ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES
Hans-Werner Henn
Dedicated to the memory of Dieter Puppe
Abstract. For odd primes p we construct finite resolutions of the trivial m*
*odule Zp
for the n-th Morava stabilizer group by (direct summands of) permutation mo*
*dules
with respect to finite p-subgroups. Furthermore we discuss the problem of r*
*ealizing
these resolutions by finite resolutions of the K(n)-local sphere via spectr*
*a which are
(direct summands of) wedges of homotopy fixed point spectra for the action *
*of these
finite p-subgroups on the Lubin-Tate spectrum En.
0. Introduction
Let p be a prime and let K(n) be the n - th Morava K-theory at p. The category *
*of
K(n)-local spectra is a basic building block of the stable homotopy category of*
* p-
local spectra and, of course, the localization of the sphere, LK(n)S0, plays a *
*central
role in this category. The homotopy of LK(n)S0 can be studied by the Adams-
Novikov spectral sequence: if En denotes the periodic Landweber exact spectrum
En whose coefficients in degree 0 classify deformations (in the sense of Lubin *
*and
Tate) of the Honda formal group law over Fpn then, up to a Galois extension, the
E2-page of the spectral sequence can be identified, by the Morava change of rin*
*gs
isomorphism, with the continuous cohomology of the Morava stabilizer group Sn
with coefficients in (En)*.
In this paper we discuss homological properties of the groups Sn, in particular
resolutions of the trivial module Zp, and show how they can be used to construct
finite resolutions of LK(n)S0 in terms of spectra which are easier to understan*
*d.
For example, if p - 1 does not divide n, then the mod-p cohomological dimension
cdp(Sn) of Sn is finite, equal to n2, the trivial module Zp admits a projective
resolution of length n2 and the E2-term of the Adams-Novikov spectral sequence
has a horizontal vanishing line. In homotopy theory this allows us to construct*
* a
finite En-resolution of LK(n)S0 in the sense of Miller [Mi] (at least up to a G*
*alois
extension in case n is still divisible by p).
If p - 1 divides n, then the mod-p cohomological dimension of Sn is infinite, so
Zp does not admit a finite projective resolution, the E2-term has no horizontal
____________
This paper is a sequel to the joint paper [GHMR1]. It is inspired by that paper*
* and the author is
happy to acknowledge the influence of numerous discussions with Paul Goerss, Ma*
*rk Mahowald
and Charles Rezk on this subject. Thanks are also due to the referee for his su*
*ggestions.
Typeset by AM S-TEX
1
2 HANS-WERNER HENN
vanishing line and a finite En-resolution for LK(n)S0 cannot exist. However, in
analogy with discrete groups of finite virtual cohomological dimension one can *
*hope
to construct resolutions of the trivial module Zp by permutation modules on fin*
*ite
subgroups of Sn and then hope to realize those by finite resolutions (which wil*
*l not
be En-resolutions) of LK(n)S0 via homotopy fixed point spectra of the form EhFn
for suitable finite subgroups of Sn. This is in fact the main subject of this p*
*aper.
There are various advantages of such resolutions. For instance, such a finite r*
*esolu-
tion gives rise to a spectral sequence with a horizontal vanishing line startin*
*g from
the homotopy of the homotopy fixed point spectra and converging to ss*(LK(n)S0).
This spectral sequence should be more manageable than the Adams-Novikov spec-
tral sequence. For example, some of the delicate differentials in the latter m*
*ight
already be accounted for by more transparent phenomena in the homotopy of the
homotopy fixed point spectra EhF2. In fact, this is essentially what happened i*
*n the
calculation of the homotopy of the Toda-Smith complex V (1) at the prime p = 3
carried out in [GHM] (completing earlier work of Shimomura [Sh]). The resolutio*
*ns
can also be used to analyze the exotic part of Hopkins' Picard groups (cf. [HMS*
*]),
i.e. the group of homotopy equivalence classes of invertible spectra in the cat*
*egory
of K(n)-local spectra whose Morava module is isomorphic to that of the sphere S*
*0.
This will be pursued in a separate paper. On a more philosophical level, one can
say that these resolutions capture to what extent the presence of finite p-subg*
*roups
in the stabilizer group influences the homotopy of ss*(LK(n)S0), very much in t*
*he
same way as finite subgroups in a discrete group of finite virtual cohomological
dimension influence the cohomology of the group.
Here is an outline of the paper. In section 1 we recall background material on
K(n)-localization, the stabilizer groups and homotopy fixed point spectra. Sect*
*ion
2 discusses algebraic and homotopy theoretic resolutions in the case n 6 0 mod*
* p-1;
there is a general existence result (Theorem 4) and beyond that we discuss the *
*few
cases n = 1 and p > 2 as well as n = 2 and p > 3 in which finite resolutions are
known in an explicit form. The results in this section are mostly reformulation*
*s or
reinterpretations of results which have been known for more than 25 years. They*
* are
included for completeness and in order to help develop our ideas on the interpl*
*ay
between homological properties of the groups Sn and homotopical properties of
LK(n)S0. In section 3 we discuss the much more difficult case n 0 mod p - 1.
We start by explaining how the well-known fibration
3-id
LK(1)S0 ! KOZ2 - ! KOZ2
(where KOZ2 is 2-adic real K-theory) can be regarded as the realization of a
particular permutation resolution of the trivial S1-module Z2. Then we use this
example as a role model which suggests possible generalizations for larger n. *
*In
section 3.3 we survey recent joint work with Goerss, Mahowald and Rezk in which
the case n = 2, p = 3 was studied. In section 3.4 we comment on joint work in
progress with the same coauthors which concerns the case n = 2, p = 2. In the f*
*inal
two sections we present new general results on the existence of finite resoluti*
*ons
of the trivial module Zp by permutation modules, at least for p > 2 (Proposition
17). If n = p - 1, we show how these algebraic resolutions can be realized by f*
*inite
resolutions of LK(n)S0 (Theorem 25 and Theorem 26).
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 3
1. Background
1.1. Localization with respect to Morava K-theory.
1.1.1. Let E be a spectrum and E* be the generalized homology theory determined
by E. We recall that Bousfield localization with respect to E* is a functor LE *
*from
the homotopy category of spectra to itself together with natural maps ~ : X !
LE X which are terminal among all E*-equivalences. By [B] LE exists for each
E. Classical examples are given by localization with respect to the Moore spect*
*ra
MZ(p), for the p-local integers Z(p), resp. MQ, for the rationals, in which cas*
*e LE
is the homotopy theoretic version of arithmetic localization with respect to Z(*
*p)
resp. with respect to Q.
1.1.2. In this paper we will be concerned with the localization functors LK(n) *
*with
respect to Morava K-theory K(n). We refer to [HS] for a good general reference.
Here we recall that K(0) = HQ is the rational Eilenberg-Mac Lane spectrum and
is independent of p. Otherwise, for any fixed prime p we have K(n)* = Fp[vn1]
with generator vn in degree 2(pn - 1). Furthermore, K(n)* is a multiplicative
periodic cohomology theory which admits a theory of characteristic classes, and
the associated formal group law n is the Honda formal group law of height n.
Localization with respect to K(n) plays a prominent role in stable homotopy the*
*ory
because the functors LK(n)are elementary "building blocks" of the stable homoto*
*py
category of finite p-local complexes in the following sense.
o The localization functor LK(n)is "simple" in the sense that the category of K*
*(n)-
local spectra contains no nontrivial localizing subcategory [HS, Theorem 7.5].
o There is a tower of localization functors
. .!.Ln ! Ln-1 ! . .!.L0
(with Ln = LK(0)_..._K(n)) together with compatible natural maps X ! LnX such
that
X ' holimnLnX
for every finite p-local spectrum X [Ra2, Theorem 7.5.7].
Furthermore, for every X there is a homotopy pullback diagram (a "chromatic
square")
LnX -! LK(n)X
?? ?
y ?y
Ln-1X -! Ln-1LK(n)X
i.e. Ln is built from LK(n) and Ln-1. (The diagram is easily established by usi*
*ng
that LK(n)Ln-1Z ' * for any Z. Its existence is implicit in [Ho].)
4 HANS-WERNER HENN
1.2 The stabilizer groups.
1.2.1. The functors LK(n)are controlled by cohomological properties of the Mora*
*va
stabilizer group Sn. We recall that Sn is the group of automorphisms of the p-t*
*ypical
formal groupnlaw n over the field Fq (with q = pn) whose [p]-series is given by
[p](x) = xp . The group Sn can be extended to a slightly larger group Gn. In fa*
*ct,
because n is already defined over Fp the Galois group Gal(Fq=Fp) of the finite
field extension Fp Fq acts on Aut( n) = Sn and Gn can be identified with the
semidirect product Sn o Gal(Fq=Fp). In the sequel we will recall some of the ba*
*sic
properties of the group Sn resp. Gn. The reader is referred to [Ha] or [Ra1] *
*for
more details (see also [He] for a summary of what will be important in this pap*
*er).
1.2.2. The group Sn is equal to the group of units in the endomorphism ring of
n, and this endomorphism ring can be identified with the maximal order On of
the division algebra Dn over Qp of dimension n2 and Hasse invariant 1_n. In more
concrete terms, On can be described as follows: let WFq denote the Witt vectors
over Fq. Then On is the non-commutative ring extension of WFq generated by
an element S which satisfies Sn = p and Sw = woeS, where w 2 WFq and woeis
the image of w with respect to the lift of the Frobenius automorphism of Fq. The
element S generates a two sided maximal ideal m in On with quotient On=m = Fq.
Inverting p in On yields the division algebra Dn, and On is its maximal order.
The action of the Galois group Gal(Fq=Fp) on Sn is realized by conjugation by S
inside Dxn, the group of units of Dn, and the semidirect product Gn can therefo*
*re
be described as the quotient of Dxnby the central subgroup generated by Sn, i.e.
Gn ~=Dxn= < Sn >.
1.2.3. Reduction mod m induces an epimorphism Oxn- ! Fxq. Its kernel will be
denoted by Sn and is also called the strict Morava stabilizer group. The group *
*Sn
is equipped with a canonical filtration by subgroups FiSn, i = k_n, k = 1, 2, .*
* .,.
defined by
FiSn := {g 2 Sn|g 1 mod Sin} .
The intersection of all these subgroups contains only the element 1 and Sn is c*
*om-
plete with respect to this filtration, i.e. we have Sn = limiSn=FiSn. Furthermo*
*re,
we have canonical isomorphisms
FiSn=Fi+_1nSn ~=Fq
induced by
x = 1 + aSin 7! ~a.
Here a is an element in On, i.e. x 2 FiSn and ~ais the residue class of a in
On=m = Fq
The associated graded object grSn with griSn = FiSn=Fi+_1nSn, i = 1_n, 2_n, . *
*. .
becomes a graded Lie algebra with Lie bracket [~a, ~b] induced by the commutator
xyx-1y-1 in Sn. Furthermore, if we define a function ' from the positive real
numbers to itself by '(i) := min{i + 1, pi} then the p - th power map on Sn ind*
*uces
maps P : griSn -! gr'(i)Sn which define on grSn the structure of a mixed Lie
algebra in the sense of Lazard [La, Chap. II.1]. If we identify the filtration *
*quotients
with Fq as above then the Lie bracket and the map P are explicitly given as fol*
*lows
(cf. Lemma 3.1.4 in [He]).
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 5
Lemma 1. Let ~a2 griSn, ~b2 grjSn. Then
a)
ni pnj
[~a, ~b] = ~a~bp- ~b~a 2 gri+jSn .
b) 8
ni+...+p(p-1)ni -1
><~a1+p ifi < (p - 1)
P ~a= > ~a+ ~a1+pni+...+p(p-1)niifi = (p - 1)-1
:
~a ifi > (p - 1)-1.
1.2.4. Next we record some basic facts about finite p-subgroups of Sn. First of
all, all finite abelian p-subgroups of Sn are cyclic. Sn is known to contain a *
*cyclic
subgroup of order pk if and only if pk-1(p - 1) divides n, and then such a cycl*
*ic
subgroup Cpk Sn is unique up to conjugacy. Furthermore, if p > 2, or p = 2 and
n is odd, then all finite p-subgroups are cyclic.
The structure of centralizers of cyclic subgroups will be of importance for us.*
* To get
at it we note that the centralizer CDn(Cpk) of Cpk in Dn is again a division al*
*gebra.
It is central over the cyclotomic extension of Qp generated by Cpk and its dime*
*nsion
over its center is m2, if n = mpk-1(p - 1). Then the centralizer CSn(Cpk) of Cpk
in Sn can be identified with the group of units in the maximal order of CDn(Cpk*
*).
1.2.5. Recall that the cohomological p-dimension cdp(G) of a profinite group G *
*is
defined as
cdp(G) = sup{n 2 N|Hn(G, M) 6= 0 for some finite continuousG - module M}
where, here and elsewhere in this paper, H*(G, M) is always continuous cohomol-
ogy. Later on M may be a continuous profinite module over the completed group
algebra Zp[[G]] := limUZp[G=U] (with U running through all open normal sub-
groups of G). We refer to [SyW] for a discussion of the relevant homologial alg*
*ebra.
