Homotopy groups of homotopy fixed point spectra
associated to En
Ethan S. Devinatz*
Department of Mathematics, University of Washington, Seattle, Washington, USA
Email: devinatz@math.washington.edu
URL: http://www.math.washington.edu/~devinatz/
Abstract
We compute the mod (p) homotopy groups of the continuous homotopy fixed
point spectrum EhH22for p > 2, where En is the Landweber exact spectrum
whose coefficient ring is the ring of functions on the Lubin-Tate moduli space *
*of
lifts of the height n Honda formal group law over Fpn, and Hn is the subgroup
W Fxpno Gal(Fpn=Fp) of the extended Morava stabilizer group Gn. We examine
some consequences of this related to Brown-Comenetz duality and to finiteness
properties of homotopy groups of K(n)*-local spectra. We also indicate a plan
for computing ss*(EhHnn^ V (n - 2)), where V (n - 2) is an En*-local Toda
complex.
AMS Classification numbers Primary: 55Q10, 55T25
Secondary:
Keywords: Brown-Peterson homology, Morava stabilizer group, K(n)*-local
homotopy theory
____________________________*
Partially supported by a grant from the NSF
1
Introduction
Let En denote the Landweber exact spectrum with coefficient ring
En* = 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
< uiu1-p i < n
r(vi) = u1-pn i = n ,
:
0 i > n
where vi2 BP* is the ith Hazewinkel generator. In particular, each ui has de-
gree 0 and u has degree -2. En is a commutative ring spectrum, and Morava
theory tells us that the group of ring automorphisms of En is isomorphic to the
profinite group Gn = Sn o Gal, where Sn denotes the group of (not necessarily
strict) isomorphisms of the height n Honda formal group law over Fpn, and
Gal is the Galois group of Fpn=Fp. A priori, Gn acts on En only in the sta-
ble category, but Hopkins and Miller (later improved by Goerss and Hopkins)
proved that this can be made an honest action in an appropriate point set cat-
egory of spectra (see [9] and [13]). "Continuous homotopy fixed point spectra"
may also be constructed [7]: if G is a closed subgroup of Gn, the continuous
homotopy G fixed point spectrum will be denoted by EhGn; if G is finite, this
spectrum agrees with the ordinary homotopy fixed point spectrum. Moreover,
EhGnn' LK(n)S0, the K(n)*-localization of S0, EhGnhas the expected functo-
rial properties, and there is a strongly convergent "continuous homotopy fixed
point spectral sequence"
H*c(G, E*nX) ) (EhGn)*X
for any spectrum X . (H*c(G, E*nX) denotes the continuous cohomology of G
with coefficients in the profinite G-module E*nX .)
The hope of this paper is to make some headway towards the computa-
tion of ss*EhGn, for G a closed subgroup of Gn. At first sight, this program
seems impossible: the formulas for the action of (most elements of) Gn on En*
are extremely complicated (see [6]), making the computation of H*c(G, En*)
apparently inaccessible. However,
H*c(Gn, N) = Ext*Mapc(G,En*)(En*, N),
where (En*, Map c(G, En*)) is the complete Hopf algebroid defined using the
action of G on En* (see for example [5]). Since Map c(G, En*) is a quotient of
Map c(Gn, En*) = En* bBP* BP*BP b BP*En* E^n*En,
2
one may try to use the Hopf algebroid structure maps in BP*BP together with
several Bockstein spectral sequences to go from, for example, H*c(G, En*=In)
to H*c(G, En*). As usual, In is the maximal ideal (p, u1, . .,.un-1) in En*.
Let Hn = W FxpnoGal Gn, where W Fxpnis the subgroup of Sn consisting
of the diagonal matricesi(see x1), andjlet M(p) denote the mod (p) Moore
spectrum. We compute ss* EhH22^ M(p) for all primes p > 2 (Theorem 3.8).
