohomology.
In the case at hand, S0nis a compact p-adic analytic group of dimension n2 an*
*d is
torsion-free if p - 1 - n ([15, Theorem 4.3.4], or see Proposition AI.4). This*
* implies
that U is a torsion-free compact p-adic analytic group ([8, Theorem 9.6]), and *
*hence,
by a result of Lazard (proved as Theorem 5.1.9 in [21]), the cohomological p-di*
*mension
of U is dimU. Thus
x
Hsc(U x Fxp, En*V (n - 2)) = Hsc(U, Fpn[[un-1]][u, u-1])Fp = 0
if s > n2. Since Fxpacts trivially on un-1 and by multiplication on u (Proposi*
*tion
AI.1), it follows that
ae s,t -1
(0.2)Hs,tc(U x Fxp, Fpn[[un-1]][u, u-1]) = Hc (U, Fp[[un-1]][u, u ])2(p - 1)*
*|t
0 2(p - 1) - *
*t.
In particular, the continuous homotopy fixed point spectral sequence collapses *
*for p
n, and information about H*c(U, Fpn[[un-1]][u, u-1]) immediately yields informa*
*tion
xp)
about ss*(Eh(UxFn ^ V (n - 2)).
The poor algebraic behavior of modules of essentially finite rank also preven*
*ts
x)
us from easily gaining information about ss*(Eh(GoKn ^ X) from information abo*
*ut
xp)
ss*(Eh(UxFn ^ X), where K is a subfield of Fpn whose units normalize G. Indeed,
x)
U x Fxpis an open subnormal subgroup of G o Kx and therefore Eh(GoKn is in the
xp)
thick subcategory generated by Eh(UxFn (Proposition AII.1). However, as we've s*
*een,
x)
this does not immediately imply that ss*(Eh(GoKn ^ X) is of essentially finite*
* rank
xp)
provided ss*(Eh(UxFn ^ X) is.
5
*(G, F n[[u ]]*
*[u, u-1])
Our approach to the main theorems will be to try to compute Hc p n-1
using a Bockstein spectral sequence; the differentials in this spectral sequenc*
*e are to
be computed using the Hopf algebroid techniques of [5]. In x1, we will establis*
*h some
general properties of the Bockstein spectral sequence and see that the question*
* of
whether Hsc(G, Fpn[[un-1]][u, u-1]) is of essentially finite rank_where is in*
*duced by
the G-equivariant multiplication of n-1 on Fpn[[un-1]][u, u-1]_has a simple an*
*swer
in terms of the behavior of this spectral sequence (Proposition 1.5). We will a*
*lso see
that if z 2 S0nis not torsion, then H*c(, Fpn[[un-1]][u, u-1]) is of essenti*
*ally finite
rank if and only if Fpn[[un-1]][u, u-1]= Fpn[vn-1] (Proposition 1.6). In x2,*
* we will
apply the techniques of [5] to the group to prove Theorem 2. From this we wi*
*ll
deduce Theorem 1, which will be carried out in x3. Finally, in x4 and x5 we wil*
*l use
some general properties of analytic pro-p groups to construct the requisite U a*
*nd prove
Theorem 3.
There are two appendices: the first describes some properties of Sn and its a*
*ction
on En* which were needed earlier, and the second gives a quick proof that EhGni*
*s in
the thick subcategory generated by EhUnwhenever U is an open subnormal subgroup
of G.
1 The Bockstein spectral sequence
Let G be a closed subgroup of S0n, where, for the rest of this paper, n 2. T*
*he
Bockstein spectral sequence is obtained from the unrolled exact couple
0 _____oeH*(En*=In)_____oeH* En*=(p, u1, . .,.un-2, u2n-1)_oe. . .
Z ` Z ` Z ` 2
Z Z vn-1 Z vn-1
(1.1) Z Z Z
Z" H*(En*=In) Z"H*(En*=In) Z" H*(En*=In)
where In = (p, u1, . .,.un-1) and we have written H*M for H*c(G, M).
i j
H** En*=(p, u1, . .,.un-2, ukn-1)
is finite in each bidegree, so this yields a strongly convergent spectral seque*
*nce
(1.2) H*c(G, Fpn[u, u-1]) ) H*c(G, Fpn[[un-1]][u, u-1]).
Since G acts trivially on Fpn[u, u-1] (Proposition AI.1),
H*c(G, Fpn[u, u-1]) = H*c(G, Fp) Fpn[u, u-1].
Of course, the E1-term is actually trigraded, but it is usually easiest to supp*
*ress the
first grading, or filtration degree, and write Es,t1= Hs,tc(G, Fpn[u, u-1]). Es*
*,tris then
defined to be the Er-term (in any filtration degree) of the half-plane spectral*
* sequence
arising from the exact couple
vn-1
H*c(G, Fpn[[un-1]][u, u-1])_oeH*c(G, Fpn[[un-1]][u, u-1])
Z `
Z
(1.3) Z
Z"H* -1
c(G, Fpn[u, u ])
6
Truncated, this exact couple yields a spectral sequence isomorphic to the Bocks*
*tein s,t
spectral sequence of 1.1; this allows us to give the following description of d*
*r : Er !
n-1-1)
Es+1,t-2r(pr .
Let (C*c(G, M), ffi) denote the continuous cobar complex whose cohomology is
H*c(G, M). If x 2 Es,tr, let us write exfor a representative of x in C*c(G, Fpn*
*[[un-1]][u, u-1]);
that is, an element whose image in the quotient C*c(G, Fpn[u, u-1]) is a cochai*
*n repre-
sentative of x. Then drx = y if and only if there are representatives exand eys*
*uch that
ffiex= vrn-1ey.
We can also give a description of Es,t1. If M is any Fp[ ]-module, let St(M) *
*denote
the quotient of the group Mt= Mt-| |by the group consisting of all u such that *
* ku = 0
in kM= k+1M for some k. Then
(1.4) Es,t1= St(Hsc(G, Fpn[[un-1]][u, u-1]).
This follows from the fact that, in the Bockstein spectral sequence, the E1 -te*
*rm in
filtration degree k is given by vkn-1H*(Fpn[[un-1]][u, u-1])=vk+1n-1H*(Fpn[[un-*
*1]][u, u-1].
The next result shows what all this has to do with determining whether or not
Hsc(G, Fpn[[un-1]][u, u-1]) is of essentially finite rank.
PropositionL1.5Hsc(G, Fpn[[un-1]][u, u-1]) is of essentially finite rank if and*
* only if
Es,t1is finite.
t
Proof. First observe that, by an argument similar to the proof of Proposition 4*
*.5 in
[5], Hsc(G, Fpn[[un-1]][u, u-1]) is good for each s 0.
Now using the decomposition 0.1, it's easy to see that,Lif M is a good Fp[ ]-*
*module,
then M is of essentially finite rank if and only if tSt(M) is finite. By 1.4*
*, this
completes the proof.
Now suppose G = , where z 2 S0nis non-torsion. Then Zp and
H*c(, Fpn[u, u-1]) = E(_z*) Fpn[u, u-1],
where _z*2 H1c(, Fp) = Hom c(, Fp) is the homomorphism with _z*(z) = 1. I*
*n this
case, the above Bockstein spectral sequence criterion simplifies a great deal.
Proposition 1.6Let z 2 S0nbe non-torsion. Then H*c(, Fpn[[un-1]][u, u-1]) is*
* of
essentially finite rank if and only if Fpn[[un-1]][u, u-1]= Fpn[vn-1].
Proof. First observe that Fpn[[un-1]][u, u-1]= Fpn[vn-1] if and only if E0,**
*1= Fpn
concentrated in degree 0.