The cohomological p-dimension of the group Sn resp. Gn is n2 unless Sn resp. Gn
contain non-trivial finite p-subgroups in which case it is infinite. By 1.2.4 t*
*his hap-
pens for Sn iff p-1 divides n, and in the case of Gn this happens iff p or p-1 *
*divides
n. However, even in these cases Sn and Gn are still virtually of finite cohomol*
*og-
ical p-dimension (i.e. they contain a finite index subgroup of finite cohomolog*
*ical
p-dimension) and its virtual cohomological p-dimension vcdp(Sn) remains n2. The
reader is referred to [La] or [SyW] for more details on these notions.
1.3 Homotopy fixed point spectra.
1.3.1. By Hopkins-Miller (cf. [Re]) the group Gn acts on the Lubin-Tate spectrum
En; we recall that En is the Landweber exact spectrum given by the 2-periodic
theory with coefficients ss*(En) = ss0(En)[u 1] (with u 2 ss-2(E)) whose associ*
*ated
formal group law over ss0(En) is a universal deformation of n in the sense of *
*Lubin
and Tate [LT]. In particular there is a (non-canonical) isomorphism between ss0*
*(En)
and WFq[[u1, . .,.un-1]], the ring of formal power series over WFq in the varia*
*bles
6 HANS-WERNER HENN
u1, . .,.un-1. We can and will choose the universal deformation f nto be p-typi*
*cal
with p-series
n-1 pn
[p]e n(x) = px +e nu1xp +e n. .+.e nun-1xp +e nx ,
in other words the classifying map BP* ! ss*(En) sends the Araki generator vi to
uiu1-pi, if i < n, vn to u1-pn, and vi to 0 if i > n.
Let OGn be the orbit category of Gn, i.e. the objects of OGn are orbits Gn=K wh*
*ere
K is a closed subgroup of Gn and morphisms are continuous Gn-equivariant maps.
By Devinatz-Hopkins [DH2] there is a contravariant functor from OGn to K(n)-
local spectra which assigns to Gn=K the homotopy fixed point spectrum EhKnand
this spectrum comes with an associated homotopy fixed point spectral sequence
Es,t2= Hs(K; sst(En)) =) sst-s(EhKn) .
Furthermore, EhGnncan be identified with LK(n)S0 and the Adams-Novikov spectral
sequence for LK(n)S0 can be identified with the associated homotopy fixed point
spectral sequence
Es,t2~=Hs(Gn, (En)t) =) sst-sLK(n)S0 .
Finally, EhGnncan be identified with the iterated homotopy fixed point spectrum
(EhSnn)hGal(Fq=Fp), the Galois group acts on the homotopy fixed point spectral *
*se-
quence
Es,t2~=Hs(Sn, (En)t) =) sst-s(EhSnn) ,
and the action on the whole spectral sequence is coinduced. Thus we get isomor-
phisms
ss*LK(n)S0 ~=ss*(EhSnn)Gal(Fq=Fp), ss*EhSnn~=ss*LK(n)S0 Zp WFpn ,
and we may therefore say that EhSnnis equal to LK(n)S0, up to a Galois extensio*
*n.
1.3.2. Hopkins and Devinatz also showed that for any closed subgroup K Gn
there is an isomorphism
ss*(LK(n)(En ^ EhKn)) ~=mapscts(Gn=K, (En)*)
(where mapctsdenotes continuous maps). The isomorphism is functorial on OGn .
It is compatible with the obvious (En)*-module structures on both sides as well*
* as
with the actions of Gn on both sides, which is via the action on En on the left*
* hand
side and via the diagonal action on the space of continuous maps on the right h*
*and
side. In other words, the isomorphism is one of Morava modules where a Morava
module M is a complete (En)*-module with a continuous action of Gn such that
g(ax) = g(a)g(x) forg 2 Gn, a 2 (En)*, x 2 M .
By abuse of notation we will also say that M is a (twisted) (En)*[[Gn]]-module.
Typical examples of such modules are given by ss*(LK(n)(En ^ X)), at least under
suitable conditions on X, e.g. if K(n)*X is evenly graded (see [HS], or [GHMR1]
for a summary of what is important for us). In order to keep our notation compa*
*ct
we will write in the sequel (En)*X instead of ss*(LK(n)(En ^ X)).
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 7
1.3.3. We will need information about maps between various homotopy fixed point
spectra. In the following F stands for function spectrum. We recall the followi*
*ng
results from [GHMR1].
1.3.3.1. Let U be an open subgroup of Gn. Functoriality of the homotopy fixed
point spectra construction of [DH2] gives us a map EhUn^ Gn=U+ ! En where as
usual Gn=U+ denotes Gn=U with a disjoint base point added. Together with the
product on En we obtain a map
En ^ EhUn^ Gn=U+ ! En ^ En ! En
whose adjoint induces an equivalence of En-module spectra
Y
LK(n)(En ^ EhUn) ! En
Gn=U
realizing the isomorphism of 1.3.2 above.
Now let FEn be the function spectrum in the category of En-module spectra (see
[EKMM] for details). If we apply FEn(-, En) to this last equivalence we obtain
another equivalence of En-module spectra
Y
FEn( En, En) ! FEn(En ^ EhUn, En)
Gn=U
which can be rewritten as an equivalence (still of En-module spectra)
En ^ Gn=U+ ' F (EhUn, En) .
The same reasoning shows that we can replace En in the target of the function
spectrum by LK(n)(En ^ I) where I is any spectrum and we obtain an equivalence
LK(n)(En ^ I) ^ Gn=U+ ' F (EhUn, LK(n)(En ^ I)) .
1.3.3.2. More generally, let K be any closed subgroup ofTGn. Then there exists
a decreasing sequence Ui of open subgroups Ui with K = iUi and by [DH2] we
have
EhKn' LK(n)hocolimiEhUin.
By passing to the limit we obtain an equivalence
LK(n)(En ^ I)[[Gn=K]] ' F (EhKn, LK(n)(En ^ I))
where we have used the convention that if E is a spectrum and X = limiXi is an
inverse limit of a sequence of finite sets with each Xi finiteQthen E[[X]] is g*
*iven as
holimiE ^ (Xi)+ , i.e. as the fibre of the usual self map of iE ^ (Xi)+ . Not*
*e that
if X is such a profinite set with continuous K-action and if E is a K-spectrum *
*then
E[[X]] is a K-spectrum via the diagonal action.
If we concentrate (for simplicity) on the case I = S0 and take homotopy fixed
points in this equivalence with respect to another finite subgroup of Gn then we
get the following result.
8 HANS-WERNER HENN
Proposition 2 ([GHMR1, Prop. 2.6]).
a) Let K1 be a closed subgroup and K2 a finite subgroup of Gn. Then there is
a natural equivalence (where the homotopy fixed points on the left hand side are
formed with respect to the diagonal action of K2)
En[[Gn=K1]]hK2 ' F (EhK1n, EhK2n) .
b) If K1 is also an open subgroup then there is a natural decomposition
Y
En[[Gn=K1]]hK2 ' EhKxn
K2\Gn=K1
where Kx = K2\xK1x-1 is the isotropy subgroup of the coset xK1 and K2\Gn=K1
is the finite (!) set of double cosets.
T
c) If K1 is a closed subgroup and K1 = iUi for a decreasing sequence of open
subgroups Ui then
Y
F (EhK1n, EhK2n) ' holimiEn[[Gn=Ui]]hK2 ' holimi EhKx,in
K2\Gn=Ui
where Kx,i= K2\ xUix-1 is, as before, the isotropy subgroup of the coset xUi.
Remark_If Ui Uj then the map
Y Y
EhKx,in! EhKx,jn
K2\Gn=Ui K2\Gn=Uj
in the inverse system of part (c) of the proposition can be described as follow*
*s: if
x 2 Gn=Ui has isotropy group Kx,iand its image x02 Gn=Uj has isotropy group
Kx0,jthen the restriction of the map to the factor determined by x sends EhKx,in
via the transfer to the factor EhKx0,jndetermined by x0. In particular, this im*
*plies
that on homotopy groups the inverse system is Mittag-Leffler.
In the next result Hom (En)*[[Gn]](-, -) denotes homomorphisms of Morava mod-
ules.
Proposition 3 ([GHMR1, Prop. 2.7]). Let K1 and K2 be closed subgroups of
Gn and suppose that K2 is finite. Then there is an isomorphism
K2 ~= hK hK
(En)*[[Gn=K1]] -! Hom (En)*[[Gn]]((En)*En 1, (En)*En 2)
such that the following diagram commutes
K2
ss*En[[Gn=K1]]hK2- ! (En)*[[Gn=K1]]
?? ?
y~= ?y~=
ss*F (EhK1n, EhK2n)-! Hom (En)*[[Gn]]((En)*EhK1n, (En)*EhK2n)
where the top horizontal map is the edge homomorphism in the homotopy fixed poi*
*nt
spectral sequence, the left-hand vertical map is the isomorphism given by Propo*
*sition
2 and the bottom horizontal map is the En-Hurewicz homomorphism.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 9
2. The case n 6 0 mod p - 1
In this section we begin our discussion of LK(n)S0. The case n = 0 is both exce*
*p-
tional and trivial: K(0) = MQ and LK(0)S0 is the rationalized sphere spectrum.
From now on we will therefore assume n > 0.
2.1 Explicit examples I: the case n = 1 and p > 2.
2.1.1. We briefly review the case n = 1 which is well understood. In this case *
*we
have E1 = KZp (p-adic complex K-theory). The group G1 = S1 can be identified
with Zxp, the group of units in the p-adic integers. If p is odd then Zxp~=Cp-1*
*xZp,
where Cp-1 denotes the cyclic group of order p - 1 given by the roots of unity *
*in
Zp. The homotopy fixed points EhG11can be formed in two steps, first with respe*
*ct
to Cp-1 and then with respect to Zp. Thus we obtain the following fibration (cf.
[HMS]) in which _p+1 is the appropriate Adams operation and KZhCp-1pcan be
identified with the Adams summand of p-adic complex K-theory
p+1-id
LK(1)S0 ! KZhCp-1p_-! KZhCp-1p.
2.1.2. This fibration can also be considered as a suitable realization of a pro*
*jective
resolution of the trivial Zp[[G1]]-module Zp, and it is this point of view whic*
*h turns
out to be useful for finding generalizations of the above fibration for larger *
*n.
To get at this projective resolution we start with the following obvious short *
*exact
sequence of modules over the power series ring Zp[[t]]
0 ! Zp[[t]] xt-!Zp[[t]] ! Zp ! 0 .
Now recall that there is an isomorphism of complete algebras Zp[[t]] ~= Zp[[Zp]]
induced by sending t to g - e 2 Zp[[Zp]] if g is a topological generator of Zp,*
* e.g. if
g is the image of p + 1 2 Zxp= G1 in the quotient group G1=Cp-1 ~=Zp. Therefore
we can consider this sequence as an exact sequence of Zp[[Zp]]-modules, or even*
* as
an exact sequence of Zp[[G1]]-modules, and as such we can write it as
0 ! Zp "G1Cp-1g-e-!ZpG"1Cp-1! Zp ! 0
where M " G1Cp-1denotes the induced module of a Zp[Cp-1]-module M, i.e. the
completed tensor product Zp[[G1]]b Zp[Cp-1]M. Because the trivial Zp[Cp-1] mod-
ule Zp is projective we find that Zp "G1Cp-1is a projective Zp[[G1]]-module and*
* the
exact sequence is a projective resolution of the trivial Zp[[G1]]-module Zp whi*
*ch
is even split as a sequence of continuous Zp-modules. So if we apply the functor
Hom cts(-, (KZp)*) of continuous homomorphisms into (KZp)* to this sequence
then we obtain another short exact sequence which, by 1.3.2, can be identified *
*with
the (`a priori long) exact sequence which is associated to our fibration:
0 ! (KZp)*(LK(1)S0) ~=(KZp)* ! (KZp)*KZhCp-1p! (KZp)*KZhCp-1p! 0 .
10 HANS-WERNER HENN
2.2 The general case.
2.2.1. Following Miller [Mi] we say that a K(n)-local spectrum I is En-injectiv*
*e if
the map I = S0^I ! LK(n)(En^I) induced by the unit in En is split. Furthermore,
a sequence X0 ! X ! X00of K(n)-local spectra is En-exact if the composition
X0! X00is trivial and if
[X0, I] - [X, I] - [X00, I]
is an exact sequence of abelian groups for each En-injective spectrum I. Finall*
*y an
En-resolution of a K(n)-local spectrum X is an En-exact sequence of K(n)-local
spectra
* ! X ! I0 ! I1 ! . . .
(i.e. every three term subsequence is En-exact) such that each Is, s 0, is E*
*n-
injective.
2.2.2. The following result is a folk theorem whose roots can be traced back to*
* the
work of Morava [Mo].