Of course, ss*LK(2)M(p) is known ([14], [15]) for p > 2, so it is unclear if our
computation yields any new homotopy information. Our computation is, how-
ever, much simpler and already indicates the necessity of "p-adic suspensions"
in the Gross-Hopkins work on Brown-Comenetz duality (Remark 3.9). More-
over, we believe that computations such as ss* EhHnn^ V (n - 2) _ recall that
the Toda complex V (n - 2) exists En*-locally whenever p is sufficiently large
compared to n _ should be accessible to more skilled calculators.
Even when a complete calculation of ss*EhGnis unattainable, partial infor-
mation can lead to interesting consequences. For example, it is a long-standing
conjecture that ss*LK(n)S0 is a module of finite type over the p-adic integers
Zp. (This conjecture is known to be true for n = 1 and, if p 3, for n = 2
[16], [17].) By a thick subcategory argument _ see [3] for a discussion of this
in the En*-local category _ if ss*LK(n)X is of finite type for some X in the
En*-local category, then ss*LK(n)Y is of finite type for any finite Y such th*
*at
{m n : K(m)*Y 6= 0} {m n : K(m)*X 6= 0}.
This in turn only requires that we prove that ss*(EhGn^ X) is of finite type for
some closed subgroup G of Gn for which there exists a chain
G = K0 C K1 C . .C.Kt= Gn
of closed subgroups. Indeed, assume inductively that ss*(EhKin^ X) is of finite
type. Then, since Ki+1=Ki is a p-adic analytic profinite group (see [8, Theorem
9.6]), we have that H*c Ki+1=Ki, ss*(EhKin^ Y )is also of finite type. (This
follows from the fact that any p-adic analytic profinite group is of type p-F P1
in the language of [19].) But, in [7], we constructed a strongly convergent
spectral sequence
i j
H*c Ki+1=Ki, ss*(EhKin^ X) ) ss*(EhKi+1n^ X)
and showed that its E1 term has a horizontal vanishing line. This implies
that ss*(EhKi+1n^ X) is of finite type and hence by induction so is ss*LK(n)X =
ss*(EhGnn^ X).
These considerations are unfortunately not applicableito Gj= Hn, since the
normalizer of Hn in Gn is Hn, and, moreover, ss* EhH22^ M(p) is not even
3
of finite type. Yet it is, in some sense, "almost" of finite type (see x4), alt*
*hough
the significance of this property is not clear.
1 H*c(Hn, En*=In) and its Hopf algebroid description
Recall that the group Sn may be described in several ways. If n denotes
the height n Honda formalPgroup law over Fpn, then Sn consists of all formal
power series of the form in 0bixpi with each bi2 Fpn and bi6= 0. The ring of
endomorphisms of n may also be described as the ring obtained by adjoining
an indeterminate S _ which corresponds to the endomorphism f(x) = xp _
to W Fpn along with the relations Sn = p and Sw = woeS , wherePoe : W Fpn !
W Fpn denotes the frobenius automorphism. The automorphism in 0bixpi
P n-1
corresponds to the element i=0 aiSi with
X
ai= e(bi+nk)pk,
k 0
where e(b) is the multiplicative representative of b in W Fpn. The subgroup
W Fxpnof Sn is then the group of automorphisms with ai= 0 for all i > 0. In
terms of matrices, Sn is the subgroup of GLn(W Fpn) consisting of matrices of
the form
2 3
a0 pan-1 pan-2 . . . pa1
66 aoe-11 aoe-10 paoe-1n-1. . .paoe-127
66 .. .. .. 77
66 . . . 777 ,
4 ... ... ... paoe-(n-2)n-15
aoe-(n-1)n-1aoe-(n-1)n-2aoe-(n-1)n-3.a.o.e-(n-1)0
and W Fxpnis the subgroup of diagonal matrices in Sn.