If E0,*16= Fpn, then ujL2 E0,*1survives to E1 for some j 6= 0. This implies t*
*hat ujk
survives for all k, so tE0,t1is not finite and therefore H0c(, Fpn[[un-1]]*
*[u, u-1]) is
not of essentially finite rank. L
Conversely, suppose that E0,*1= Fpn. We will show that dimFpn( tE1,t1) 1, *
*and
thus that H*c(, Fpn[[un-1]][u, u-1]) is of essentially finite rank. Indeed, *
*we must have
n-1-1)_*
dj0u `=u1+j0(p z
7
for some j0 1, where the `=denotes that this equality is only true up to mult*
*iplication
by a unit in Fpn. The differentials in the Bockstein spectral sequence are deri*
*vations;
therefore, n-1
dj0uk `=kuk+j0(p -1)_z*.
This implies that every ui_z*with i 6= -j0mod (p) is hit by a differential. Sin*
*ce up does
not survive, we have n-1
dj1up `=up+j1(p -1)_z*
for some j1 pj0 with j1 = j0mod (p). Again it follows that
n-1-1)_*
dj1ukp=`kupk+j1(p z,
and we conclude that ui_z*is hit by a differential of length j1 whenever i 6=*
* j1(pn-1-
1) mod(p2). In general,
m pm+j (pn-1-1)_*
djmup `=u m z
with jm pjm-1 and jm = jm-1 mod(pm ). Let j = limm!1 jm 2 Zp. Then ui_z*is
a boundary if and only if i 6= j(pn-1 - 1). If j(pn-1 - 1) 62 Z,Lthis shows tha*
*t ui_z*is
always a boundary; if j(pn-1 - 1) 2 Z, this shows that dimFpn( tE1,t1) = 1. In*
* any
event, the proof is complete.
2 The case G = .
Let z 2 S0nand suppose that z is non-torsion in S0n=C0. In this section, we wil*
*l compute
the Bockstein spectral sequence
(2.1) E(_z*) Fpn[u, u-1] ) H*c(, Fpn[[un-1]][u, u-1])
and see that (Fpn[[un-1]][u, u-1])= Fpn[vn-1]. We will then use this to dedu*
*ce that
the ring of units in (En*)is W Fxpn.
Our methods will be those of [5]; namely, we consider the complete Hopf algeb*
*roid
(En*, ), where
= Mapc(, En*),
and observe that
H*c(, N) = Ext* (En*, N)
for any appropriate twisted -En*module N (cf. [1, x5]). If N = Fpn[[un-1]][u*
*, u-1],
the Bockstein spectral sequence becomes
E(_z*) Fpn[u, u-1] = Ext*_=vn-1__(Fpn, Fpn[u, u-1]) ) Ext*_(En*, En*
**=In-1),
where __
= =In-1 = Mapc(, En*=In-1).
We will compute the differentials in this spectral sequence by using the usual *
*Hopf
algebroid formulas in
E^n*En = En*bBP*BP*BP bBP*En*
8
together with information about the quotient map
__
E^n*En = Mapc(Gn, En*) ! Mapc(, En*) ! .
We begin our work by observing that z may be assumed to have a special form.
In Appendix I, we define f(z), the reduced filtration of z, and observe that, i*
*f f(z) =
m < 1, there exists g 2 C0 such that
i
(2.2) zg = x + n nbixp ,
i m
where bm 6= 0, and if n|m, bm 62 Fxp. As in the first Remark following Theorem *
*1 in
the Introduction, the Bockstein spectral sequences 2.1 for and are iso*
*morphic;
we will thus assume from now on that z is already in the form 2.2.
H*c(,_Fpn[u,_u-1]) can be described explicitly in terms of the cobar const*
*ruction
for =vn-1 , where this Hopf algebroid is regarded as a quotient of E^n*E*
*n. Indeed,
the map E^n*En ! Map c(S0n, En*=In) is well understood; the image of ti is_the *
*map__
sending x+ n jn 1cjxpjto u1-pici[1, Proposition 2.11]. tiis thus zero in =v*
*n-1
for i < m, and tm is non-zero and primitive. This implies that
(2.3) H*c(, Fpn[u, u-1]) = E(hm ) Fpn[u, u-1],
where m
hm = up -1tm .
Alternatively, we may regard
tm 2 Hom c(, Fpn[u, u-1]).
Since m
tm (z) = u1-p bm ,
it follows that m
up -1tm = bm _z*.
One last point: We have chosen our conventions (cf. [4]) so that if x 2 En*,
x jL(x) 2 Mapc(Gn, En*)
is given by
jL(x)(g) = g-1x.
It is jR(x) which sends all of Gn to x. However, u = jR(u) in Map c(S0n, Fpn[u,*
* u-1])
and a fortiori in Map c(, Fpn[u, u-1]).
We will need a bit more notation in our computation of the differentials. Wri*
*te
__ i __
(i) = vn-1 ,
__ __ __
and for x 2 , write OE(x) = i if x 2 (i) but x 62 (i+1). (If x = 0,*
* OE(x) = 1.)
We call OE(x) the valuation of x.
9
j for all j;*
* this
The only differentials which need to_be_determined are those on u
requires an understanding of jR(u) in . In what follows, assume that f(z) >*
* 1, so
that OE(t1) 1. We have [16, 4.3.21]
n-1 p
(2.4) jR(vn) = vn + vn-1tp1 - vn-1t1
in BP*BP=In-1BP*BP , and hence
n 1-pn __
jR(u1-p ) = u mod (p + OE(t1)).
This implies that __
jR(u) = u mod (p + OE(t1)),
whence n n __
jR(up ) = up mod (pn+1 + pnOE(t1)).
Multiplying both sides of 2.4 by upn thus yields
n pn-1 p pn __ n+1 n
(2.5) jR(u) = u + vn-1up t1 - vn-1u t1 mod (p + p OE(t1)).
__
We must therefore relate t1 to tm in in order to compute our differentials.
Our main tool will be the relation
n X F pi pn+1-i X F pi pn+1
(2.6) jR(vn)xp +F vn-1tix +F jR(vn) tix
i 1 i 1
n X F pn-1 pn+i-1 X F pn pn+i
= vnxp +F vn-1ti x +F vnti x
i 1 i 1
in BP*BP=In-1 [16, A2.2.5], where F denotes the usual formal group law associat*
*ed
to BP . Although this is of course a relation between power series, the powers *
*of x in
this relation are normally suppressed, since the coefficients of xi have gradin*
*g 2i - 2.
We will also need a small amount of information about F .
Proposition 2.7F (x, y) = x+y-vn-1Cp(xpn-2, ypn-2) mod(x, y)pn-1+1in BP*=In-1,
where Cp(x, y) = 1_p[(x + y)p- xp-.yp]
i
Proof. Recall that logF(x) = i 0~ixpiin BP* Q, where p~k = 0 i.
Proof. We proceed by induction. Suppose that k m - 1 and OE(t1) OE(t2) . *
*. .
OE(tk). Since __
jR(vn) = vn mod (OE(t1) + p),
equation 2.6 becomes
X F pi X i X pn-1 X pn __
vn-1ti+F Fvpnti= Fvn-1ti +F Fvnti mod (OE(t1) + p).
i 1 i 1 i 1 i 1
Examining the coefficients of xpn+kyields
k+1 pk pn-1 __
vpn-1tk+1+ vn tk = vn-1tk+1 mod (OE(tk) + 2)
and thus
k pn-1 pk+1 __
(2.9) vpntk = vn-1tk+1 - vn-1 tk+1 mod (OE(tk) + 2).
__
If OE(tk+1) > OE(tk), we would then have OE(tk) < 1 and tk = 0 mod (OE(tk)*
* + 2), a
contradiction.
Finally, if k = m - 1, 2.9 becomes
m-1 pn-1 pm __
vpn tm-1 = vn-1tm - vn-1tm mod (3)
n-1 __
= vn-1tpm mod (3).
n-1
Since OE(tpm ) = 0, OE(tm-1) = 1.