Theorem 4. If n is neither divisible by p - 1 nor by p then LK(n)S0 admits an
En-resolution of length n2. In fact, each of the En-injectives in the resolutio*
*n can
be chosen to be a direct summand of a finite wedge of En's.
Proof. The idea of the proof is to start with information in homological algebra
and use this to construct an En-resolution.
If neither p - 1 nor p divides n then cdp(Gn) = n2. Therefore the trivial Zp[[G*
*n]]-
module Zp admits a projective resolution
Po : 0 ! Pn2 ! . .!.P0 ! Zp ! 0
of length n2, and by [La] we may assume that each projective is finitely genera*
*ted
as Zp[[Gn]]-module.
We want to construct an En-resolution Xo of LK(n)S0
Xo : * ! X-1 ! X0 ! . .!.Xn2 ! *
with X-1 = LK(n)S0 such that the complex Hom cts(Po, (En)*) is isomorphic, as a
complex of Morava modules, to the complex (En)*Xo.
For this we note that if F ris a free Zp[[Gn]]-module of rank r then we have an
isomorphism of Morava modules
r`
(1) Hom cts(F r, (En)*) ~=(En)*( En)
j=1
and the En-Hurewicz homomorphism
(2) [En, En] ! Hom (En)*[[Gn]]((En)*En, (En)*En)
is an isomorphism. In fact, (1) resp. (2) follow immediately from 1.3.2 resp. f*
*rom
Proposition 3.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 11
Property (2) allows us now to construct both the spectra Xs, s = 0, 1, . .,.n2,
(by lifting idempotents on finitely generated free Zp[[Gn]]-modules to homotopy
idempotents on corresponding wedges of En's) as well as the required maps betwe*
*en
these spectra. Because En is K(n)-local and En is a ring spectrum, the spectra *
*Xs
will all be En-injective. It remains to show that the sequence is En-exact.
For this it is enough to show that for any spectrum Z of the form Z := LK(n)(En*
*^I)
with some spectrum I the complex [Xo, Z] is exact. By our discussion in 1.3.3 we
know that
[En, Z] ~=ss0(Z[[Gn]]) ~=limi(ss0(Z) Zp[[Gn=Ui]]) .
Now assume first that ss0(Z) is p-complete, i.e. ss0(Z) ~=limiss0(Z)=pn. Then we
even have
[En, Z] ~=ss0(Z[[Gn]]) ~=limi,j(ss0(Z) Z=pj[[Gn=Ui]]) .
This can be restated as follows. For a fixed abelian group A let Ae - denote
the functor from profinite Zp-modules and continuous homomorphisms to abelian
groups which sends the profinite Zp-module M ~=limffMffwhere each Mffis finite
to limff(A Mff). Therefore, as long as ss0(Z) is p-complete, we can write
[En, Z] ~=ss0(Z)e Zp[[Gn]] .
L r
More generally, if P is a directWsummand in j=1Zp[[Gn]] and if X is the corre-
sponding direct summand in rj=1En then
[X, Z] ~=ss0(Z)e P
and we even obtain an isomorphism of complexes
[Xo, Z] ~=ss0Z ePo .
Because Po is split as a complex of continuous Zp-modules, [Xo, Z] is exact pro*
*vided
ss0(Z) is p-complete. In addition, under this hypothesis on Z this complex is e*
*ven
naturally split in Z.
In the general case we use that Z = LK(n)(En ^ I) can be written as homotopy
inverse limit of a sequence Zn of spectra with p-complete homotopy groups, even
bounded p-torsion homotopy groups (cf. [HS, Proposition 7.10]). Because [Xo, Zn]
is naturally split in n, we see that limin[Xo, Zn] is split for i = 0, 1, in pa*
*rticular
exact, and therefore [Xo, Z] is exact, i.e. the sequence Xo is En-exact.
Remark_1_This result is a pure existence result. It says nothing about an expli*
*cit
form of such a resolution.
Remark_2_If n is divisible by p (but not by p - 1) we can offer the two followi*
*ng
substitutes of Theorem 4.
Either we can use the existence of a finite projective resolution of the trivia*
*l Zp[[Sn]]-
module Zp to get one for the induced Zp[[Gn]]-module Zp "GnSn. This resolution *
*can
then be realized as in the proof of Theorem 4 to give an En-resolution for EhSn*
*n.
which by 1.3.1 is, up to a Galois extension, equal to LK(n)S0.
12 HANS-WERNER HENN
Or, if we insist on a resolutionSof LK(n)S0, we can consider the formal group *
*n over
the algebraic extension K := r 0 Fqprof Fq with Galois group Gal(K, Fq) ~=Zp.
This will have the effect of replacing Gn = Sn oGal(Fq=Fp) by the group Gn(K) :=
SnoGal(K=Fp). The advantage of doing this is that while Gn has elements of order
p and therefore infinite mod-p cohomological dimension, the group Gn(K) has no
elements of order p and its mod-p cohomological dimension is finite, equal to n*
*2+1.
As a consequence one gets a projective resolution of the trivial Gn(K)-module Zp
of length n2 + 1. If we also replace En by the corresponding Lubin-Tate spectrum
En(K) whose homotopy groups in degree 0 classify deformations of n over K then
our proof carries over verbatim: we only need to remark that 1.3.2 and the two
properties (1) and (2) in the proof of Theorem 4 hold with En and Gn replaced by
En(K) and Gn(K).1
2.3. Explicit examples II: the case n = 2 and p > 3.
As before we let q = pn. We start with an observation valid for all n > 1 (cf.
section 1.3 of [GHMR1]). The reduced norm Sn ! Zxpadmits a canonical extension
Gn ! Zxpx Gal(Fq=Fp) and by composing with the evident projection we obtain
a homomorphism Gn ! Zxp. Furthermore, if we identify the quotient of Zxpby its
subgroup of elements of finite order by Zp we obtain a homomorphism Gn ! Zp.
The kernel of this homomorphism will be denoted G1nand the kernel of its restri*
*ction
to Sn will be denoted by S1n. G1ncontains a cyclic group Cq-1 of order q - 1 (t*
*he
roots of unity in WFq On) and this subgroup is invariant with respect of the *
*action
of Gal(Fq=Fp). Therefore G1ncontains the semidirect product Cq-1 o Gal(Fq=Fp).
We will denote this finite subgroup by Fn(q-1)in the sequel.
For more information on ss*(EhFn(q-1)n) we refer to the appendix.
Theorem 5. Assume n = 2 and p > 3.
a) There exists a fibration
1 hG1
LK(2)S0 ! EhG22-! E2 2
and an En-resolution
1 hF2(q-1) hF2(q-1)
* ! EhG22! E2 ! X ! X ! E2 ! *
where X ' 2(p-1)EhF2(q-1)2_ 2(p2-p)EhF2(q-1)2.
b) There exists an En-resolution of the form
* ! LK(2)S0 ' EhG22! EhF2(q-1)2! EhF2(q-1)2_ X !
! X _ X ! X _ EhF2(q-1)2! EhF2(q-1)2! * .
Given Theorem 4 this is a fairly straightforward consequence of the following p*
*urely
algebraic result in which ~p-1 denotes the Zp[F2(q-1)]-module whose underlying *
*Zp-
module is WFq, on which Cq-1 WxFqacts via
Cq-1 x WFq ! WFq, (g, w) 7! gp-1w
____________
1I would like to thank Ethan Devinatz for a reassuring discussion of this point.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 13
and on which the group Gal(Fq=Fp) acts via the lift of Frobenius (cf. appendix).
The algebraic result is in turn a consequence of the calculation of the cohomol*
*ogy
of the relevant Morava stabilizer algebra in [Ra1, Theorem 6.3.22]. Proposition*
* 7
below is the group theoretic version of Ravenel's result.
Theorem 6. Assume n = 2 and p > 3.
a) There exists a short exact sequence of Zp[[G2]]-modules
0 ! Zp "G2G12! Zp "G2G12! Zp ! 0
and a projective resolution of the trivial Zp[[G12]]-module Zp
1 G1 G1 G1
0 ! Zp "G2F2(q-1)! ~p-1 "F22(q-1)! ~p-1 "F22(q-1)! Zp "F22(q-1)! Zp ! 0 .
b) There exists a projective resolution of the trivial Zp[[G2]]-module of the f*
*orm
0 ! Zp "G2F2(q-1)!(~p-1 Zp) "G2F2(q-1)! (~p-1 ~p-1) "G2F2(q-1)!
! (Zp ~p-1) "G2F2(q-1)! Zp "G2F2(q-1)! Zp ! 0 .
Proposition 7. Assume n = 2 and p > 3.
a) H*(S12; Fp) is a 3-dimensional Poincar'e duality algebra and
H3(S12; Fp) ~=Fp
is trivial as module over F2(q-1).
b) There is a canonical isomorphism of Zp[F2(q-1)]-modules
H1(S12; Fp) ~=~p-1 Zp Fp .
c) The Bockstein homomorphism induces an isomorphism of Zp[F2(q-1)]-modules
H1(S12; Fp) ~=H2(S12; Fp) .
Proof of the Proposition. We start by proving (b). S12is a torsionfree p-adic *
*Lie
group of dimension 3; so by [La, V.2.5.8] H*(S12; Fp) is a 3-dimensional Poinca*
*r'e
duality algebra which is therefore additively determined by H1(S12; Fp) resp. *
*its
dual H1(S12; Fp).
The filtration of S2 described in section 1.2.3 induces one of S12. From the sp*
*litting
S2 ~=S12x Zp (with Zp being central) we deduce that griS12= griS2 if i = k_2with
k odd and griS12= Fq=Fp if i = k_2with k even. In particular, if i = k_2with k *
*odd,
_p2i _
there is an element in griS12with (b - b) 6= 0 and then Lemma 1a shows that
_ ___p2i _
the commutator map griS12x gr1S12! gri+1S12given by [__a, b] = a(b - b) is on*
*to.
Furthermore, if i = k_2with k odd and j = l_2with l odd then the commutator map
14 HANS-WERNER HENN
_ ___p ___
griS12x grjS12! gri+jS12given by [__a, b] = ab - bapis again onto. This implies
that there is a canonical isomorphism
H1(S12; Zp) ~=F1=2S12=F1S12~=Fq ~=H1(S12; Fp) .
Furthermore, the conjugation action of ! 2 Cq-1 is induced by (!, 1 + aS) 7!
!(1+aS)!-1 = 1+!1-paS while the action of Frobenius is induced by (S, 1+aS) 7!
S(1 + aS)S-1 = 1 + aoeS and this implies that our isomorphism is an isomorphism
of Zp[F2(q-1)]-modules: H1(S12; Zp) ~=H1(S12; Fp) ~=~1-p Zp Fp. By dualizing we
get an isomorphism of Zp[F2(q-1)]-modules H1(S12; Fp) ~=(~1-p)* Zp Fp.
Now ~p-1 (and ~1-p) are self-dual (the isomorphism is induced by the pairing
(x, y) 7! xy-1 + (xy-1)oe) and they are both isomorphic (the isomorphism is giv*
*en
by x 7! xoe) and therefore we obtain (b).
Because of H1(S12; Zp) ~=H1(S12; Fp) ~=Fq we see that the Bockstein homomorphism
fi : H2(S12; Fp) ! H1(S12; Fp) is onto, and hence fi : H1(S12; Fp) ! H2(S12; Fp)
is mono. On the other hand Poincar'e duality gives an additive isomorphism
H1(S12; Fp) ~=H2(S12; Fp) and hence the Bockstein gives an isomorphism H1(S12, *
*Fp)
~=H2(S12; Fp) which is clearly Zp[F2(q-1)]-linear and thus (c) is proved.
To prove (a) we consider the subgroup F1S12of S12and note that this subgroup is
invariant by the conjugation action of F2(q-1). Using Lemma 1a as above shows
that the closure of the commutator subgroup of F1S12is F5_2S12and then Lemma 1b
gives that the closure of the subgroup generated by p-th powers and commutators
is F2S12. It follows that H1(F1S12; Zp) is torsion (~=Z=p2 Z=p Z=p) and that*
* we
have an isomorphism of Zp[F2(q-1)]-modules
H1(F1S12; Fp) ~=gr1S12 gr3_2S12.
Identifying the Zp[F2(q-1)]-module structure on gr1S12 gr3_2S12as before shows *
*that
we have an isomorphism of Zp[F2(q-1)]-modules
H1(F1S12; Fp) ~=Fq=Fp ~p-1
where Cq-1 acts trivially on Fq=Fp and Frobenius acts by multiplication by -1.
Again this module is self-dual so that we have an isomorphism of Zp[F2(q-1)]-
modules
H1(F1S12; Fp) ~=Fq=Fp ~p-1 .