Now let S0nbe the p-Sylow subgroup of Sn consisting of strict automor-
phisms of n. There is a split extension
S0n! Sn ! Fxpn;
P n-1 __
the map Sn ! Fxpnis given by i=0aiSi 7! a0, and the splitting sends a 2 Fpn
to e(a) 2 W Fxpn Sn. This map also gives us a splitting of the short exact
sequence
0 ! W F0pn! W Fxpn! Fxpn! 0,
and hence an isomorphism W Fxpn! W F0pnx Fxpn. Since the order of Fxpnis
prime to p, it follows that
x
H*c(W Fxpn, N) -! H*c(W F0pn, N)Fpn
4
x
whenever N is a discrete Zp [W Fpn] -module. If, in addition, N is a W Fpn-
module and Hn-module in such a way that oe(c) = coeoe(n) for all c 2 W Fpn and
n 2 N , then it follows from [1, Lemma 5.4] that Hi Gal , H*c(W Fxpn, N)= 0
for all i > 0, and hence
H*c(Hn, N) -! H*c(W Fxpn, N)Gal.
Now Sn acts on En*=In = Fpn[u, u-1] via Fpn-algebra homomorphisms,
and the action on u is given by
`n-1 '
P i __
aiS (u) = a0u, (1.1)
i=0
where, once again, __a0is the mod (p) reduction of a0. From this it follows
that
H*c(W Fxpn, Fpn[u, u-1]) = Fpn[vn, v-1n] FpnH*c(W F0pn, Fpn).
Moreover, since Gal acts trivially on vn,
H*c(Hn, Fpn[u, u-1]) = Fp[vn, v-1n] H*c(W F0pn, Fpn)Gal.
It is also easy to compute H*c(W F0pn, Fpn)Gal. Let gi 2 Hom c(W F0pn, Fpn)
be defined by _ !
P j pi oei
gi 1 + e(cj)p = c1 = c1 , (1.2)
j 1
0 i n - 1. Since the Galois automorphisms id, oe, . .,.oen-1 are linearly
independent over Fpn, so are the gi's. Now, and for the rest of this section,
assume that p > 2. Then Znp W Fpn- ! W F0pn(via the map sending x 2
P pjxj
W Fpn to exp(px) = 1 + j 1____j!2 W F0pn), so that H*c(W F0pn, Fpn) is the
exterior algebra over Fpn on n generators in H1c(W F0pn, Fpn). This implies
that these generators may be taken to be g0, g1, . .,.gn-1. Each gi is Galois
invariant, so
H*c(Hn, Fpn[u, u-1]) = Fp[vn, v-1n] E(g0, g1, . .,.gn-1).(1.3)
Next consider the complete Hopf algebroid En*, Map c(W F0pn, En*) (En*, *
*n).
We explicitly identify n=In n as a quotient of E^n*En=InE^n*En and give cobar
representatives for
gi2 H1,0c(W F0pn, Fpn[u, u-1]) = Ext1,0 n=In(nFpn[u, u-1], Fpn[u, u-1]).
First recall that the maps jL, jR : En* ! Map c(Gn, En*) are given by
jR (x)(s) = x, jL(x)(s) = s-1x. Since W F0pn Gn acts trivially on W Fpn
5
fi fi
En*, it follows that jR fifi = jLfifi in n, so that n is a Hopf algebra
WFpn WFpn
over W Fpn and is a quotient of
W Fpn Zp Map c(Sn, En*)Gal= W Fpn Zp (EGaln*bBP*BP*BP b BP*EGaln*)
= W Fpn Zp (EGaln)^*EGaln,
where EGaln is the Landweber exact spectrum with coefficient ring
Zp[[u1, . .,.un-1]][u, u-1].