__
Remark 2.10 We only need to work mod (OE(tk)+1) to prove the above lemma.
We will, however, need these sharper congruences for our work in x5. The reader*
* should
also note that Proposition 2.7 is useful in establishing 2.9.
We are now ready to prove our main technical result. In what follows, we will
sometimes obtain equalities in which u appears on one side of the equation with*
* a
complicated exponent. We will write this term as u(o)when we do not wish to str*
*ess
the exact value of the exponent; of course, by comparing gradings, this can alw*
*ays be
determined.
Lemma 2.11 Suppose f(z) = m with 1 < m < 1 and write m = nk + r with k, r
nonnegative integers and 0 r n - 1. Then
0 1
Xk m-k-1X
OE(t1) = @ pj(n-1)+ pjA (m)
j=0 j=2
and __
t1 = cvOE(t1)n-1u(o)hpm mod (OE(t1) + 2)
for some c 2 Fxpn.
11
Proof. Start with the case m = nk + r with 1 r n - 1. By the proof of the
preceding lemma,
m-1 pn-1 __
vpn tm-1 = vn-1tm mod (OE(tm-1) + 2)
and OE(tm-1) = 1. Next, we have
m-2 pn-1 pm-1 __
vpn tm-2 = vn-1tm-1 - vn-1 tm-1 mod (OE(tm-2) + 2).
n-1 pm-1
Since OE(vn-1tpm-1) = pn-1 + 1 and OE(vn-1 tm-1) = pm-1 + 1, it follows that
8 pn-1 __
m-2 < vn-1tm-1 mod (OE(tm-2) + 2) ifm > n
vpn tm-2 =
: -vpm-1 __
n-1 tm-1 mod (OE(tm-2) + 2) ifm < n.
In the first case, OE(tm-2) = 1+pn-1, and in the second, OE(tm-2) = 1+pm-1. Con*
*tinue
using 2.9 in this way to obtain
8 pn-1 __
m-i < vn-1tm-i+1 mod (OE(tm-i) + 2) ifi k + 1
vpn tm-i =
: -vpm-i+1 __
n-1 tm-i+1 mod (OE(tm-i) + 2) ifi > k + 1,
and, concurrently,
8 P i-1
< j=0pj(n-1) ifi k + 1
OE(tm-i) =
: P k j(n-1) P m-k-1 j
j=0p + j=m-i+1p ifi > k + 1.
Now substitute downwards; this yields
s __
(2.12) t1 = vOE(t1)n-1u(o)tpm mod (OE(t1) + 2)
for some s n - 1. But tm and tpm are both group homomorphisms from to
Fpn[u, u-1]; therefore __
tpm= ffu(o)tm mod vn-1 ,
s p __
where ff = hm (z)p=hm (z). From this we obtain that tpm=`u(o)tm mod (p), w*
*here
`=is used as in the proof of Proposition 1.6. Substituting into 2.12 yields the*
* desired
conclusion in this case.
Now suppose m = nk. As before, we obtain that
m-i pn-1 __
vpn tm-i = vn-1tm-i+1 mod (OE(tm-i) + 2)
P i-1
and that OE(tm-i) = j=0pj(n-1)if 1 i k. However,
m-k-1 pn-1 pm-k __
vpn tm-k-1 = vn-1tm-k - vn-1 tm-k mod (OE(tm-k-1) + 2)
and n-1
OE(vn-1tpm-k) = 1 + pn-1 + . .+.pk(n-1)= OE(vm-kn-1tm-k ).
12
Substituting as before, we have
k(n-1) __
tm-k = v(o)nvOE(tm-k)n-1tpm mod (OE(tm-k ) + 2)
and hence
m-k-1 (o)fl p(k+1)(n-1)pk(n-1) __
vpn tm-k-1 = u vn-1[hm - hm ] mod (fl + 2),
__ *
* __
where fl = 1 + pn-1 + . .+.pk(n-1). Once again regarding primitives in =vn-*
*1
as group homomorphisms from to Fpn[u, u-1], we have
(k-1)(n-1) p(k-1)(n-1)
hpm (z) = bm 62 Fp
__ __
in =vn-1 and thus
i (k-1)(n-1)jpn-1 (k-1)(n-1)
hpm (z) - hpm (z) 6= 0.
P i
(Recall that z = x + n in mbixp .) This implies that
k(n-1) p(k-1)(n-1) __
hpm - hm = ffhm mod (vn-1 )
and (k+1)(n-1) k(n-1) n-1 n-1 __
hpm - hpm = ffp hpm mod (pn-1)
P k
for some ff 2 Fxpn. Therefore, OE(tm-k-1) = j=0pj(n-1)and
m-k-1 (o)OE(tm-k-1)pn-1 __
(2.13) vpn tm-k-1 =` u vn-1 hm mod (OE(tm-k-1) + 2)
__
=` u(o)vOE(tm-k-1)n-1hpm mod (OE(tm-k-1) + 2).
We then continue as before to obtain
m-i pm-i+1 __
vpn tm-i = -vn-1 tm-i+1 mod (OE(tm-i) + 2)
and
Xk m-k-1X
OE(tm-i) = pj(n-1)+ pj
j=0 j=m-i+1
for i > k + 1. Once again, substituting downwards from 2.13 completes the proof.
Corollary 2.14Continue_with the hypotheses of Lemma 2.11, and set a(m) = (m)+ *
* __
p. Let (C*c( , Fpn[[un-1]][u, u-1]), ffi) denote the cobar construction fo*
*r with
coefficients in Fpn[[un-1]][u, u-1]. Then, if p - s,
n-1-1)p __
ffi(us) `=va(m)n-1us+a(m)(p hm mod (a(m) + 2).
13
Proof. By 2.5 and our expression for OE(t1), we have
n __
jR(u) = u - vpn-1up t1 mod (a(m) + 2)
provided m > 2 or n > 2. If m = 2 and n = 2, we get, by an argument similar to *
*one
in the previous proof, that
n __
jR(u) = u - ffvpn-1up t1 mod (a(m) + 2)
for some ff 2 Fxp2. Since ffi(us) = jR(us) - us, the desired result now follows*
* easily from
the preceding lemma together with the binomial theorem.
We are now in position to compute the differentials in the Bockstein spectral*
* se-
quence 2.1, provided that the reduced filtration of z is sufficiently large.
Theorem 2.15 Suppose that z 2 S0nand that f(z) = m with 1 < m < 1. If m >
n=(p - 1), then uspi, p - s, i 0, supports a nontrivial differential of lengt*
*h a(m + in).
Moreover, uihm is hit by a differential except when i = -(1 + p + . .+.pn-1).
Proof. We first show that this pattern of differentials is possible; that is, s*
*ince we are
claiming that i i n-1
da(m+in)usp `=usp +a(m+in)(p h-1)m
provided p - s, we must check that uspi+a(m+in)(pn-1-1)hm is not hit by one of *
*the lower
claimed differentials. Indeed, write m = nk + r with 0 r n - 1. Then
spi+ a(m + in)(pn-1 - 1) = spi- (1 + p + . .+.pn-1) mod (p(m+in)-(k+i)).
As s ranges over all integers not divisible by p, spi+ a(m + in)(pn-1 - 1) ther*
*efore
ranges over all integers
ae
= -1 - p - . .-.pi-1 mod (pi)
6= -1 - p - . .-.pi-1- pi mod (pi+1)
if i n - 1, and over all integers
ae
= -1 - p - . .-.pn-1 mod (pi)
6= -1 - p - . .-.pn-1 mod (pi+1)
if i > n - 1. It follows from these congruences that uspi+a(m+in)(pn-1-1)hm can*
*'t be
hit by one of the lower claimed differentials. These congruences also show tha*
*t if
i 6= -(1 + . .+.pn-1), then uihm is hit by a differential.