Now we use that F1S12is 'equi-p-valu'e in the sense of Lazard, hence its coho-
mology is the exterior algebra on H1(F1S12; Fp) [La, Proposition V.2.5.7.1)]. *
*So
if ff is any endomorphism of H1(F1S12; Fp) then the induced homomorphism on
H3(F1S12; Fp) ~=Fp is given by multiplication with the determinant of ff. In pa*
*rtic-
ular, for the action of ! 2 Cq-1 the determinant is 1; it is obviously 1 on Fq=*
*Fp and
it is 1 on ~p-1 because ! acts via multiplication by !p-1 and hence its determi*
*nant
is a (p - 1)-st power in Fxp. For the action of Frobenius the determinant is ag*
*ain 1
because it is -1 on both Fq=Fp and on ~p-1. Therefore H3(F1S12; Fp) is trivial *
*as
Zp[F2(q-1)]-module.
Because H1(F1S12; Zp) is a torsion group we deduce from the mod-p calculation a*
*nd
the universal coefficient theorem an isomorphism H3(F1S12; Zp) ~= Zp. Likewise
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 15
we find H3(S12; Zp) ~= Zp. Now a restriction-transfer argument shows that the
Zp[F2(q-1)]-module structure on H3(S12; Zp) ~= Zp is trivial if and only if it *
*is
trivial on H3(F1S12; Zp). We have already seen that the latter is trivial after*
* mod-p
reduction and this implies that it was trivial before.
Proof of Theorem 6. a) The existence of the exact sequence of Zp[[G2]]-modules
(3) 0 ! Zp "G2G12! Zp "G2G12! Zp ! 0
is an immediate consequence of the isomorphism G2=G12~=Zxp=Cp-1 ~=Zp. (Note
that this sequence is essentially the same exact sequence as that in section 2.*
*1.2.)
The projective resolution of Zp as Zp[[G12]]-module is now constructed by using
1
Proposition 7 as follows. The map Zp "G2F2(q-1)! Zp is just the G12-linear exte*
*nsion
of the identity of Zp (considered as an F2(q-1)-linear map). If N0 is its kerne*
*l then
we can compute H0(S12; N0=(p)) from the long exact homology sequence associated
to the short exact sequence
1
0 ! N0 ! Zp "G2F2(q-1)! Zp ! 0
of Zp[F2(q-1)]-modules and identify it with H1(S12; Fp) ~=~p-1 Zp Fp. Because
~p-1 is projective as Zp[F2(q-1)]-module (the order of F2(q-1)is prime to p!) we
can lift the resulting map from ~p-1 ! H0(S12; N0=(p)) to an Zp[F2(q-1)]-linear
map ~p-1 ! N0.
1
Let N1 be the kernel of the Zp[[G12]]-linear extension ~p-1 "G2F2(q-1)! N0. By*
* a
Nakayama Lemma type argument with H0 we see that this extension is onto (cf.
Lemma 4.3 of [GHMR1]) and then we find an isomorphism of Zp[F2(q-1)]-modules
H0(S12; N1=(p)) ~=H2(S12; Fp).
1
By iterating the procedure we construct a Zp[[G12]]-linear surjection ~p-1 "G2F*
*2(q-1)!
N1 whose kernel N2 satisfies H0(S12; N2=(p)) ~= H3(S12; Fp) ~= Fp as Zp[F2(q-1)*
*]-
module and Hi(S12; N2=(p)) = 0 if i > 0.
Finally by using1the Nakayama Lemma once more we see that the G12-linear ex-
tension Zp "G2F2(q-1)! N2 of the F2(q-1)-linear projection Zp ! H0(S12; N2=(p))*
* is
an isomorphism. By splicing together the short exact sequences we obtain the
projective resolution of Theorem 6a.
b) We take the projective resolution obtained in (a) and induce it up to get on*
*e of
the Zp[[G12]]-module Zp "G2G12. Then we use the exact sequence (3) and construc*
*t the
obvious double complex whose columns are these induced projective resolutions.
The resulting double complex gives the projective resolution of (b).
Proof of Theorem 5. a) For the fibration we can refer to Proposition 7.1 in [DH*
*2]. In
fact, the fibration "realizes" (in the same sense as before)1the short exact se*
*quence
of Zp[[G2]]-modules in Theorem 6a. The En-resolution of EhG2nis now obtained as
in the proof of Theorem 4 as the realization of the projective resolution of Zp*
* "G2G12
which is induced from the one given in Theorem 6a. To finish the proof of (a) it
16 HANS-WERNER HENN
1
remains to identify the spectrum which corresponds to the module ~p-1 "G2F2(q-1*
*).
For this we refer to the appendix.
b) The En-resolution of part (b) is nothing but the realization of the resoluti*
*on of
Theorem 6b obtained via (the proof of) Theorem 4.
Remarks_a) Ravenel [Ra1, Theorem 6.3.31] resp. Yamaguchi [Y] have also studied
H*(S3, Fp) for p 5 resp. p 3. In principle this can be used to obtain an ex*
*plicit
resolution for LK(3)S0 if p 3.
b) No other explicit resolutions seem to be known if n 6 0 mod p - 1.
3. The case n 0 mod p - 1
3.1. Explicit examples III: the case n = 1 and p = 2.
This case is again well understood. The isomorphism G1 = S1 ~=Zx2~= C2 x Z2
allows, as before, to form the homotopy fixed points in two stages and we obtain
the following fibration (cf. [HMS]) in which _3 is again given by the appropria*
*te
Adams operation:
3-id
(4) LK(1)S0 ! KZhC22_-! KZhC22.
The homotopy fixed points KZhC22can be identified with 2-adic real K-theory
KOZ2. Note, however, that this is not an example of Theorem 4. In fact, an
En-resolution of finite length cannot exist in this case because the cohomologi*
*cal
dimension cd2(S1) is infinite. Nevertheless this is a very good substitute of s*
*uch a
resolution.
3.2. The general problem.
The natural question arises whether there are generalizations of the fibre sequ*
*ence
(4) for higher n. What should they look like? In other words, can we explain the
appearance of KOZ2 in (4) so that it fits into a more general framework?
A good point of view is again provided by homological algebra as follows: the f*
*ibre
sequence (4) is a homotopy theoretic analogue of the exact sequence of Z2[[G1]]-
modules
0 ! Z2 "G1C2! Z2 "G1C2! Z2 ! 0 .
This is not a free (neither a projective) resolution of the trivial module Z2 b*
*ut
rather a resolution by permutation modules.
This suggests that we should look for a resolution of the trivial Gn-module Zp *
*in
terms of something like permutation modules on finite subgroups and try to real*
*ize
those by appropriate homotopy fixed point spectra where realization is again in*
* the
sense of the isomorphism of Morava modules of 1.3.2 which gives us for each fin*
*ite
subgroup F of Gn a canonical isomorphism En*En hF~=Hom cts(Zp "GnF, En*). This
leads to the following questions.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 17
Questions.
(Q1) Are there resolutions of finite length of the trivial Gn-module Zp by (fin*
*ite)
direct sums of permutation modules on finite (p)-subgroups of Gn?
(Q2) Can these algebraic resolutions be realized by "resolutions" of spectra? W*
*hat
do we mean by a "resolution" of a spectrum?
(Q3) If the answers to (Q1) and (Q2) are yes, how unique are these resolutions?
Remark_The group Sn resp. Gn is of finite virtual cohomological p-dimension. If
G is a discrete group which is virtually of finite cohomological dimension, the*
*n a
permutation resolution of finite length can be obtained from the cellular chains
of a contractible finite dimensional G-CW -complex on which G acts with finite
stabilizers. Such spaces always exist and hence such resolutions always exist [*
*Se].
In case G is profinite and vcdp(G) < 1 then some sort of positive answer to
(a) may be given by algebraically mimicking Serre's construction: one considers
a finite index open subgroup H with cdp(H) < 1, then one takes a projective
resolution of finite length of the trivial Zp[[H]]-module Zp and finally one ob*
*tains
the desired resolution by tensor induction from H to G. This ensures existence,
but the drawback of this construction is that it tends to be not very efficient*
*. In
particular, the length of the resolution would be much larger than necessary (t*
*he
vcd?) and the modules in the resolution would not be finitely generated.
In the case of the stabilizer groups we will describe a construction which give*
*s better
qualitative (and in favorable cases quantitative) information on the form of su*
*ch
resolutions. However, before we turn to the general theory we will survey recent
joint work with Goerss and Mahowald [GHM] resp. with Goerss, Mahowald and
Rezk [GHMR1] in which we construct explicit and efficient resolutions in the ca*
*se
n = 2 and p = 3.
3.3. Explicit examples IV: the case n = 2 and p = 3.
Throughout this section we assume n = 2 and p = 3. In this case there are two
different explicit resolutions which we will call duality resolution resp. cent*
*ralizer
resolution. We will see in sections 3.5 and 3.6 below that the centralizer reso*
*lution
can be generalized both algebraically and homotopy theoretically. The justific*
*a-
tion for its name will become clear in section 3.5. The duality resolution has *
*the
advantage of being more efficient and having an intriguing symmetry.
In the sequel we describe both resolutions. For more details in the case of the
duality resolution we refer to [GHMR1]. The centralizer resolution is implicit*
* in
[GHM] and is a special case of Theorem 26 below.
3.3.1. In the following results we use the phrase "resolution of spectra" in the
following weak sense: we call a sequence of spectra
* ! X-1 ! X0 ! X1 ! . . .
a resolution of X-1 if the composite of any two consecutive maps is null-homoto*
*pic
and if each of the maps Xi ! Xi+1, i 0 can be factored as Xi ! Ci ! Xi+1
such that Ci-1! Xi! Ci is a cofibration for every i 0 (with C-1 := X-1). We
say that the resolution is of length n if Cn ' Xn and Xi' * if i > n.
18 HANS-WERNER HENN
3.3.2. Before we can describe our resolutions we need to introduce certain fini*
*te
subgroups of G12and some of their representations (cf. [GHMR1] for more details*
*).
The group G12contains a group G24of order 24 which is isomorphic to the semidir*
*ect
product C3 o Q8 such that the quaternion group Q8 acts non-trivially on C3. If
! is a primitive 8-th root of unity in WFq then C3 is generated by the element
s = -1_2(1 + !S) while Q8 is generated by !2 and !S.
G12contains also the semidirect product C8 o Gal(F9=F3) generated by ! and S.
This group which was denoted F2(32-1)in section 2.3 can be identified with the
semidihedral group SD16 of order 16. It has a unique non-trivial one dimensional
representation O over Z3 which is trivial on the subgroup < !2, !S >. (O agrees
with the representation ~4,-that is discussed in the appendix.)
3.3.3 The duality resolution. The following results are proved in [GHMR1].
Theorem 8 (Algebraic duality resolution). There exists a short exact sequence
of Z3[[G2]]-modules
0 ! Z3 "G2G12! Z3 "G2G12! Z3 ! 0 ,
an exact complex of Z3[[G12]] - modules
1 G1 G1 G1
0 ! Z3 "G2G24! O "S2D16! O "S2D16! Z3 "G224! Z3 ! 0 ,
and an exact complex of Z3[[G2]] - modules
1 G G
0 ! Z3 "G2G24!O "G2SD16 Z3 "2G24! (O O) "2SD16!
1 G
! Z3 "G2G24 O "G2SD16! Z3 "G224! Z3 ! 0 .
Theorem 9 (Homotopy theoretic duality resolution). There exists a fibra-
tion 1 1
LK(2)S0 ! EhG22-! EhG22
and resolutions of spectra of length 3 resp. 4
1 hG hSD hSD hG
* ! EhG22! E2 24 ! 8E2 16 ! 40E2 16 ! 48E2 24 ! *
resp.
* ! LK(2)S0 ! EhG242! EhG242_ 8EhSD162! 8EhSD162_ 40EhSD162!
! 40EhSD162_ 48EhG242! 48EhG242! * .
Remarks_a) The spectrum EhG242is a version of the Hopkins-Miller higher real K-
theory spectrum EO2 at p = 3. Its coefficients are described in detail in [GHMR*
*1].
The coefficients of EhSD162are given by the completion of Z3[v1][v21] with resp*
*ect
to the ideal generated by v41v-12(cf. the discussion in the appendix).
b) The appearance of the 8-fold suspension is forced by the character O, while
the 40-fold suspension is there for purely asthetic reasons (note that 40EhSD1*
*62'
8EhSD162by periodicity), so that the homotopy theoretic resolution displays a
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 19
similar kind of duality as the algebraic resolution. The appearance of the 48-f*
*old
suspension, however, is a genuinely homotopy theoretic phenomenon and cannot
be avoided.
c) The resolution appears to be self dual but there is no satisfactory explanat*
*ion of
this duality yet. And it is not at all clear whether there are generalizations,*
* say to
the case n = p - 1, p > 3, and what they could look like.
3.3.4 The centralizer resolution. For describing the centralizer resolution we *
*will
make use of the Z3[SD16]-module ~2 (cf. 2.3 and appendix) and the unique non-
trivial one dimensional representation eOof G24over Z3 which is trivial on s an*
*d on
!S.
The following results are implicit in [GHM]. We will give a proof in section 3.*
*6.6.