Now let u jR (u) and w jL(u) in (EGaln)^*EGaln. By 1.1, we have that
u = w in n=In n. Moreover, the image of tj 2 BP*BP in (EGaln)^*EGaln_
also denoted tj _ satisfies
0 1
X i i
tj@ n bixp A= u1-p b-10bi mod In(EGaln)^*EGaln
i 0
(see [1, Proposition 2.11]), and thus
n=In n = Fpn[u, u-1][tn, t2n, . .].=Jn, (1.4)
with
n pn-1 pn p2n-1 pn pjn-1
Jn = (tpn - vn tn, t2n - vn t2n, . .,.tjn - vn tjn, . .)..
Finally, let g = v-1ntn 2 n. These considerations imply that gpi 2
n=In n is a cobar representative for gi2 H1(W F0pn, Fpn).
2 The Bockstein spectral sequence
Fix a prime p and integer n 2, and let N be a complete (En*, Map (Hn, En*))-
comodule. Write
x oGal
H*N H*c(Hn, N) = H*c(W F0pn, N)Fpn
x oGal
= Ext * n(En*, N)Fpn .
The Bockstein spectral sequence we will use is defined by the exact couple
vn-1
H*(En*=(p, u1,P. .,.un-2))oo___ H*(En*=(p,7u1,7. .,.un-2)) (2.1)
PPPP nnnnn
PPPP nnnn
PPP'' nnn
H*(En*=In).
6
Truncated, this spectral sequence is isomorphic to the spectral sequence of the
unrolled exact couple
022oo____H*(En*=In)_Eoo_____H*(En*=(p,_.<.,.un-2,Pu2n-1))oo___.<.6.6DD
22 yyyy EEEE nnnnnn PPPPPP
22 yy EE nnvn-1n PPPP v2
ssss2yy ""E nnn P(( n-1
H*(En*=In) H*(En*=In) H*(En*=In)
Since H*.*(En*=(p, u1, . .,.un-2, ukn-1)) is finite in each bidegree, this spec*
*tral
sequence converges strongly to
H*(En*=(p, . .,.un-2)) = limH*(En*=(p, u1, . .,.un-2, ujn-1)).
j
3 Computation of ss*(EhH2 ^ M (p))
In this section, we specialize the above spectral sequence to the case n = 2,
p > 2. Write ~ 2= 2=p 2, and let g = v-12t2 2 ~ 2as in x1.
We will need the following congruences for our calculation of the different*
*ials
in the Bockstein spectral sequence.
2 p2 p+1
Lemma 3.1 v1-p2t2 = t2 mod v1 ~ 2.
Proof Begin with the formula (see [12, Theorem A2.2.5])
X F i X pi
tijR (vj)p = Fvitj
i,j 0 i,j 0
in BP*BP=pBP*BP , where F is the universal p-typical formal group law on
BP*. Up through power series degree p4 we then have
2 p3
v1 +F vp1t1 +F vp1t2 +F v1 t3 +F jR (v3) +F t1jR (v2)p +F t2jR (v2)p2
2 p2 (3.2)
= v1 +F v1tp1+F v1tp2+F v1tp3+F v2 +F v2tp1+F v2t2
in E^2*E2=pE^2*E2. But jR (v2) = v2 + v1tp1- vp1t1 in BP*BP=pBP*BP , and
therefore, since t1 2 v1~ 2, jR (v2) = v2 mod vp+11~ 2. t3 is also in v1~ 2; h*
*ence,
mod vp+11~ 2, equality reduces to
2 p p2
vp2t1 +F vp2t2 = v1t2 +F v2t2 .
The desired result follows immediately from this equation. __|_|
Lemma 3.3 t1 = v1gp - vp+21v-12g + vp+21v-12gp mod v2p+31~ 2.
7
Proof From [12, Corollary 4.3.21],
2 p p p2 p 1+p2 p2 1+p p p
jR (v3) = v3+v2tp1+v1t2-v2t1-v1 t2-v1t1 +v1 t1 +v1w1(v2, v1t1, -v1t1)
in BP*BP=pBP*BP , where w1(x, y, z) 1_p[xp+ yp+ zp - (x + y + z)p]. Hence
0 = v1tp2- vp2t1 - v21vp-12tp1+ vp+11vp-12t1 mod v2p+31~,2(3.4)
and thus
t1 = v-p2v1tp2 mod vp+21~ 2.