To show that these differentials actually occur, it now suffices to construct*
*, for each
i 0, an element euiin En*=In-1 such that eui= upi mod (vn-1) and such that
i+a(m+ni)(pn-1-1) __
(2.16) ffi(z)eui`=va(m+in)n-1up hm mod (a(m + in) + 1).
__
Here we write ffi(z) for the differential in C*c( , Fpn[[un-1]][u, u-1]); l*
*ater in the proof
we will also need to consider ffi(zpi).
14
If i = 0,Qletpeu0=iu.-Then1equationi2.16 holds by the previous corollary. In *
*general,
let eui= j=0zj(u). eui= up in En*=In and
i -1
z(eui)eu-1i= zp (u)u .
Now f(zpi) = f(z) + in (Proposition AI.3) and therefore, by Corollary 2.14 agai*
*n,
i pi -pi
u - zp u= ffi(z )(u)(z )
n-1-1) a(m+ni)+1
=` va(m+ni)n-1u1+a(m+ni)(p mod (vn-1 ).
(At first glance, this may be confusing, but remember that ffi(zpi)(u) is an el*
*ement of
Mapc(, Fpn[[un-1]][u, u-1]).) Thus
i -1
eui- z(eui)= (u - zp (u)) . (u eui)
i+a(m+ni)(pn-1-1) a(m+ni)+1
`= va(m+ni)n-1up mod (vn-1 ).
This implies that
i+a(m+ni)(pn-1-1) -1 a(m+ni)+1
ffi(z)(eui)(z-1) `=(va(m+ni)n-1up hm )(z ) mod (vn-1 )
__ *
*__
and hence, since both ffi(z)(eui) and v(a(m+ni)n-1u(o)hm are primitive in =*
* (a(m +
ni) + 1), that
i+a(m+in)(pn-1-1) __
ffi(z)(eui) `=va(m+in)n-1up hm mod (a(m + in) + 1).
This completes the proof.
Remark 2.17 The lengths of the differentials on the prime powers of u satisfy *
*the
recursion relation
a(m + (i + 1)n) = pn-1a(m + in) + (1 + p + . .+.pn-1).
This will be useful to us in our proof of Lemma 4.6, a main ingredient in the p*
*roof of
Theorem 3.
Proof of Theorem 2. Suppose that z 2 S0nis non-torsion in S0n=C0. Then by
Proposition AI.2, f(z) + N f(zpN) < 1. In particular, we may choose N large
enough so that f(zpN) > n=p - 1. The preceding result gives us that
pN>
(Fpn[[un-1]][u, u-1])= Fpn[vn-1].
This establishes part a.
15
The proof of part b will be accomplished-in1three easy steps. Observe that t*
*he
quotient Fpn[[ui, . .,.un-1]][u, u ] of En* is acted on by Gn since (p, u1, . *
*.,.ui-1) is
an invariant ideal.
Step 1. If 1 i n - 1, then (Fpn[[ui, . .,.un-1]][u, u-1])has no nonzero *
*elements
in negative degree. Similarly, (En*)is concentrated in nonnegative degrees.
Proof of Step 1. We use downward induction. The i = n - 1 case is just part a.
Now suppose that 2 j n - 1 and that (Fpn[[uj, . .,.un-1]][u, u-1])is con*
*cen-
trated in nonnegative degrees. If y 2 (Fpn[[uj-1, uj, . .,.un-1]][u, u-1])ha*
*s negative
degree and y 6= 0, then y = vtj-1x, where t > 0 and vj-1 - x. Since vj-1 is in*
*vari-
ant mod (p, v1, . .,.vj-2), x is invariant. But x has negative degree and is n*
*onzero
in Fpn[[uj, . .,.un-1]][u, u-1], contradicting the inductive hypothesis. Using*
* v0 = p,
the same proof works to show that if (Fpn[[u1, . .,.un-1]][u, u-1])is concen*
*trated in
nonnegative degrees, then so is En*.
Step 2. If 1 i n - 1, the ring of units in (Fpn[[ui. . .,.un-1]][u, u-1])is Fxpn.
Proof of Step 2. Again we use downward induction. As before, the case i = n - 1
follows from part a. Now suppose that (Fpn[[uj, . .,.un-1]][u, u-1])x = Fxpn*
*, and let
y be a unit in Fpn[[uj-1, . .,.un-1]][u, u-1]. By Step 1, y must have degree*
* 0, and
by the inductive hypothesis, y = c + vj-1x, where c 2 Fxpn. But x has negative *
*degree
and is invariant, so x = 0 by Step 1. This completes the inductive step.
Step 3. (En*)x = W Fxpn.
Proof of Step 3. Suppose that y is a unit in En*. By Step 1, y must be in de*
*gree 0.
Write y = c + x, where c 2 W Fxpnand x 2 (u1, . .,.un-1). Since
(Fpn[[u1, . .,.un-1]][u, u-1])x = Fxpn,
it follows that p | x. If x 6= 0, then x = ptw with p - w. w is invariant, so c*
* + w is a
unit in En*. It, however, does not reduce in Fpn[[u1, . .,.un-1]][u, u-1] to*
* an element
of Fxpn. This contradicts Step 2; therefore, x = 0 and y 2 W Fxpn.
3 Modules of essentially finite rank.
We have seen (Proposition 1.6 together with Theorem 2) that if z 2 S0nis non-to*
*rsion
in S0n=C0, then H*c(, Fpn[[un-1]][u, u-1]) is of essentially finite rank. Us*
*ing more or
less standard algebraic arguments, we will generalize this as follows.
Proposition 3.1Let M be an En*En-comodule, finitely generated as an En*-module,
such that pM = 0 and v-1iM = 0 for all i n - 2. Then, with z as above, H*c(, M)
is of essentially finite rank.
N pN
Remark 3.2 The hypotheses imply that (p, vp1, . .,.vn-2)M = 0 for N sufficient*
*ly
N
large; this in turn implies that the action of vpn-1on M is Gn-equivariant and *
*therefore
induces an action on H*c(, M).
16
Before giving the proof of Proposition 3.1, we observe that Theorem 1 is an i*
*mme-
diate consequence.
Proof of Theorem 1. The homotopy fixed point spectral sequence
H*c(, En*X) ) ss*(Ehn^ X)
is a spectral sequence of Fp[ ]-modules and collapses since Hsc(, En*X) = 0 *
*for s > 1.
Therefore, there exists an Fp[ ]-module F1 ss*(Ehn^ X) such that
H0c(, En*X) ss*(Ehn^ X)=F1
and
H1c(, En*X) F1.
By Proposition 3.1, ss*(Ehn^ X)=F1 and F1 are of essentially finite rank, an*
*d we will
see in Proposition 3.7 that this implies that ss*(Ehn^ X) is as well.
We begin our work on the proof of Proposition 3.1 with some general results on
good Fp[ ]-modules and on modules of essentially finite rank.
Lemma 3.3 Suppose that f : A ! B is a map of complete Fp[ ]-modules.
a. If A is compact in each degree_e.g. if A= A is of finite type_then cokerf *
*is
-adically complete.
b.If N B = 0 for N sufficiently large, then kerf is -adically complete.
Remark 3.4 By "complete," we always mean what other authors might refer to as
"complete Hausdorff."
Proof. It's standard (see e.g. [12, 23.D]) that every Cauchy sequence in cokerf*
* (with
the -adic topology)Tconverges. (This doesn't require compactnessTof A.) We thu*
*s only
need show that i i(cokerf) = {0}. Suppose then that __x2 i i(cokerf). If x 2*
* B
is a preimage of __x, we have that, for each i, there exists yi2 B and ai2 A su*
*ch that
x = iyi+ f(ai). But A is compact in each degree, so {ai}Tcontains a subsequen*
*ce
converging to some a 2 A. This implies that x - f(a) 2 i iB, and thus x - f(a)*
* = 0.