Theorem 10 (Algebraic centralizer resolution). There exists a short exact
sequence of Z3[[G2]]-modules
0 ! Z3 "G2G12! Z3 "G2G12! Z3 ! 0 ,
an exact complex of Z3[[G12]] - modules
1 G1 G1 G1 G1
0 ! Z3 "G2SD16! ~2 "2SD16! O "S2D16 eO"2G24! Z3 "G224! Z3 ! 0
and an exact complex Z3[[G2]] - modules
1 G1 G1 G1
0 ! Z3 "G2SD16! (Z3 ~2)2"SD16! (~2 O)S"2D16 eO"2G24!
1 G1 G1
! O "G2SD16 (eO Z3) "2G24! Z3 "G224! Z3 ! 0 .
Theorem 11 (Homotopy theoretic centralizer resolution). There exists a
fibration 1 1
LK(2)S0 ! EhG22-! EhG22
and resolutions of spectra of length 3 resp. 4
1 hG hSD hG
* ! EhG22! E2 24 ! 8E2 16 _ 36E2 24 !
! 4EhSD162_ 12EhSD162! EhSD162! *
resp.
* ! LK(2)S0 ! EhG242! 8EhSD162_ 36EhG242_ EhG242!
! 4EhSD162_ 12EhSD162_ 8EhSD162_ 36EhG242!
! EhSD162_ 4EhSD162_ 12EhSD162! EhSD162! * .
20 HANS-WERNER HENN
3.4. Work in progress (the case n = p = 2).
In this case we do have an algebraic duality resolution but we have not yet com-
pletely succeeded in realizing it. However, there is an algebraic centralizer r*
*esolution
which can be realized.
To describe the homotopy theoretic resolutions we note that, for p = 2, S2 cont*
*ains
a subgroup of order 24, also denoted G24(but not isomorphic to the group with t*
*he
same label that we used in the last section). For p = 2 the group G24is isomorp*
*hic
to the semidirect product Q8oC3 of the quaternion group Q8 with the cyclic group
C3 or order 3 which cyclically permutes i, j and k. G24 contains cyclic subgrou*
*ps
of order 2, 4 and of order 6, We fix such subgroups and denote them by C2 resp.
C4 resp. C6.
Then we have the following results which, up to Galois extension, give resoluti*
*ons
of LK(2)S0 at p = 2. Details will appear in [GHMR2].
Theorem 12 (Centralizer resolution). There exists a fibration
1 hS1
EhS22-! EhS22-! E2 2
and a resolution of spectra of length 3
1 hG hG hC hC hC hC
* ! EhS22! E2 24_ E2 24 ! E2 6 _ E2 4 ! E2 2 ! E2 6 ! * .
Theorem 13 (Duality resolution). There exists a fibration
1 hS1
EhS22-! EhS22-! E2 2
and a resolution of spectra of length 3
1 hG hC hC
* ! EhS22! E2 24 ! E2 6 ! E2 6 ! X3 ! *
together with an isomorphism of Morava modules E*(X3) ~=E*(EhG242).
Remarks_a) The spectrum EhG242is a version of the higher real K-theory spectrum
EO2 at p = 2. We have not been able yet to further identify the spectrum X3.
b) We expect that the resolutions described in this and the previous section wi*
*ll
help to better understand the Shimomura-Wang calculation [ShW] of ss*LK(2)S0 at
p = 3 and that they will be crucial for calculating ss*LK(2)S0 at p = 2.
3.5. Permutation resolutions in the case n = k(p - 1) for p odd.
In this section we will give a positive answer to question (Q1) and the algebra*
*ic
part of question (Q3) of section 3.2 above, at least if p is odd.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 21
3.5.1. We start by introducing some relative homological algebra (cf. [EM]) in a
form which parallels Miller's discussion of En-injective spectra and En-injecti*
*ve
resolutions of spectra (cf. section 2.2.1).
Let p be a fixed prime. If G is a profinite group we denote the collection of f*
*inite
p-subgroups of G by Fp(G), or simply by F(G) or even F if G and p are clear from
the context. Throughout this section we will make the following
Assumption_: G contains only finitely many conjugacy classes of finite p-subgro*
*ups.
We recall that all our modules will be profinite continuous modules for the com-
pleted group algebras and that induced modules are formed by using the completed
tensor product.
A Zp[[G]]-module P will be called F-projective if the canonical Zp[[G]]-linear *
*map
M
P "GF! P
(F)2F
is a split epimorphism (where the sum is taken over conjugacy classes of finite
p-subgroups). It is clear that each Zp[[G]]-module which is induced from a Zp[F*
* ]-
module for some F 2 F is F-projective, and that a Zp[[G]]-moduleLP is F-project*
*ive
if and only if P is a retract of some module of the form (F)2FMF "GFwhere each
MF is a Zp[F ]-module.
The class of F-projectives determines in the usual way a class of F-exact seque*
*nces:
a sequence of Zp[[G]]-modules M0 ! M ! M00is called F-exact if the composition
M0 ! M00is trivial and
Hom Zp[[G]](P, M0) ! Hom Zp[[G]](P, M) ! Hom Zp[[G]](P, M00)
is an exact sequence of abelian groups for each F-projective Zp[[G]]-module P .*
* It
is obvious that the category of Zp[[G]]-modules has enough F-projectives.
Finally an F-resolution of a Zp[[G]]-module M is a sequence of Zp[[G]]-modules
. .!.P1 ! P0 ! M ! 0
where each Ps is F-projective and each 3-term subsequence is F-exact. Then it is
clear that each module M admits an F-resolution and that any homomorphism of
Zp[[G]]-modules M ! N is covered by a map of F-resolutions which is unique up
to chain homotopy.
We will be interested in constructing F-resolutions of finite length of the tri*
*vial
module Zp such that all modules in the resolution are finitely generated.
3.5.2. The construction of our F-resolutions will rely on the following three r*
*esults.
Lemma 14. Suppose G is a profinite group and H is a normal finite p-subgroup.
a) If P is F(G=H)-projective, then considered as a Zp[[G]]-module via the canon*
*ical
projection ss : G ! G=H, P is F(G)-projective.
b) If M0 ! M ! M00is a sequence of Zp[[G=H]]-modules which is F(G=H)-
exact, then considered as a sequence of Zp[[G]]-modules via the canonical proje*
*ction
ss : G ! G=H, M0 ! M ! M00is F(G)-exact.
22 HANS-WERNER HENN
Proof. a) This follows immediately from the following observation. If F is a fi*
*nite
p-subgroup of G=H and M is a Zp[F ]-module then M "G=HF, considered as Zp[[G]]-
module via ss, is isomorphic to M "Gss-1F.
b) If F is a finite p-subgroup of G and N is a Zp[F ]-module then there is a na*
*tural
isomorphism (N "GF) Zp[H]Zp ~=(N Zp[F\H]Zp) "G=HF=F\H) of Zp[[G=H]]-modules.
This implies that - Zp[H]Zp sends F(G)-projectives to F(G=H)-projectives which
in turn implies (b).
Lemma 15. Suppose G is a profinite group and K is a closed subgroup of G which
contains only a finite number of conjugacy classes of finite subgroups.
a) If P is F(K)-projective, then P "GKis F(G)-projective.
b) Assume thatTthere is a decreasing sequence of open normal subgroups Un of G
such that nUn = {1}. If M0 ! M ! M00is a sequence of Zp[[K]]-modules which
is F(K)-exact, then M0 "GK! M "GK! M00"GKis F(G)-exact.
Proof. a) This is trivial.
b) It is enough to show that for each F 2 F(G) and each Zp[F ]-module L the fun*
*ctor
Hom Zp[F](L, -) sends the sequence M0 "GK! M "GK! M00"GKto an exact sequence
of abelian groups. This is in fact a consequence of the Mackey decomposition
formula which describes the restriction of an induced module. In the case of pr*
*ofinite
groups this requires some care so that it seems appropriate to give some detail*
*s.
To simplify notation we let Kn = K=K \ Un, Gn = G=Un and Fn = F=F \ Un.
Note that if F is finite then F = Fn if n is sufficiently large. Now let N be a*
*ny
profinite Zp[[K]]-module. Then we can write N = limiNi (where i runs through
some directed set I, not necessarily a countable sequence) with Ni finite and a*
*cted
on trivially by K \ U~(i)for some increasing function ~ : I ! N. Then we have
the classical Mackey decomposition formula for the Zp[K~(i)]-modules Ni (where
as usual (-) #GFdenotes the restriction of a Zp[[G]]-module to a Zp[F ]-module)
M gK g-1 F
Ni"G~(i)K~(i)#G~(i)F~(i)~= (gNi) #gK~(i)~(i)g-1\F~(i)"g~(i)K~(i)g-*
*1\F~(i)
g2F~(i)\G~(i)=K~(i)
and by passing to the limit we obtain
N "GK#GF~=(Zp[[G]]b Zp[[K]]N) #GF~=limiNi"G~(i)K~(i)#G~(i)F~(i)
~=limi M (gNi) #gK~(i)g-1gK "F~(i) .
~(i)g-1\F~(i)gK~(i)g-1\F~(i)
g2F~(i)\G~(i)=K~(i)
Now we consider our F(K)-split_sequence M0 ! M ! M00and_factor the first
homomorphism via the kernel_M of M ! M00as M0 ! M ! M. It is enough to
show that M0 "GK#GF! M "GK#GFinduces a surjection on Hom Zp[F](L, -).
___
First we note that p : M0 ! M ! 0 is F(K)-exact. This implies that_for any
finite p-subgroup H of K there exists an H-linear splitting s : M ! M0, i.e. th*
*ere
exist increasing functions ff_:_I ! I, fi : I ! I and compatible_families of Zp*
*[H]-
linear maps pi : (M0)ff(i)! M i representing_p and sff(i):_(M )fiff(i)! (M0)ff(*
*i)
representing s such that the composition (M )fiff(i)! (M )i is the map in the g*
*iven
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 23
___
system for M . By explicitely choosing conjugations we may assume that we have
such a splitting (with the same ff and fi) for all finite p-subgroups in the co*
*njugacy
class of H. And because we assume that there are only finitely many conjugacy
classes of finite p-subgroups in K we may assume that the functions ff and fi w*
*ork
for all H 2 F(K).
___
The Mackey decomposition formula for (M )fiff(i)and M0ff(i)gives us therefore, *
*for
any choice of double coset representatives in G~fiff(i)resp. G~ff(i), well def*
*ined
Zp[F ]-linear maps
___ G~fiff(i) G~fiff(i) 0 G~ff(i) G~ff(i)
esff(i): (M )fiff(i)"K~fiff(i)#F~fiff(i)! Mff(i)"K~ff(i)#F~ff(i).
Now the elementary theory of profinite sets (cf. Lemma 5.6.7 in [RZ]) tells us *
*that
the quotient map G ! F \G=K admits a continuous section and any such section
gives us a compatible choice of double coset representatives in Gn for all n. (*
*We
note that it is here that we use the assumption on the existence of the decreas*
*ing
sequence of subgroups Un.) For any such choice the associated Zp[F ]-linear maps
esff(i)are compatible with respect to i so that they patch together and_define_*
*a Zp[F ]-
linear map on the level of inverse limits. This shows that M0 "GK#GF! M "GK#GF*
*is
split as a map of Zp[F ]-modules and hence we are done.
Lemma 16. Suppose G is a profinite group and M is a Zp[[G]]-module which
admits a finite projective resolution and which is projective as a Zp-module. T*
*hen
M is projective as a Zp[F ]-module for every F 2 F.
Proof. By induction on the length of a finite projective resolution it is enoug*
*h to
show that for a finite p-group F a short exact sequence 0 ! M1 ! M2 ! M3 ! 0
of Zp[F ]-modules splits if M1 is projective as a Zp[F ]-module and if the sequ*
*ence
splits as a sequence of Zp-modules.
The existence of a Zp-splitting of the inclusion M1 ! M2 implies that any Zp[F *
*]-
linear map ' from M1 to the coinduced module Hom (Zp[F ], M1) can be extended
to an Zp[F ]-linear map e': M2 ! Hom (Zp[F ], M1). Next, if M1 is projective as
a Zp[F ]-module then it is a direct summand in the induced module M1 "F{1}, and
because F is finite the induced module is isomorphic to the coinduced module.
Now we take for ' any Zp[F ]-split inclusion of M1 into Hom (Zp[F ], M1). Then *
*the
composition of e'with a Zp[F ]-linear splitting of ' provides the desired split*
*ting.
3.5.3. Here is the promised answer to question (Q1) and the algebraic part of (*
*Q3).
Proposition 17. Suppose G is a virtually profinite p-group and S is a closed
normal subgroup which is a profinite p-group. Furthermore assume that
(1) H*(S; Fp) is a finitely generated Fp-algebra,
(2) all finite p-subgroups of S are cyclic and there is a bound on their order,
(3) the trivial module Zp admits a projective resolution of finite type and fin*
*ite
length over the p-completed group algebra of G=S and all its closed subgroups.
T
(4) There is a sequence of open subgroups Un of G such that nUn = {1}.
Then the trivial Zp[[G]]-module Zp admits an F-resolution of finite length.