Plug this relation for t1 back into the last two terms of 3.4 to get
2+p-1 p2 p+2 -1 p 2p+3
vp2t1 = v1tp2- vp+21v-p2 t2 + v1 v2 t2 mod v1 ~ 2.
By the previous lemma,
2+p-1 p2 p+2 p-2 2p+3
vp+21v-p2 t2 = v1 v2 t2 mod v1 ~ 2.
We then get the desired result. __|_|
The next propositions will allow us to compute the Bockstein differentials *
*on
vk22 H0(Fp2[u, u-1]). In what follows, we will often suppress left multiplicati*
*on
by powers of v2 on one side of an equation, since the appropriate power can
always be determined by examining gradings. For example, we might write the
conclusion of the previous lemma as
t1 = v1gp - vp+21g + vp+21gp mod v2p+31~.2
Proposition 3.5 In ~ 2=v3p+31~ 2,
2 p 2(1+p) p s - 1 2(1+p) p2 p 2
jR (v2)s - vs2= s[v1+p1(gp - g ) + v1 (g - g ) + _____v1 (g - g ) ].
2
Proof Compute in ~ 2=v3p+31~:2
jR (v2)s - vs2= vs2[(v-12jR (v2))s - 1]
= vs2[(1 + v-12v1tp1- v-12vp1t1)s - 1]
s - 1 p p 2
= s[(v1tp1- vp1t1) + _____(v1t1 - v1t1) ]
2
2 p+1 p 2(p+1) p s - 1 2(p+1) p2 p 2
= s[vp+11gp - v1 g + v1 (g - g ) + _____v1 (g - g ) ]
2
by the previous lemma. __|_|
8
Proposition 3.6 Suppose that p 6 |s and k 0. Then there exists zspk2 E2*
k
such that zspk= vsp2 mod I2 and
k+pk-1+...+1)(p+1)p (pk+pk-1+...+p+2)(p+1)
dzspk jR (zspk)-zspk= cv(p1 (g -g) mod v ~ 2
for some c 2 Fxp.
Proof Proceed by induction on k. If k = 0, let zs = vs2. Then by the
preceding proposition,
2 p 2(p+1)
jR (zs) - zs = svp+11(gp - g ) mod v1 ~ 2.
But gp2 = g mod vp+11~ 2, so we get the desired result.
Suppose now that zspk-1has been chosen. Then
k+pk-1+...+p)(p+1)p2 p (pk+pk-1+...+p2+2p)(p+1)
d(zspk-1)p = cv(p1 (g - g ) mod v1 ~ 2.
k-pk-1-...-p+1
Next consider dv(s-1)p2 . Since the exponent of v2 is equal to 1
mod (p), we have that
k-pk-1-...-p+1 1+p p2 p 2(1+p) p 3p+3
dv(s-1)p2 = v1 (g - g ) + v1 (g - g ) mod v1 ~ 2.
Then take
k+pk-1+...+p-1)(p+1)(s-1)pk-pk-1-...-p+1
zspk= (zspk-1)p - cv(p1 v2 .
__|_|
Corollary 3.7 Suppose that p 6 |s and k 0. Up to multiplication by a unit
in Fp,
k (s-1)pk-pk-1-...-p-1p
d(pk+pk-1+...+1)(p+1)vsp2= v2 (g - g)
in the Bockstein spectral sequence 2.1. In particular, vt2(gp - g) is a boundary
for all t 2 Z.
Now, essentially by their definition, g and gp are permanent cycles; more-
over,
H*(Fp2[u, u-1]) = Fp[v2, v-12] E(g + gp, g - gp).
Using the preceding corollary, it's easy to read off the remaining differential*
*s,
yielding our main result.