Hence __x= 0.
b. Clearly, kerf = limkerf=(kerf \ iA), so it suffices to show that, for ea*
*ch
i
k, there exists an i with kerf \ iA k kerf. But N A kerf and therefore
k+N A k kerf. This completes the proof.
Lemma 3.5 Suppose that 0 ! A g!B f!C ! 0 is a short exact sequence of Fp[ ]-
modules and that A and C are complete. If A= A and ker( : C ! C) are of finite
type, then B is complete.
17
T i T i i
Proof. We first show that iT Bi= {0}. Suppose b 2 i B and write b = bi
for each i 0. Since f(b) 2 i C, f(b) = 0. Now fix a positive integer N. Th*
*en
f( k-N bk) 2 ker( N : C ! C) for k N. But ker N is of finite type; thus th*
*ere
exists a sequenceTk0 < k1 < k2 < . .s.uch that f( ki-Nbki) is constant. In part*
*icular,
f( k0-Nbk0) 2 i iC and is therefore 0. Hence
g(a) = b = N ( k0-Nbk0) = N g(aN )
T
for some a, aN 2 A. Since N was arbitrary, a 2 i iA = {0}. This proves that b *
*= 0.
Next suppose that {xi} is a sequence in B such that xi+1- xi2 iB for all i. *
*Then
there exists x 2 B such that f(x) - f(xi) 2 iC for all i. This means that ther*
*e exist
bi2 B and ai2 A such that x - xi= vibi+ g(ai) for all i. But A is compact, so {*
*ai}
has a subsequence converging to some a. It then follows that limi!1xi = x - g(a*
*),
completing the proof.
Lemma 3.6 Suppose that 0 ! A g!B f!C ! 0 is a short exact sequence of complete
Fp[ ]-modules. If two of the three modules are good, then so is the third.
Proof. If M is any Fp[ ]-module, then
8
< M= M i = 0
TorFp[i](M, Fp) = ker( : M ! M) i = 1
: 0 i > 1.
The desired result now follows from the long exact sequence obtained by applying
TorFp[*]( , Fp).
Proposition 3.7Suppose that 0 ! A g!B f!C ! 0 is a short exact sequence of good
Fp[ ]-modules. If A and C are of essentially finite rank, then so is B.
Proof. We will show that if A is of essentially finite rank but B isn't, then n*
*either is
C. __ __
Let TA,_TB denote_the closure of the torsion submodules of A and B respective*
*ly.
Then (B=T B)=(A=T A) is of essentially finite rank if and only if C is of essen*
*tially
finite rank, since these modules are isomorphic modulo the closure of their tor*
*sion
submodules. Now write
__ Mk n
A=T A = iFp[ ]
i=1
__ Y n
B=T B = ffFp[ ],
ff2J
where the indexing set J is (countably) infinite. An easy induction shows that *
*it suffices
to consider the case where k = 1; that is, it suffices to show that
Y Y
M = nffFp[ ]=im( m Fp[ ] ! nffFp[ ])
ff2J ff2J
18
is not of essentially finite rank. But upon consideration of the groups St(M) (*
*see the
proof of Proposition 1.5 and its preceding discussion), this follows without di*
*fficulty.
Proof of Proposition 3.1. We first show that H*c(, M) is good. If M is torsi*
*on-
free as an Fp[ ]-module, this follows as in Proposition 1.5. In general, let T *
*denote the
-torsion submodule of M, and consider the short exact sequence
0 ! Cs-1! HsM ! Ks ! 0,
where Cs-1 = coker(Hs-1(M=T ) ! Hs(T )), Ks = ker(Hs(M=T ) ! Hs+1(T )), and
we write H*c(, ?) as H*(?). By Lemmas 3.5 and 3.6, it suffices to show that *
*Cs-1
and Ks are both good. Since T is of finite type and annihilated by a power of *
*, the
same holds true for HsT ; this immediately implies that the quotient Cs-1 is go*
*od.
Moreover, by Lemma 3.3b, Ks is complete. It is also good, by Lemma 3.6, since i*
*t sits
in the short exact sequence
0 ! Ks ! Hs(M=T ) ! ker(Hs+1T ! Hs+1M) ! 0.
Now let
0 = M0 M1 . . .Mt= M
be a Landweber filtration of M ([11, Theorem D]]; that is, each Mi is an En*En-
comodule and Mi+1=Miis isomorphic to either a suspension of Fpn[[un-1]][u, u-1]*
* or a
suspension of Fpn[u, u-1]. In the first case, Hs(Mi+1=Mi) is of essentially fin*
*ite rank
by Theorem 2 together with Proposition 1.6, and in the second case, Hs(Mi+1=Mi)*
* is
a module of finite type annihilated by and therefore of essentially finite ra*
*nk. We
will prove by induction on i that HsMiis of essentially finite rank.
First consider the short exact sequence
0 ! B ! H1(Mi) ! H1(Mi=Mi-1) ! 0,
where B = coker(H0(Mi=Mi-1) ! H1(Mi-1)). Since H0(Mi=Mi-1) is good, B is
complete (Lemma 3.3) and is good by virtue of the above exact sequence. If H1Mi*
*-1
is of essentially finite rank, it now follows that B is also. Therefore, by Pro*
*position
3.7, H1Miis of essentially finite rank.
Finally, we have the short exact sequence
0 ! H0(Mi-1) ! H0(Mi) ! D ! 0,
where D = ker(H0(Mi=Mi-1) ! H1(Mi-1)). H0(Mi=Mi-1) is either a suspension of
Fpn[u, u-1] or a suspension of Fpn[vn-1]; thus any submodule of H0(Mi=Mi-1) is *
*of
essentially finite rank. If H0(Mi-1) is of essentially finite rank, it now foll*
*ows as above
that H0(Mi) is as well. This completes the induction and the proof.
4 The Bockstein spectral sequence for some stan-
dard groups.
We begin with some generalities on analytic pro-p groups; our main reference wi*
*ll be
[8].
19
___p
Recall_that a pro-p group G is said to be powerful if ptishodd and G=G is ab*
*elian,
where Gp denotes the closure_of the group generated by p powers of elements of*
* G. If
p = 2, we require that G=G4 be abelian. If G is also (topologically) finitely g*
*enerated,
then every element of Gp is a pthpower of an element of G and Gp is open (and h*
*ence
closed) in G ([8, Lemma 3.4]). Similar statements apply to Gpi: Every element i*
*s a
(pi)thpower of an element of G and it is an open normal subgroup of G. In addit*
*ion,
the map
i pi+1 pi+1 pi+2
(4.1) Gp =G -! G =G
sending x to xp is a surjective homomorphism for all i 0 ([8, Theorem 3.6]). *
* G
is uniformly powerful, or just uniform, if it is powerful, finitely generated, *
*and the
homomorphism of 4.1 is an isomorphism for all i 0.
If G is a finitely generated powerful pro-p group, then G=Gp is a finite dime*
*nsional
Fp-vector space_addition is just the group multiplication_which we denote Gel.a*
*b..
Let E((Gel.ab.)*) denote the exterior Fp-algebra on the (vector space) dual of *
*Gel.ab..
If p is odd, there is a map
E((Gel.ab.)*) -! H*c(G, Fp)
which sends h 2 (Gel.ab.)* to the element of H1(G, Fp) given by the homomorphism
G -! Gel.ab.h-!Fp.
If p = 2, such a map exists provided G is uniform. For then G=G4 is a free Z=(*
*4)-
module and any basis for the F2-vector space Gel.ab.lifts to a basis for G=G4. *
*This can
be used to show that the cup square of the above cohomology class is 0.