Furthermore, all F-projectives in this resolution can be chosen to be summands *
*in
finite direct sums of modules of the form Zp "GFwith F 2 F.
24 HANS-WERNER HENN
Remarks_a) Assumption (1) implies by a recent result of Minh and Symonds [MS]
that S has only a finite number of conjugacy classes of finite p-subgroups. In
particular the bound in assumption (2) is already a consequence of assumption
(1). Furthermore recent work of Symonds suggests that finite length F-resolutio*
*ns
should exist even if we skip condition (2).
b) Assumption (3) implies that G=S has no elements of order p, in other words
every finite p-subgroup of G is already contained in S.
c) It is well-known that H*(Sn; Fp) is a finitely generated algebra. As already
mentioned in 1.2.4 and 1.2.5, Sn has finite p-subgroups if and only if n is div*
*isible
by p-1. Furthermore, if p is odd, or p = 2 and n is odd, then all finite p-subg*
*roups
are cyclic and their conjugacy class is unique. The p-Sylow subgroup Sn of Sn is
normal in Gn of index (pn - 1)n which is of order prime to p if n = k(p - 1) wi*
*th
k 6 0 mod p. Therefore, for such n the assumptions of the proposition hold with
G = Gn and S = Sn.
If n = k(p - 1) with k divisible by p thenSwe replace Gn as in remark 2 of sect*
*ion
2.2 by Gn(K) := Sn o Gal(K=Fp) where K := rFqpr. This has the effect that
all p-torsion elements of Gn(K) are already contained in Sn resp. its normal Sy*
*low
subgroup Sn. In this case it is clear that the assumptions hold with G = Gn(K)
and S = Sn. In fact, G=S has a finite normal subgroup of order prime to p with
quotient Zp. The somewhat artificially looking assumptions on the pair (G, S) in
Proposition 17 have been introduced in order to cover this case.
d) If n is even and p = 2 then Sn has finite 2-subgroups which are not cyclic. *
*Thus
for n even and p = 2 the proposition does not give any information. Nevertheles*
*s,
if n = p = 2 the algebraic centralizer resolution mentioned in section 3.4 give*
*s an
explicit finite length F-resolution of the trivial Z2[[S2]]-module Z2.
Proof. The proof will be by induction over the order of the largest finite p-su*
*bgroup
of G. We distinguish the following cases.
Case_1:_G has no non-trivial finite p-subgroups.
In this case it follows from Quillen's F -isomorphism theorem [Q] that H*(S; Fp*
*) is
a finite Fp-algebra. Because S is a profinite p-group this implies cdp(S) < 1 a*
*nd
then assumption (3) implies cdp(G) < 1. Finally, (3) and the spectral sequence
of the group extension 1 ! S ! G ! G=S ! 1 show that Hi(G, M) is a finite
group for every finite discrete continuous Zp[[G]]-module M and this ensures the
existence of a projective resolution of Zp of finite length in which all projec*
*tives are
finitely generated (cf. Proposition 4.2.3 in [SyW]).
Case_2:_G has a normal finite p-subgroup F .
By [MS] H*(S=F, Fp) is still a finitely generated Fp-algebra and hence (G=F, S=*
*F )
still satisfies our assumptions. Therefore, by induction hypothesis, the trivia*
*l G=F -
module Zp admits an F(G=F )-resolution of finite length such that all F(G=F )-
projectives are of the required form. Then Lemma 14 implies that the very same
resolution is also an F(G)-resolution of finite length for the trivial G-module*
* Zp,
and all F(G)-projectives are as required.
Case_3:_The general case.
We may suppose that G does not contain any finite normal p-subgroups. In this
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 25
case we consider the short exact sequence of Zp[[G]]-modules
M "
(5) 0 ! K -f! Zp "GNG(E)-! Zp ! 0
(E)
where the sum is over conjugacy classes of non-trivial elementary abelian p-sub-
groups of G (which by our assumption are all of order p), " is the canonical au*
*g-
mentation and K is its kernel.
First we note that the exact sequence (5) is F-exact. In fact, if F is a finite
p-subgroup of G then it has a non-trivial central element of order p and F is
contained in the normalizer of the elementary abelian p-subgroup E0 generated by
this element of order p. This means, that the action of F on G=NG (E0) has a fi*
*xed
point. Such a fixed point determines a Zp[F ]-linear splitting of the surjecti*
*on in
(5). In other words, the exact sequence splits upon restriction to every F 2 F *
*and
this is equivalent to saying that the sequence is F-exact.
L
It will therefore be enough to construct F-resolutions for (E)Zp "GNG(E)and f*
*or
K where the F-projectives have the required form. In fact, if we have two such
resolutions we can lift f to a map of resolutions and then the resulting double
complex will be an F-resolution for Zp with all F-projectives as required.
The pair (NG (E), S\NG (E)) satisfies the same assumptions as (G, S): assumptio*
*ns
(2), (3) and (4) are obvious. For (1) we note that S \NG (E)) = NS(E) agrees wi*
*th
the centralizer CS(E) because E is of rank 1. In fact, because the p-rank of E *
*is
always 1, the p-group NS(E)=CS(E) injects into Aut(E) ~=Z=(p - 1) and hence it
is trivial. Now CS(E) satisfies (1) because it is given by a component of Lanne*
*s'
T -functor (by Theorem 2.6 of [He]) and Lannes' T -functor takes unstable finit*
*ely
generated Fp-algebras to unstable finitely generated Fp-algebras ([DW, Theorem
1.4]). Furthermore, by [MS] the groups NS(E) and hence also NG (E) have only a
finite number of conjugacy classes of finite p-subgroups. Therefore we can app*
*ly
case 2 and deduce that for each E 2 F(G) the trivial Zp[[NG (E)]]-module Zp adm*
*its
an F(NG (E))-resolution of finite length. Inducing this resolution gives, by Le*
*mma
15, an F(G)-resolution of Zp "GNG(E)with all F-projectives as required.
Finally consider K. The group G=S acts on the finite set of conjugacy classes of
elementary abelian p-subgroups of S and the stabilizer of such a subgroup E is *
*the
image of NG (E) in G=S. In particular this image is of finite index in G=S and *
*this
implies that for each E G we get an isomorphism of Zp[[S]]-modules
M
Zp "GNG(E)~= Zp "SNS(E0)
(E0)~~ 1.
3.6.1. Let us start by looking at the F-resolution of the trivial Zp[[Gn]]-modu*
*le
constructed in the previous section. By 1.2.4 there is a unique conjugacy class*
* of
subgroups of Sn of order p. We pick a representative and denote it by E and its
normalizer in Gn simply by N. So the exact sequence of the proof of Proposition
17 takes the following form
(6) 0 ! K -f! Zp "GnN-"!Zp ! 0 .
By 1.3.2 this sequence can be realized by the cofibration
LK(n)S0 ' EhGnn-'! EhNn! C
where ' is induced by the inclusion of N into Gn and C is the cofibre of '. Of *
*course,
as before realizing means that the map Hom cts(", (En)*) agrees with the map
Hom cts(Zp, (En)*) ~=(En)*LK(n)S0 -'*!(En)*EhN2~=Hom cts(Zp "GnN, (En)*)
induced by ' in (En)*. This map is injective and therefore E*C can be identified
with Hom cts(K, (En)*).
This suggests that we should start by constructing resolutions for EhNnand C.
3.6.2. We begin with C.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 27
Lemma 18. C admits an En-resolution of finite length in which each spectrum is
a summand of a finite wedge of En's.
Proof. We know from section 3.5 that K has a projective Zp[[Gn]]-resolution
Po : 0 ! Pd ! . .P.0! K
of finite length d in which each Ps is finitely generated. Let Qo be the follow*
*ing
exact complex of Zp[[Gn]]-modules
Qo : 0 ! Pd ! . .!.P0 ! Zp "GnN! Zp ! 0
obtained by splicing the resolution of K and the exact sequence (6). By Proposi*
*tion
3 there is a sequence of maps between spectra of the following form
Yo : * ! LK(n)S0 ' EhGnn-'! EhNn! X0 ! . .!.Xd ! *
where each Xi is a summand in a finite wedge of En's such that the complex
Hom cts(Qo, (En)*) is isomorphic, as a complex of Morava modules to the complex
(En)*(Yo). Again by Proposition 3 the composite LK(n)S0 ! EhNn! X0 is null
and we obtain a sequence of maps between spectra
Xo : * ! C ! X0 ! . .!.Xd ! *
such that Hom cts(Po, (En)*) is isomorphic, as a complex of Morava modules to t*
*he
complex (En)*Xo. We claim that Xo is, in fact, an En-resolution of C in the sen*
*se
of 2.2.1. In fact, it is clear that all Xs for s 0 are En-injective and En-ex*
*actness
of Yo is seen as in the proof of Theorem 4. En-exactness of Xo follows immediat*
*ely
from that of Yo.
3.6.3. The case of EhNnis substantially more difficult and unlike in the case o*
*f C
the resolution will not be an En-resolution although it will still have some go*
*od
properties. In this subsection we will make the algebraic resolution of Zp "GnN
explicit.
3.6.3.1. First we investigate the structure of the group N = NGn (E).
We choose a primitive (pn - 1)-st root of unity ! 2 WxFq Sn and note that the
element X := ! p-1_2S 2 Dn satisfies Xn = -p.
Lemma 19. The subfield Qp(X) of Dn is isomorphic to the cyclotomic extension
Qp(ip) generated by a primitive p-th root of unity ip and this isomorphism rest*
*ricts
to an isomorphism Zp[X]=(Xn + p) ~=Zp[ip].
Proof (outline). In fact, this is a straightforward consequence of local class *
*field
theory but for the convenience of the reader we outline an elementary and direct
proof.
R := Zp[X]=(Xn +p) is a local ring with maximal ideal generated by X. The powers
of the maximal ideal induce, as in section 1.2.3, a decreasing complete filtrat*
*ion Fi
with i = k_n, k = 1, 2, . .,.of the group of units of R via
Fi:= {g 2 Rx |g 1 mod Xin}
28 HANS-WERNER HENN
and with filtration quotients gri := Fi=Fi+_1nwhich are all canonically isomorp*
*hic
to Fp. Furthermore the p-th power map induces, as in the case of the groups Sn,
maps gri! gr'(i)which via these isomorphisms are given by the identity if i 6= *
*1_n
and by __a7! __ap- __aif i = 1_n(with notation as in section 1.2.3). The comple*
*teness of
the filtration shows now that R contains a p-th root of unity, in fact for any *
*a 2 R
with ap a mod (X) there is a p-th root of unity of the form 1 + aX mod (X2).
This shows that Qp(X) contains an isomorphic copy of the field Qp(ip) and becau*
*se
both are extensions of degree n = p - 1 of Q they have to be isomorphic. The fi*
*nal
statement of the Lemma is now obvious.
We choose a primitive p-th root of unity in Qp(X) and we still denote it by ip *
*thus
identifying Qp(X) with Qp(ip). We choose our subgroup E Sn to be generated
by ip. If n = k(p - 1) then (cf. 1.2.4) the centralizer CDn(E) is a division al*
*gebra
which is central over Qp(ip) of dimension k2. In particular, if n = p - 1, then
CDn(E) = Qp(ip), and CDxn(E) = Qp(ip)x .
We need to know the structure of the group of units in Qp(ip) = Qp(X). This is
well known in algebraic number theory. For the convenience of the reader we give
some details.
Consider the filtration by subgroups
(7) 0 F_2n F_1n Zp[ip]x Qp(ip)x
where F_2nand F_1nare the filtration subgroups of Zp[ip]x introduced above. The
valuation v, normalized by v(p) = 1, is a split epimorphism Qp(ip)x ! 1_nZ ~=Z
with kernel Zp[ip]x . Next, the description of the p-th power map given above s*
*hows
that F_2nis a free Zp-module of rank n and the torsion-subgroup of Zp[ip]x iden*
*tifies
with Zp[ip]x =F_2n~=E x Z=n. In particular, we see that the filtration (7) is s*
*plit
and we obtain an isomorphism
(70) Qp(ip)x ~=Z x Z=n x Z=p x Znp.
Furthermore, the discussion above shows that we can choose generators as follow*
*s:
o The element X satisfies v(X) = 1_nand can therefore be chosen as a generator *
*of
the factor Z.
pn-1_
o The subgroup Z=n is the subgroup generated by ! p-1.
o Z=p ~=E is the subgroup generated by ip,
o The elements jj := 1 + Xj 2 Zp[ip]x ~=Zp[X]x , j = 2, . .,.n + 1, qualify as a
system of topological generators of Znp.
We will need to understand the action of the Galois group Gal(Qp(ip)=Qp) on
Zp[ip]x . It is clear that the three factors F_2n, E and Z=n are invariant with*
* respect
to the action of the Galois group and that the action on Z=n Zxpis trivial.
Furthermore, the Galois group can be canonically identified with the group Aut(*
*E)
of automorphisms of E, in other words the action on E is the tautological actio*
*n.