Theorem 3.8 If p 3
8
>>>:Q Fp[v1] 1
t2Z _____(vnt1){cti} i = 2
9
as Fp[v1]-modules, where i = g+gp, ct reduces to vt2(g-gp) 2 H1(Fp2[u, u-1]),
and
ae
nt= p(+p1i+ pi-1+ . .+.1)(p + 1) tt6==-1(smod- (p)1)pi- pi-1- . .-.p - 1, s*
* 6= 0 mod (p).
By sparseness,
sst-s(EhH22^ M(p)) Hs,t(Fp2[[u1]][u, u-1]).
Remark 3.9 Let In denote the Brown-Comenetz dual of LnS0, the En*-
localization of S0. In is characterized by
ss0F (X, In) = [X, In]0 = Hom (ss0LnX, Q=Z(p))
for any spectrum X . In [10] (see also [18]), Gross and Hopkins establish a
remarkable relationship between Brown-Comenetz and Spanier-Whitehead du-
ality: they prove that if p is sufficiently large compared to n 2, and if X
is a K(n - 1)*-acyclic finite complex with pEn*X = 0 and with vn self-map
2pN (pn-1)X ! X , then
F (X, In) ' ffLnDX, (3.10)
where ff is any integer with
ff = 2pnN (pn - 1=p - 1) + n2 - n mod (2pN (pn - 1)).
(As usual, DX denotes the Spanier-Whitehead dual of X .) There is, however,
no integer ff for which 3.10 is satisfied for all X . This contrasts with the
situation when n = 1: here we have I1 ' 2L1(S0p) (if p > 2), where S0p
denotes S0 completed at p, and thus F (X, I1) ' 2L1DX whenever X is a
rationally acyclic finite spectrum.
Historically, it was Shimomura's calculation [14] of ss*L2M which shattered
the hope that I2 might also be an integral suspension of L2(S0p). Our calculati*
*on
of ss*(EhH22^ M(p)) yields this result as well; a sketch of the proof follows.
Suppose there existed an integer c with
F (M(p, vk1), I2) ' cL2DM(p, vk1) (3.11)
for a cofinal set of k, where M(p, vk1) denotes a finite spectrum with BP*M(p, *
*vk1) =
BP*=(p, vk1). In addition, we may assume that DM(p, vk1) ' -2k(p-1)-2M(p, vk1).
Let
E2*M(p, vk1)~ = Hom (E2*M(p, vk1), Q=Z(p)),
and recall that
4(E2*M(p, vk1)~ ) E2*F (M(p, vk1), I2)
10
as modules over E2* and G2. (See [18, Proposition 17] or [2] for p 5; note,
however, that we are using Strickland's definition of E2*M(p, vk1)~ .) Then 3.11
implies that
E2*M(p, vk1)~ c-2k(p-1)-6E2*M(p, vk1),
and, by the theory of Poincar'e pro-p groups (cf. [6, Sections 5, 6]), there is*
* a
map
H2,6+2k(p-1)-c(E2*M(p, vk1)) ! Q=Z(p)
such that
Hi(E2*M(p, vk1)) H2-i(E2*M(p, vk1)) ! H2(E2*M(p, vk1)) ! Q=Z(p)
is a perfect pairing. Hence there must exist, for each k, an element dk in
H2,6+2(p-1)-c(E2*M(p, vk1)) such that vk-11dk 6= 0. But the computation of
H2(Fp2[[u1]][u, u-1]) together with the exact sequence
vk1 2 -1 2 k
H2(Fp2[[u1]][u, u-1] -! H (Fp2[[u1]][u, u ]) ! H (E2*M(p, v1)) ! 0
shows that this is impossible.
4 Some remarks on finiteness
In this section, we work in the En*-local stable category, so that by a finite
spectrum, we mean an object of the thick subcategory generated by LnS0.
Let G be a closed subgroup of Gn, n 1.