The following remarkable result is due to Lazard (see also [21, Theorem 5.1.5*
*]).
Theorem 4.2 If G is a uniformly powerful pro-p group, then
E((Gel.ab.)*) -! H*c(G, Fp).
We will use this theorem in the following way. A set {y1, . .,.yd} is a minim*
*al set of
generators of the uniformly powerful pro-p group G if and only if the image of *
*this set in
G=Gp is a basis. Then H*c(G, Fp) is the exterior algebra on the dual basis {_y**
*1, . .,._y*d}.
We will construct the subgroup U of Theorem 3 to be uniformly powerful and so t*
*hat
one element, say y1, of a minimal generating set has reduced filtration smaller*
* than
the reduced filtrations of the other elements. We will then see that the differ*
*entials in
the Bockstein spectral sequence
H*c(U, Fp2[u, u-1]) ) H*c(U, Fp2[[u1]][u, u-1])
are determined by the differentials in
H*c(, Fp2[u, u-1]) ) H*c(, Fp2[[u1]][u, u-1]),
20
s(U, F 2*
*[[u ]][u, u-1])
and we will use our knowledge of this spectral sequence to prove that Hc p *
* 1
is of essentially finite rank. From here, we will use arguments given in the In*
*troduc-
xp)
tion to conclude that ss*(Eh(UxF2 ^ M) is of essentially finite rank. U will in*
* fact be
constructed as a standard group, whose definition and properties we now recall.
A p-adic analytic group G is standard (of dimension d over Qp) if the analytic
structure on G is defined by a homeomorphism
( d
(pZp) p odd
_ = (_1, . .,._d) : G -!
(4Z2)d p = 2
sending the identity to 0, and if there exists a d-dimensional formal group law
M(X, Y ) = (M1(X, Y ), . .,.Md(X, Y ))
over Zp such that
_j(xy) = Mj(_(x), _(y))
for all j and x, y 2 G. Every p-adic analytic group contains an open subgroup w*
*hich is
standard with respect to the induced analytic structure ([8, Theorem 8.29]), an*
*d every
standard group of dimension d over Qp is a uniform pro-p group of the same dime*
*nsion
([8, Theorem 8.31]). Moreover, if G is standard and p is odd, then
N-1 N d
_ : Gp - ! (p Zp)
is a bijection and the quotient map
__ GpN-1 (pN Zp)d ` Z 'd
_: ______N-! _________= ___
Gp (pN+1 Zp)d (p)
is a group isomorphism for all N 1. If_p_= 2, then _ : G2N-1 ! (2N+1 Z2)d is*
* a
bijection and the analogous quotient map _ is a group isomorphism. As mentioned
above, U will be a standard group; however, some further care will be required *
*for its
construction. The following result will be needed.
Lemma 4.3 Suppose that G is a standard group as above, and set ~ = 0 if p > 2 *
*and
~ = 1 if p = 2. Let
V = _-1[pN+~ Zpx (pN+a+~)Zp)d-1],
where N 1 and a 0. If N a, V is a normal subgroup of GpN-1, and if N a+*
*1,
V is standard.
Proof. We first recall some notation. We will write ff for a d-tuple (ff1, . *
*.,.ffd)
of nonnegative integers and set = ff1 + . .+.ffd. If X = (X1, . .,.Xd), X*
*ff=
Xff11.X.f.fdd. With these conventions,
X
(4.4) Mj(X, Y ) = Xj+ Yj+ bj,fffiXffY fi.
1
1
21
There are also power series Pj(X, Y ) over Zp, j = 1, . .,.d, such that
_j(xy-1) = Pj(_(x), _(y)),
and
X
(4.5) Pj(X, Y ) = Xj- Yj+ aj,fffiXffY fi,
(ff,fi)2I
where
I = {(ff, fi) 2 Nd x Nd : + 2 and 6= 0}.
Using 4.5, it's easy to check that if N a, then xy-1 2 V whenever x, y 2 V *
*, and
hence V is a subgroup. As for normality, we have that
N-1 pN-1 2N+2~
Pj(_(x), _(yp )) = _j(x) - _j(y ) mod (p )
whenever x 2 V , and hence
N-1 pN-1 -1 pN-1 pN-1
_j(yp x(y ) )= Mj(_(y ), P (_(x), _(y )))
= _j(x) mod (p2N+2~).
Since 2N N + a, this implies that ypN-1x(ypN-1)-1 2 V .
Finally, we prove that V is standard. There is certainly an analytic diffeomo*
*rphism
_0: V ! (p1+~Zp)d given by
_0(x) = (p1-N _1(x), p1-N-a_2(x), . .,.p1-N-a_d(x)).
The multiplication on V is given by
_0(xy) = M0(_0(x), _0(y)),
where (
p1-N Mj(ae(X), ae(Y ))j = 1
M0j(X, Y ) =
p1-N-aMj(ae(X), ae(Y ))j > 1,
and
ae(X1, . .,.Xd) = (pN-1 X1, pN+a-1X2, . .,.pN+a-1Xd).
It now follows easily from 4.4 that M0(X, Y ) has coefficients in Zp provided N*
* a + 1.
Continue with the assumptions of the previous lemma and assume N a + 1. If
y1, . .,.yd are the generators of GpN-1 with _(yi)a= pN+~affli, where ffliis th*
*e element of
Zdpwith only a 1 in the ithplace, then y1, yp2, . .,.ypdform a minimal set of g*
*enerators
of V . This follows from the fact_an easy consequence of 4.4_that
a a 2N+2~
_(yp ) = p _(y) mod (p )
whenever y 2 GpN-1.
22
0. Since taking pth powers rais*
*es
We are of course interested in subgroups of Sn
reduced filtrations (Proposition AI.2), this observation allows us to construct*
* an abun-
dance of uniformly powerful subgroups, one of whose generators has relatively l*
*ow
reduced filtration. The next result describes exactly how the Bockstein spectr*
*al se-
quence is controlled by such a generator.
Lemma 4.6 Suppose that U is a uniformly powerful subgroup of S02, not containe*
*d in
C0, with a minimal set of generators {x1, . .,.xd} such that f(xi) > f(x1) 3 *
*for all
i > 1 if p is odd, and f(xi) > f(x1) 5 if p = 2. Then
H*c(U, Fp2[[u1]][u, u-1]) H*c(, Fp2[[u1]][u, u-1]) E(__x*2, . .,*
*.__x*d)
as Fp[v1]-modules, where {__x*1, . .,.__x*d} H1c(U, Fp) is the basis dual to *
*the basis {__x1, . .,.__xd}
of the Fp-vector space Uel.ab.. In particular, Hsc(U, Fp2[[u1]][u, u-1]) is of *
*essentially fi-
nite rank.
Remark 4.7 Conditions on the reduced filtrations willnbe usedoin two places in*
* the
course of the proof. We will need that f(x1) > max 1, _2_p-1to allow us to com*
*pute the
differentials in the Bockstein spectral sequence for as in Theorem 2.15. W*
*e will also
need that a(f(xi)) a(f(x1)) + 2p + 2; this will allow us to prove that the di*
*fferentials
on ui in the Bockstein spectral sequence for U look the same as the differentia*
*ls in
the Bockstein spectral sequence for . We conjecturenthatothe evident analog*
*ue of
this theorem is true for n > 2 provided f(x1) > max 1, _n_p-1and the other f(x*
*i)'s
are sufficiently large compared to f(x1). Exactly how large they need to be is *
*however
unclear.
We defer the proof of Lemma 4.6 until x5. As remarked earlier, subgroups sati*
*sfying
the hypotheses of this lemma are ubiquitous.