To understand the action on F_2nwe observe that the Galois automorphisms can
be realized by the conjugation action (in Dn) of the quotient of the cyclic gro*
*up
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 29
pn-1_
generated by o := ! (p-1)2by the subgroup generated by on . Then an elementary
calculation shows j(1-pn)
o*(jj) = 1 + ! ______p-1Xj ,
j(1-pn)_
i.e. jj is an eigenvector in gr_jnwith eigenvalue ! p-1. To obtain a genuine
j(1-pn)_
eigenvector with eigenvalue ! p-1 in Zp[ip]x we note that
Xn ij(pn-1)
ej := 1_n ! ______p-1oi
i=1
j(1-pn)_
is an idempotent in Zp[Z=n] which satisfies oej = ! p-1ej. If we replace jj by
j0j:= (ej)*jj
then the elements j0j, j = 2, . .,.n + 1, generate one-dimensional characters, *
*say
O(j), and as a Galois-module F_2nis isomorphic to the direct sum of all the dif*
*ferent
one-dimensional characters of Aut(E) over Zp.
We note that if we rename O(n + 1) as O(1) then we get O(j) = O(1) j for j =
2, . .,.n + 1. In the sequel we will interpret O(j), for any integer j 2 Z, as*
* the
tensor product O(1) j.
After these preparations we will now desribe NGn (E). By abuse of notation we
continue to denote the image of elements of Dxnin Gn ~=Dxn= < Sn > by the same
name.
Proposition 20. Let n = p - 1 and p be odd.
a) There is an exact sequence
1 ! CGn (E) ! NGn (E) ! Aut(E) ! 1 .
b) There is an isomorphism
CGn (E) ~=H x E x Znp
where H ~=Z=2n x Z=n_2is generated by X and on X2, E is generated by ip, and
the elements jj = 1 + Xj, j = 2, . .,.n + 1, are a system of topological genera*
*tors
of Znp.
c) The action of Aut(E) leaves H, E and Znpinvariant:Lthe action of Aut(E) on
E is the tautological action while Znp~= nj=1O(j) and O(j) is generated by jj0*
*for
j = 2, . .,.n and by jn+10for j = 1.
d) The subgroup generated by ip, H and o is a finite subgroup F of N of order p*
*n3
which contains H x E as a normal subgroup with quotient Aut(E).
Proof. a) We have seen above that there is an exact sequence
1 ! CDxn(E) ! NDxn(E) ! Aut(E) ! 1
30 HANS-WERNER HENN
pn-1_
where ! (p-1)2projects to a generator of Aut(E). In particular all of the autom*
*or-
phisms of the group E are realized by conjugation in Dxnand hence also in the
central quotient Gn ~=Dxn= < Sn >. This shows (a).
b) We claim that the epimorphism Dxn! Gn induces an epimorphism CDxn(E) !
CGn (E) with kernel the central subgroup generated by Sn. In fact, if we lift *
*an
element g 2 CGn (E) to an element eg2 Dxnthen egipeg-1= ipz with z 2< Sn >.
But then we obtain
1 = egippeg-1= (egipeg-1)p = (ipz)p = zp
and hence z = 1, in other words eg2 CxDn(E). Part (b) follows now from the
_n-1_
observation that Sn = p = ! p 2Xn, i.e. in terms of the decomposition (7') of
CDxn(E), the element Sn has components (n, n_2) in Z x Z=n and trivial componen*
*ts
in Z=p x Znp.
(c) and (d) are clear from the discussion above.
Remark_The subgroup H is intrinsically defined as the subgroup of the abelian
profinite group CGn (E) generated by elements of finite order prime to p. As su*
*ch
it is necessarily invariant with respect to the action of Aut(E). The splitting*
* H ~=
Z=2n x Z=n_2, however, is visibly not invariant with respect to this action.
3.6.3.2. In the next step we will_construct an explicit F(N)-resolution_of the *
*trivial
Zp[[N]]-module Zp. If we set N := N=(H x E) then we have N = Znpo Z=n where
the semidirect product is taken with respect to the Z=n-module structure on Znp
given by the direct sum of the n different characters of Z=n over Zp.
Proposition 21. __
a) The trivial Zp[[N ]]-module Zp admits a projective resolution
Po : 0 ! Pn ! . .!.P1 ! P0 ! P-1 = Zp
with M __
Pr ~= O(i1 + . .+.ir)"NZ=n
(i1,...,ir)
and where the direct sum is taken over all sequences (i1,_._.,.ir) with n - 1 *
* i1 >
i2 > . .>.ir 0 (For r = 0 there is a unique summand 1 "NZ=ncorresponding to
the empty sequence).
__
b) Considered as a complex of Zp[[N]]-modules, via the projection N ! N , this
complex is an F(N)-resolution in which all modules are summands in a finite dir*
*ect
sum of modules Zp "NE.
Proof. a) We start by observing that
n-2M
H*(Znp; Zp) ~= ( O(i))
i=0
as a module over Zp[Z=n]. The resolution is now constructed step by step as in *
*the
proof of Theorem 6.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 31
__
The identity of Zp considered as_a map of Zp[Z=n]-modules extends to a Zp[[N ]]-
linear homomorphism P0 := Zp "NZ=n! Zp. Let N0 be its kernel. Then we can
compute H*(Znp, N0=(p)) and in particular we find
n-1M
H0(Znp; N0=(p)) ~= O(i) Zp Fp ,
i=0
L n-1
as a module over Zp[Z=n]. The Zp[Z=n]-moduleL i=0O(i) is projective and hence
we can lift the resulting homomorphism n-1i=0O(i) ! H0(Znp, N0=(p)) to N0 and
__ L n-1 __
by Nakayama's Lemma the Zp[[N ]]-linear extension P1 := i=0O(i) "NZ=n! N0 is
onto. Then we repeat the game with the kernel N1 of this epimorphism, and so on.
When we finally arrive at Nn-1 we see that
n-1X
H0(Znp, Nn-1=(p)) ~=O( i) Zp Fp, Hi(Znp, Nn-1=(p)) = 0 ifi > 0 .
i=0
P n-1 __
Then we can construct in the same manner a map Pn := O( i=0 i) "NZ=n! Nn-1
which by Nakayama's lemma is now even an isomorphism.
b) This is an immediate consequence of Lemma 14.
3.6.4. Now we turn towards the problem of realizing the resolution Po construct*
*ed
in Proposition 21b, or rather the induced resolution Po "GnNof the induced modu*
*le
Zp "GnN, by a sequence of maps between spectra.
Let F be the subgroup of N of order pn3 described in Proposition 20. The charac*
*ters
O(i) of the last section can be considered as characters of F via the canonical
projection F ! F=(H x E) = Aut(E) ~=Z=n. We will also need to consider the
group F1 := F \ Sn which is of order pn2 and is generated by ip and o, and the
group F2 which is cyclic of order pn and is generated by ip and on .
Lemma 22. For any i 2 Z there is an isomorphism of Morava modules
(En)*( 2pniEhFn) ~=Hom cts(O(i) "GnF, (En)*) .
Proof. For i = 0 this is nothing but 1.3.2. More generally, for every k 2 Z the*
*re is
an isomorphism of Morava-modules
(En)*( kEhFn) ! Hom cts(Zp "GnF, (En)*(Sk)) .
We will show that there is an invertible element (i) in (En)* of degree 2pni on
which F acts via O(i). Then we will get the desired isomorphism by composing
with the isomorphism of Morava modules
Hom cts(Zp "GnF, (En)*(Sk)) ! Hom cts(O(i) "GnF, (En)*)
given by
' 7! g 7! oe-1('(g))g*( (i))
32 HANS-WERNER HENN
~=
where oe is the suspension isomorphism E* = E*(S0) -! E*+2pni(S2pni).
We recall that F is generated by o, ip and X = ! p-1_2S. We need some informati*
*on
about the action of these elements on (En)*. For this we recall that the action*
* of
an element g 2 Sn is determined as follows (cf. [DH1]): if we lift g to a power
series eg(x) 2 (En)0[[x]] then there is a unique continuous ring homomorphism
g* : (En)0 ! (En)0 and a unique * -isomorphism h 2 (En)0[[x]] from the formal
group law g*(f n) to the formal group law H defined by H(x, y) = eg-1f(neg(x), *
*eg(y))
The action of g on u is then given by g*(u) = eg0(0)h0(0)u.
In particular, if g(x) = ax with a 2 Fxqand if the Teichm"uller lift of a will,*
* by
abuse of notation, still be denoted by a then we can take as lift eg(x) = ax an*
*d the
[p]-series of the formal group law H satisfies
[p]H (x) =a-1([p]e n(ax)) = a-1([p]e n(ax))
p pn-1 pn
= a-1 p(ax) +e nu1(ax) +e n. .+.e nun-1(ax) +e n(ax)
n-1-1 pn-1 pn
= px +H u1ap-1xp +H . .+.Hun-1ap x +H x ) .
This shows that the *-isomorphism h is the identity, i.e. h(x) = x, and that g**
* is
given by
i-1
(8) g*(ui) = ap ui, g*(u) = au .
pn-1_
In the case of o we have a = ! (p-1)2so that
pn-1_
(9) o*(u) = ! (p-1)2u .
The action of ip is more difficult. However, we only need to know that ip acts
trivially modulo the maximal ideal m = (p, u1, . .,.un-1) (En)*, and in parti*
*cular
that
(10) ip*(u) u mod m .
In fact, this holds for any element g in the p-Sylow subgroup Sn of Sn.
The universal deformation e nis already defined over Zp[[u1, . .,.un-1]] and th*
*ere-
fore the subgroup of Gn given by Galois automorphisms of Fq acts trivially on t*
*he
ui and on u. Together with (8) this implies
p-1_
(11) X*(u) = ! 2u .
The action of o and of ip on (En)* is WFq-linear while the action of X is only
Zp-linear and satisfies X*(wx) = woeX*(x) if w 2 WFq and x 2 (En)*.
Now consider the element Y
0:= g*(u) .
g2F2
This is clearly fixed by the subgroup F2. Furthermore by (9) and (10) we have
Yn Yn i(pn-1) p(p-1)pn-1
0 oin*(up) = !p_____p-1up = !p_____2___p-1upn = -upn mod m
i=1 i=1
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 33
so that 0is an invertible element of degree -2pn. Now F2 is normal in F and the
quotient F=F2 is isomorphic to Z=n x Z=n with generators o and X. This quotient
acts on the F2-fixed points and by (11) we find for any ~ 2 WFq
Y Y p-1_ p-1_
X*(~ 0) = ~oeX*( 0) = ~oe g*X*(u) = ~oe g*(! 2u) = ~oe! 2pn 0.
g2F2 g2F2
In particular we get X*(!-pn_2 0) = !-pn_2 0, and thus
pn_ 0
00:= !- 2
is still an invertible element in (En)-2pn which is fixed by on , ip and X, and*
* which
satisfies
pn_ 0 -pn_ Y
o*( 00) =!- 2 o*( ) = ! 2 g*o*(u)
g2F2
pn_ Y pn-1_2 -pn_ pn-1_p0 pn-1_ 00
= !- 2 g*(! (p-1)u) = ! 2 ! p-1 = ! (p-1) .
g2F2
Therefore, for any i 2 Z, the class ( 00)-iis an invertible element in (En)2pni
on which F acts via O(i) (with the convention adopted in the discussion before
Proposition 20).
We will2need some partial information about the homotopy groups of the spectra
2p niEhGn, i 2 Z, where G runs through various subgroups of F which contain the
central subgroup Z F generated by on .
Lemma 23. Suppose G is a subgroup of F which contains Z.
a) If G is of order prime to p, then for any i 2 Z the homotopy fixed point spe*
*ctral
sequence converging to ss*( 2p2niEhGn) satisfies Es,t2= 0 if s > 0. The spectr*
*al
sequence collapses at E2 and ss*(EhGn) ~=(En)G*is concentrated in degrees divis*
*ible
by 2n.
b) If G contains an element of order p then for any i 2 Z the homotopy fixed po*
*int
spectral sequence converging to ss*( 2p2niEhGn) satisfies
8
>>>0 ifq is even, 0 < q < 2n
2nihG < E0,02 ifq = 0
ssq( 2p En ) ~=>
>>:0 ifq is odd,q 6 {1, 3, . .,.2p - 3} mod 2pn
Es(q),s(q)+q2ifq is odd,q {1, 3, . .,.2p - 3} mod 2p2n
where s(2q+1) = 2(p-2-q0)+1 if 2q+1 = 2q0+1+2p2nl and q02 {0, 1, . .,.p-2}.
Proof. a) If G contains no elements of order p then it is clear that Es,t2= 0 i*
*f s > 0
and the spectral sequence collapses. Furthermore, G contains the central subgro*
*up
Z and this subgroup acts trivially on (En)0 (by (8)) and because of (on )*(u) =
pn-1_ 0,t
! p-1u (again by (8)) we see that E2 = (En)Gtis trivial if t 6 0 mod 2n.
b) Because G always contains Z and because E is normal in F , the assumption
implies that G contains F2 = Z x E. Because the index of F2 in G is prime to
34 HANS-WERNER HENN
p the homotopy fixed point spectrum EhGnis a direct summand in EhF2nand it is
therefore enough to discuss the case G = F2.