Proposition 4.1 Let Y be a K(n-1)*-acyclic finite spectrum. Then ss*(EhGn^
Y ) is of finite type (as a graded abelian group).
Proof The proof is just as we argued in the Introduction: use the strongly
convergent spectral sequence
H**c(G, En*Y ) ) ss*(EhGn^ Y )
whose E1 term has a horizontal vanishing line. Since En*Y is of finite type, so
is H**c(G, En*Y ). The horizontal vanishing line then implies that ss*(EhGn^ Y )
is also of finite type. __|_|
Now suppose n 2 and X is a K(n-2)*-acyclic finite spectrum with vn-1
self-map . Let X( k) denote the cofiber of k : k| |X ! X , and let X( 1 )
denote the cofiber of X ! -1X , so that X( 1 ) = holim -k| |X( k). There
!k
are also canonical maps X( k) ! X( k-1) and X ! holimX( k). We will
k
need the following well-known result (cf. [11, Section 2]).
11
Lemma 4.2 If Z is any (En*-local) spectrum, the map Z ^ X ! holimZ ^
k
X( k) is the K(n)*-localization of Z ^ X .
Proposition 4.3 -1ss*(EhGn^ X) is countable if and only if ss*(EhGn^ X) is
of finite type.
Proof Proposition 4.1 implies that ss*(EhGn^ X( 1 )) is countable, and there-
fore -1ss*(EhGn^ X) is countable if and only if ss*(EhGn^ X) is countable. But
EhGn^ X ' holimEhGn^ X( k); it therefore again follows from Proposition 4.1
k
that ssi(EhGn^ X) is profinite and is thus countable if and only if it's finite*
*. __|_|
Remark 4.4 The chromatic splitting conjecture (see [11]) actually identifies
-1(EhGnn^ X) = -1LK(n)X as Ln-1X _ -1Ln-1X .
Although ss*(EhH22^ M(p)) is not of finite type, this proposition suggests
to us the sense in which it is "almost" of finite type. The details follow.
We will consider graded modules over the graded ring Fp[ ], where has
positive even (unless p = 2) degree, satisfying the following two conditions:
i. M is complete in the sense that M = limM= iM .
i
ii. M= M is an Fp vector space of finite type.
Proposition 4.5 Let X and be as above and suppose that p : X ! X is
trivial. Then ss*(EhGn^ X) is an Fp[ ]-module satisfying conditions i and ii.
Proof Since
_ss*(EhGn^_X)_ hG k
,! ss*(En ^ X( )),
kss*(EhGn^ X)
we have the requisite finiteness. Moreover, it follows from the commutative
diagram
hG ^ X( k))
ss*(EhGn^ X) _____//_limkss*(En
| n77n
| nnnn
| nnnn
fflffl|)nn
hG^X)
lim _ss*(En____ kss*(EhG^X)
k n
that ss*(EhGn^ X) is complete. __|_|
12
In [20, Proposition 4.10], Torii shows that such a module M may be written
as Y Y
M nffFp[ ] x mfiFp[ ]=( ifi). (4.6)
ff fi
If M is of finite type, then the nff's are bounded below and
Y M
-1M -1 nffFp[ ] = nffFp[ , -1].
ff ff
Q
In general, the torsion submoduleQT of M is a submodule of fi mfiFp[ ]=( ifi);
its closure T~is equal to fi mfiFp[ ]=( ifi). Let us say that M is essentially
of finite rank if there are only a finite number of ff in the decomposition 4.6;
that is, if and only if M=T~ is a finitely generated Fp[ ]-module.
Our main theorem shows that ss*(EhH22^ M(p)) is essentially of finite rank
for p > 2. We do not know, however, whether this property is generic; that is,
whether, given G, ss*(EhGn^ X) is essentially of finite rank for all X satisfyi*
*ng
the hypotheses of Proposition 4.5 if it is for one such X with K(n - 1)*X 6= 0.
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14