Lemma 4.8 Let G be a closed subgroup of S0nsuch that the image of G in S0n=C0 *
*is
not all torsion. Given constants a and b, G then contains an open subnormal sub*
*group
U such that
i.U is uniformly powerful
ii.U has a minimal set of generators {x1, . .,.xd} with 1 > f(x1) b and f(xi)
f(x1) + a for all i > 1.
Proof. G is a p-adic analytic group, so let H be an open standard subgroup. L*
*et
_ : H ! (p1+~Zp)d, where ~ is as in Lemma 4.3, be an analyticndiffeomorphism, a*
*nd
N-1 pN-1o
let {y1, . .,.yd} be a minimal set of generators. Then yp1 , . .,.yd genera*
*tes
HpN-1; by Proposition AI.2, this implies that the generators z1, . .,.zd of HpN*
*-1 with
_(zi) = pN ffli each have reduced filtration at least N. Assume that f(z1) f(*
*zi) for
all i. By the assumptions on G, f(z1) < 1. If N max{b, a + 1}, then, by Lemma
4.3, i j
U = _-1 pN+~ Zpx (pN+a+~ Zp)d-1
23
pN-1. Moreover, as we saw bef*
*ore,
is an openpuniformlyapowerfulpnormalasubgroup of H
{z1, z2 , . .,.zd } is a minimal set of generators of U satisfying condition ii*
*. If N is in
T N-1
additionTchosen large enough so that HpN-1 g2G gHg-1, then Hp is normal in
-1
g2GgHg , which in turn is normal in G. Therefore, U is subnormal, completing *
*the
proof.
Proof of Theorem 3. By Remark AI.4, G satisfies the hypotheses of the previ-
ous lemma and therefore G contains an open subnormal subgroup U satisfying the
hypotheses of Lemma 4.6. This lemma, together with 0.2, implies that Hsc(U x
Fxp, Fp2[[u1]][u, u-1]) is of essentially finite rank. Moreover, Hs,tc(UxFxp, F*
*p2[[u1]][u, u-1])
is sparse, again by 0.2, and vanishes for s > 4. The continuous homotopy fixed *
*point
spectral sequence therefore collapses to give
xp)
sst-s(Eh(UxF2 ^ M) Hs,tc(U x Fxp, Fp2[[u1]][u, u-1])
xp)
and hence that ss*(Eh(UxF2 ^ M) is of essentially finite rank.
5 Proof of Lemma 4.6.
We begin the proof with the following sharpening of Theorem 2.15.
n *
* o
Lemma 5.1 Suppose that z 2 S02is not in C0 and that f(z) = m with m > max 1, *
*_2__(p-1).
Let ai = a(m + 2i), so that upisupports a nontrivial differential of length ai *
*in the
Bockstein_spectral sequence 2.1, and let ffi denote the differential in the cob*
*ar complex
C*c( , Fp2[[u1]][u, u-1]). For each i 0, there exists an element wiin Fp2*
*[[u1]][u, u-1]
such that i __
ffiwi=`vai1up +ai(p-1)hm mod (ai+ 1)
and such that
i i-1XX ff -pjafi pj
wi= up + cjsv js u0 js,
j=0 s
where cjs2 Fp2, p - fijs, and ffjs pj+1ai-j-1. In particular, wi= upi mod (v1).
Remark 5.2 The numbers cjs, ffjs, and fijs all depend upon i as well, but this*
* is
omitted from the notation.
Proof. We proceed by induction. First take w0 = u. Next suppose that wi-1has be*
*en
constructed and write
i-1 i-2XX ffjs-pja0fi pj
wi-1= up + cjsv1 u js
j=0 s
with cjs2 Fp2, p - fijs, and ffjs pj+1ai-j-2. Then
i i-1XX p pffj-1,s-pja0fi pj
wpi-1= up + cj-1,sv1 u j-1,s,
j=1 s
24
j+1a . We also have
and pffj-1,s p i-j-1
i+pa (p-1)p __
ffi(wpi-1) `=vpai-11up i-1 hm mod (pai-1+ p),
and, by Corollary 2.14,
i+pa (p-1)p __
ffi(vpai-1-a01ufi0) `=vpai-11up i-1 hm mod (pai-1+ 2),
where fi0 = 1 mod (p). Thus
i+(pa +2)(p-1)
ffi(wpi-1+ c0vpai-1-a01ufi0) = fl0vpai-1+21up i-1 hm
__
mod (pai-1+ 3) for some scalars c0 and fl0. (Of course, fl0 may be 0.) B*
*ut
upi+(pai-1+2)(p-1)hm is hit by a differential of length a0on ufi1, where fi1 = *
*-1 mod (p);
hence
i+(pa +3)(*
*p-1)
ffi(wpi-1+ c0vpai-1-a01ufi0+ c1vpai-1+2-a01ufi1) = fl1vpai-1+31up i-1 *
* hm
__
mod (pai-1+ 4) for some scalars c1 and fl1. We may continue in this manner *
*to
obtain
0 1
p-1X
i+(pa +p+1)(p-1)
ffi @wpi-1+ cjvffj-a01ufijA= flvpai-1+p+11up i-1 hm
j=0
__
mod (pai-1+ p + 2), with each ffj pai-1and each fij not divisible by p. S*
*ince
upisupports a differential of length ai= pai-1+ (p + 1) (see Remark 2.17), we m*
*ust
therefore have that fl 2 Fxp2, and we may take
p-1X
wi= wpi-1+ cjvffj-a01ufij.
j=0
This completes the induction.
Proof of Lemma 4.6. By Theorem 4.2 and Proposition AI.1, the initial term in the
Bockstein spectral sequence for U is
H*c(U, Fp2[u, u-1]) = Fp2[u, u-1] E(__x*1, . .,.__x*d).
Since Fp2 Fp2[u, u-1], each __x*iis a permanent cycle. We will show that the d*
*ifferen-
tials on Fp2[u, u-1] agree with the corresponding differentials in the Bockstei*
*n spectral
sequence for and prove that the Bockstein spectral sequence for U is the t*
*ensor
product of the Bockstein spectral sequence for with E(__x*2, . .,.__x*d). *
*Let us write dr
for the rthdifferential in the Bockstein spectral sequence for U and ffi for th*
*e differential
in the continuous cobar complex C*c(U, Fpn[[u1]][u, u-1]). We will also write *
*ffi(i) for
the differential in C*c(, Fp2[[u1]][u, u-1]).
25
-1] guaranteed by Lemma 5.1 with z = x *
*. I
Let wj be an element in Fp2[[u1]][u, u 1
claim that
j+a (p-1)_* aj+1
(5.3) ffiwj `=vaj1up j x1 mod (v1 ),
where aj = a(f(x1) + 2j). It now follows easily that, if p - s,
j spj+a (p-1)_*
(5.4) dajusp `=u j x1
and is nontrivial.
To prove the claim, begin by writing
ffiwj = vk1fl,
where fl 2 C1c(U, Fp2[[u1]][u, u-1]) is a cocycle which is nontrivial in the qu*
*otient
C1c(U, Fp2[u, u-1]). Then
Xd
fl = ciu(o)_x*imod(v1),
i=1
and at least one ci2 Fp2is nonzero. Moreover,
ffi(i)wj = civk1u(o)_x*imod(vk+11),
whence k a(f(xi) + 2j). In particular, k aj. But, with the notation of Lemm*
*a 5.1,
ffi(i)wj = 0 mod (vNij1),
where
Nij = min {pja(f(xi)), ffts- pta0+ pta(f(xi)) : 0 t j - 1}
min {pja(f(xi)), pt+1aj-t-1- pta0+ pta(f(xi)) : 0 t j - 1}.
If a(f(xi)) a0 + 2(p + 1), it follows without difficulty from the relation at*
*+1 =
pat+ (p + 1) that this minimum is at least aj+ 1. By the construction of wj, we*
* must
thus have k = aj and hence equation 5.3.
By 5.4 and Theorem 2.15, each uj__x*1is a boundary_and thus a permanent cycle_
in the Bockstein spectral sequence for U, except when j = -(p+1). However, u-p-*
*1__x*1
is still a permanent cycle. To see this, simply observe that multiplication by *
*upk acts
isomorphically on the Ek-term of this spectral sequence, commutes with dk, and *
*that
upk-p-1__x*1is a permanent cycle. All the differentials now follow easily from *
*5.4, and
the Bockstein spectral sequence for U behaves as claimed.
26
Appendix I
We begin by fixing our notation concerning the Morava stabilizer group. Recall *
*that
Sn is the group of automorphisms over Fpn of the height n formalPgroup law n w*
*ith
p-series [p] n = xpn. An element of Sn is then of the form in 0bixpi, where b*
*i2 Fpn
for all i and b0 2 Fxpn. Such an element is in S0nif and only if b0 = 1. The *
*center
P ni
C Zxpconsists of power series of the form in 0aixp with each ai2 Fp and a0*
* 2
P ni P
Fxp. More precisely, in 0aixp corresponds to i 0e(ai)pi, where e(ai) denot*
*es the
multiplicativePrepresentative of aiin Zp.
If z = in 0bixpi2 Sn, define f(z), the reduced filtration of z, to be the s*
*mallest
value of i for which (
bi6= 0 ifn - i
bi62 Fpifn | i.
If z 2 C, set f(z) = 1. Alternatively, if z 2 S0n, f(z) is the maximum value of*
* i for
which there exists g 2 C with
X j
zg = x + n nbjxp .
j i
Note that bf(z)6= 0 if n - f(z) and bf(z)62 Fp if n | f(z).
We need some basic properties of the action of Sn on En*. The next result is *
*easy
and well-known.
P i
Proposition AI.1 a. Let g = in 0bixp 2 Sn. Then gu = b0u in En*=In.
b.If g 2 C = Zxp, then g(u) = gu and g(ui) = ui for 1 i n - 1.
Proof. Part a is immediate from the definition of the action.
Part b follows easily from the formulas of [6]; however, a more elementary pr*
*oof is
possible. First observe that if F is any formal group law over a p-adically com*
*plete
ring, there is a unique continuous ring homomorphism Zp ! End(F ) extending the*
* ho-
momorphism Z ! End(F ) mapping n to [n]F(x). An examination of Cartier modules
(see for example [6]) shows that, in the case F = n, the restriction Zxp! Aut(*
* n)
of this map is just the above identification Zxp-! C Aut( n). If Fn denotes *
*the
universal lift of n to (En)0 and [g]Fn(x) denotes the image of g 2 C = Zxpunde*
*r the
homomorphism Zxp! Aut(Fn), it follows by naturality that
Fn____________________-[g]Fn(x)Fn
| |
| |
| |
| |
| |
|? g |?
n____________________- n
commutes. This implies that g(ui) = uifor all i. One can also check that [g]0Fn*
*(0) = g,
which implies that g(u) = gu and completes the proof.
We next turn to establishing some basic properties of the reduced filtration.
27
0. Then
Proposition AI.2Suppose y, z 2 Sn
i.f(yz) min{f(y), f(z)}
ii.f(yp) f(y) with equality if and only if y 2 C0
iii.f(y-1) = f(y)
Proof. If either y or z is in C0, then i, ii and iii are immediate, so suppose *
*y 62 C0,
z 62 C0. Write f(y) = t, f(z) = s and choose g, h 2 C0 such that
X i
yg = x + n nbixp
i t
X i
zh = x + n ncixp .
i s
Then min{s,t}
(yz)(gh) = (yg)(zh) = x mod (xp ),
so f(yz) min{s, t}, proving i. As for ii,
t pt+1
ypgp = (yg)p= x + pbtxp mod (x )
t+1
= x mod (xp ),
so that f(yp) t + 1. Finally, part iii is proved using the same strategy.
If f(y) is not too small, part ii of the preceding proposition can be conside*
*rably
sharpened.
Proposition AI.3Suppose y 2 S0nand f(y) > n=(p - 1). Then f(yp) = f(y) + n.
Proof. We may assume that y 62 C0. Write f(y) = t and choosePg 2 C0 so that
yg = (e + ffi) in the ring End( n), where e = x and ffi = inbtixpi. Then
p-1X
(yg)p = (e + ffi)p = e + ffip + cj. pffij,
j=1
p
where cj = j =p. Since multiplication in End( n) is composition of power seri*
*es,
the leading term of ffij is in degree ptj; moreover, since [p] n(x) = xpn, the *
*leading
term ofPpffij is in degree ptj+n. Finally, sincentp > t + n and c1 = 1, it fol*
*lows that
ffip + p-1j=1cj. pffij has leading term bptxpt+n= btxpt+n. Thus
f(yp) = f((yg)p) = f(yg) + n = f(y) + n.
Remark AI.4 The above proposition implies that y is non-torsion if f(y) > _n_p*
*-1, and
that S0n=C0 is torsion-free if p > n + 1. More generally, an argument similar t*
*o the
proof of Theorem 4.3.4 in [15] shows that S0n=C0 is torsion-free if neither p n*
*or p - 1
divide n.
28
Appendix II
Proposition AII.1Let G be a closed subgroup of Gn, and suppose that U is an open
subnormal subgroup of G. Then EhGnis in the thick subcategory generated by EhUn.
Proof. Without loss of generality, we may assume that U is normal in G. In [3],*
* we
used Adams spectral sequence techniques to construct a spectral sequence
(AII.2) E**2(Z) = H*(G=U, (EhUn)*(Z)) ) (EhGn)*(Z).
This allowed us to establish strong convergence and to prove in addition that t*
*here
exists s0 such that Es,*1(Z) = 0 for all s > s0 and spectra Z. (This follows fr*
*om [3,
Theorem 3.3]; see Proposition 2.3 of the same paper.) Of course, since G=U is f*
*inite,
one can form an ordinary homotopy fixed point spectral sequence, and we showed *
*in
[3, Theorem A.1] that this spectral sequence agrees with the spectral sequence *
*AII.2.
Let Yk = F (E(G=U)k, EhUn), where E(G=U)k is the k-skeleton of the bar constr*
*uc-
tion for the free contractible G=U-space E(G=U). The ordinary homotopy fixed po*
*int
spectral sequence is obtained by mapping Z into the diagram
*_______oe_______oe_______oe_______oeY0Y1Y2. . .
@ ` @ ` @ `
@@R @@R @@R
F0 F1 F2 .
Fiis the fiber of Yi! Yi-1; it is a suspension of a finite number of copies of *
*EhUn. In
addition ([7, Proposition 7.1]),
holimY-k= (EhUn)h(G=U)~=EhGn.
k
By strong convergence, {ker([Z, EhGn]* ! [Z, Ys])} is a complete Hausdorff filt*
*ration of
[Z, EhGn]* and
ker([Z, EhGn]* ! [Z, Ys-1]*)
Es,*1(Z) = _______________________.
ker([Z, EhGn]* ! [Z, Ys]*)
The horizontal vanishing line thus implies that ker([Z, EhGn]* ! [Z, Ys0]*) = 0*
* for all Z
and therefore that EhGnis a retract of Ys0. But by virtue of our description of*
* the Fi's,
Ys0is in the thick subcategory generated by EhUn; hence so is EhGn.
Remark AII.3 The same proof works to show that EhGnis in the thick subcategory
generated by EhUnin the stable category of EhGn-modules.
References
[1]E. S. Devinatz, Morava's change of rings theorem, The C~ech Centennial, Cont*
*emp.
Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 83-118.
29
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