In the case of G = F1 the homotopy fixed point spectral sequence has been analy*
*zed
by Hopkins and Miller. Their account remains unpublished. A summary of this
analysis is given in section 2 of [N]. If p = 3 a rather detailed discussion wh*
*ich
includes the case of F2 can be found in [GHM] and [GHMR1]. The approach used
in these papers generalizes without much problems to the case of any p > 2. In *
*the
following we will describe the E2-term and the differentials of this spectral s*
*equence.
First of all, let ae be the (p-1)-dimensional WFq[F2]-module which restricted t*
*o E is
the reduced regular representation and on which the central element on acts by *
*mul-
pn-1_
tiplication by ! p-1. Then, as a graded Zp[F2]-algebra, (En)* is isomorphic to*
* the
completion of S*(ae)[N-1 ] at its maximal ideal, where S*(ae) isQthe graded sym*
*metric
algebra on ae with ae in degree -2, and we have inverted N := g2F2g*(e) 2 S*(*
*ae)
where e is a suitable generator of ae (cf. [GHMR1, Lemma 3.2]). In fact, the is*
*o-
morphism identifies N, up to a scalar, with the element 00of the proof of Lemma
22. This isomorphism can be used to calculate the E2-term of the homotopy fixed
point spectral sequence as follows (cf. section 2 in [N], or Theorem 3.7 in [GH*
*MR1]
if p = 3):
o The invariants E0,t2are trivial unless t 0 mod 2n and periodic of period 2p*
*n with
periodicity generator 002 (En)-2pn. In the sequel we let := ( 00)-1 2 (En)2p*
*n.
o Multiplication with p annihilates Es,*2if s > 0. Furthermore, there are eleme*
*nts
2-2p
ff 2 E1,2p-22and fi 2 E2,2p2 such that, as a module over Fq[ 1],
o E2k,*2, for k > 0, is free of rank 1 with generator fik,
o E2k+1,*2, for k 0, is free of rank 1 with generator fffik.
The elements ff and fi are infinite cycles and represent the images of the elem*
*ents
ff1 2 ss2p-3(S0), fi1 2 ss2(p2-p-1)(S0) with respect to the unit S0 ! EhF2nof t*
*he
ring spectrum EhF2n.
The only non-trivial differentials in this spectral sequence are d2p-1 and d2n2*
*+1.
They are forced by Toda's relations ff1fip1= 0 and fipn+11= 0 and are determined
by 2 2
d2p-1( n) = cfffip-1 , d2n2+1( n ff) = c0fin +1 ,
i.e. d2p-1( ) = -c 1-nfffip-1, and d2n2+1( ff) = c0 -p(p-2)fin2+1 where c, c0
are suitable units in Fq.2 Then we end up with the following result which is mo*
*re
precise than Lemma 23 above.
Proposition 24 (cf. [N, Proposition 2.1, 2.2]).
a) ss*(EhF2n) is periodic of period 2p2n and with periodicity generator p.
b) Es,*1is trivial if s is even and s > 2n2, or if s is odd and s > 2n - 1.
c) If 2n2 s = 2k > 0 then Es,*1is a free module over Fq[ p] with generator f*
*ik,
of total degree 2k(p2 - p - 1).
d) If 2n-1 s = 2k+1 > 0 then Es,*1is a free module over Fq[ p] with generato*
*rs
lfikff, 2 l p of total degree 2pnl + 2k(p2 - p - 1) + 2p - 3.
e) E0,t1= 0 if t 6 0 mod 2n.
____________
2Note that the element in [N] corresponds to p-1= n in this paper.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 35
After these preparations we can continue with the proof of part (b) of Lemma 23.
First we investigate which of the generators in part (c) and (d) of Proposition*
* 24
can contribute to ssq for q as in Lemma 23b. We distinguish two cases according*
* to
the parity of q.
1) If 0 q = 2q0< 2n then it is enough to show that there is no k with 0 < k *
* n2
such that
2k(p2 - p - 1) 2q0 mod 2p2n .
Calculating modulo 2pn gives
-2k 2q0 mod 2pn
and because of 0 < 2k 2n2 and 0 2q0 < 2n this is clearly impossible. In
particular, we see that ssq = 0 if q is even and 0 < q < 2n, and ss0 ~=E0,02.
2) If q = 2q0+ 1 then we have to consider the congruence
(12) 2pnl + 2k(p2 - p - 1) + 2p - 3 2q0+ 1 mod 2p2n .
Reducing mod 2pn gives
-2k + 2p - 3 2q0+ 1 mod 2pn .
and thus k p - 2 - q0 mod pn. In view of 0 k p - 2 this implies that
for (12) to have a solution we must have q0 2 {0, 1, . .,.p - 2} modulo pn, i.e.
q 2 {1, 3, . .,.2p - 3} modulo 2pn. Furthermore, for such a q there is a unique*
* k
with 0 k p - 2 and a unique l such that lfikff is of total degree q.
It remains to check that the elements lfikff with 0 k p-2 which are not pe*
*rma-
nent cycles cannot be of total degree q 2q0+1 mod 2p2n with q02 {0, 1, . .,.p*
*-2}.
In fact, not being a permanent cycle is equivalent to l 1 mod p. Calculating *
*mod-
ulo 2p2n gives
2pnl+2k(p2-p-1)+2p-3-(2q0+1) = 2pn(l+k)+2(-k +p-2-q0) 2pn(l+k)
and this cannot be 0 modulo 2p2n if l 1 mod p and 0 k p - 2.
We can finally state and prove the following realization result.
Theorem 25. There is a resolution of length n (in the sense of 3.3.1)
Xo : * ! EhNn:= X-1 ! X0 ! . .!.Xn ! *
such that the complex E*(Xo) is isomorphic, as a complex of Morava modules,
to the complex Hom cts(Po "GnN, (En)*) where Po is the complex of Proposition 2*
*1.
Furthermore, for r > 0 we have
` 2
Xr ' 2p n(i1+...+ir)EhFn
(i1,...,ir)
where F is the finite subgroup of N of Proposition 20d and where the wedge is t*
*aken
over all sequences (i1, . .,.ir) with n - 1 i1 > i2 > . .>.ir 0 (For r = 0 *
*there
is a unique summand EhFncorresponding to the empty sequence).
36 HANS-WERNER HENN
Proof. Because of O(pi) ~=O(i) Lemma 22 implies that the spectra Xr realize the
Morava modules Hom cts(Pr "GnN, (En)*). So it remains to realize the maps and
show that the resulting sequence of maps of spectra can be refined into a resol*
*ution
in the sense of 3.3.1.
Because the index of F2 in F is prime to p, we have, for each i, that 2p2niEhFn
is a direct wedge summand in 2p2niEhF2n. Furthermore, it follows easily from
Proposition 24 that EhFnis periodic of period 2p2n and thus 2p2niEhFnis, for e*
*ach
i, a direct wedge summand in EhF2n. Therefore, in order to realize the maps it *
*is
enough to show that the (En)*-Hurewicz homomorphism
hF hF
(13) ss0(F (EhF2n, EhF2n)) ! Hom (En)*[[Gn]](En)*(En 2), (En)*(En 2)
is an isomorphism (onto the group of degree preserving homomorphisms). By
Proposition 3 this happens iff the canonical map
F2
(14) ss0(En[[Gn=F2]]hF2) ! (En)0[[Gn=F2]]
is an isomorphism.
T
Now we choose a decreasing sequence Uj of open subgroups of Gn with F2 = j Uj.
Then
En[[Gn=F2]] ' holimjEn[[Gn=Uj]] .
By Proposition 2 we have
Y
En[[Gn=Uj]]hF2 ' EhFx,jn
F2\Gn=Uj
where Fx,jis the isotropy group of the coset xUj with respect to the action of *
*F2 on
Gn=Uj. Because Z F2 is central each Fx,jcontains Z. Then Lemma 23b implies
that the canonical maps ss0(EhFx,jn) ! (En)Fx,j0are isomorphisms for all x and j
and therefore F
ss0(En[[Gn=Uj]]hF2) ! (En)0[[Gn=Uj]] 2
is an isomorphism for each j. Furthermore, it is clear that (En)1[[Gn=Uj]] = 0 *
*for
all j, and by the remark on the Mittag-Leffler condition following Proposition *
*2 the
relevant lim1-terms for the homotopy groups of holimj(En)[[Gn=Uj]]hF2 are also
trivial. Therefore we obtain the desired isomorphism (14) by passing to the lim*
*it.
We have now proved that all maps Xr ! Xr+1 can be (uniquely) realized and the
compositions of two successive maps are trivial.
It remains to construct the factorizations Xr ! Cr ! Xr+1, 0 r n - 1, such
that Cr-1 ! Xr ! Cr is a cofibration. This will be done inductively. We note
that these factorisations will realize the splitting of the exact complex of Mo*
*rava
modules E*(Xo) into the usual short exact sequences. In particular, this will s*
*how
that Cn ' Xn so that the resolution will be automatically of length n.
For r = 0 (where we take C-1 = X-1) this is just a consequence of the fact that
the composition X-1 ! X0 ! X1 is null. Now suppose that we have already
constructed the factorizations
Xr ! Cr ! Xr+1, 0 r k < n - 1 .
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 37
We need to show that the composition
Ck ! Xk+1 ! Xk+2
is null so that we can factor it through the cofibre Ck+1 of the map Ck ! Xk+1.
For this it is enough to show that the induced map [Ck, Xk+2] ! [Xk, Xk+2] is
injective.
Now the inductively already constructed part of the resolution
* ! X-1 ! X0 ! . .!.Xk ! Ck ! *
can be viewed as a tower of (co)fibrations for Ck which we can use to compute
ss0(F (Ck, Xk+2)). In fact, there is an Adams type spectral sequence associated*
* to
this tower which has the form
Ep,q1=) ssq-p(F (Ck, Xk+2))
with ae
ssq(F (Xk-p, Xk+2))if0 p k
Ep,q1~=
0 ifp > k .
We will use this spectral sequence to show that
[Ck, Xk+2] ~=Ker([Xk, Xk+2] ! [Xk-1, Xk+2])
thus finishing off the proof. We note that in terms of the spectral sequence t*
*his
claim says that ss0(F (Ck, Xk+2)) is isomorphic to the kernel of the differenti*
*al d1 :
E0,01! E1,01. It is therefore enough to show that Eq,q2= 0 = Eq+1,q2for q > 0.
Now Proposition 2 (including the remark on the Mittag Leffler condition followi*
*ng
it) together with Lemma 23 implies already Eq,q1= 0 = Eq+1,q1if q > 0 and q is
even.
W 2
Now let q > 0 be odd. We let Yk+2 := (i1,...,ik+2) 2p n(i1+...+ik+2)En so th*
*at
Xk+2 = (Yk+2)hF. We claim that for p = q k and p = q + 1 k there are natural
isomorphisms
Ep,q1~=ssq(F (Xk-p, (Yk+2)hF)) ~=ssq(F (Xk-p, Yk+2)hF)
~=Hs(q),s(q)+q(F, Hom (En)*[[Gn]](En)*(Xk-p), (En)*(Yk+2)
~=Hs(q),s(q)+qF, Hom (En)*Hom cts(Pk-p "GnN, (En)*), ss*(Yk+2)
where s(q) is as in Lemma 23. If we accept these isomorphism for the moment
then we can finish off the proof because Po "GnNis F(Gn)-exact and therefore
Hom cts(Pk-o "GnN, (En)*) is a split exact complex of Zp[F ]-modules which in t*
*urn
implies that Ep,q2= 0 if p > 0.
It remains to justify the chain of isomorphisms which identify Ep,q1. The first*
* two
of the claimed isomorphisms are obvious and the last one holds because by 1.3.2
(En)*(Yr) is a coinduced module, i.e.
(En)*(Yr) ~=Hom (En)*[[Gn]](Zp[[Gn]], ss*(Yr)) .
38 HANS-WERNER HENN
To get the third isomorphism it suffices to show that the homotopy fixed point
spectral sequence gives, for q odd and 0 < q k < n, an isomorphism
hF
ssq(F ((EhFn, En)hF) ~=Hs(q),s(q)+qF, Hom (En)*[[Gn]](En)*(En ), (En)*(En) .
In fact, Proposition 3 allows us to identify Hom (En)*[[Gn]](En)*(EhKn), (En)*(*
*En)
with ss*(F (EhKn, En)) whenever K is a closed subgroup of Gn. Now we replace fi*
*rst
EhFnin the source by EhUnwhere U is an open subgroup of Gn. Then Proposition
2 together with Lemma 23 give the required identification. Finally we write EhF*
*n'
LK(n)hocolimj(En)hUj and pass to the limit once more using the remark on the
Mittag-Leffler condition following Proposition 2.
3.6.5. The resolution of LK(n)S0.
It remains to construct a resolution of LK(n)S0. For this we start with an F(Gn*
*)-
resolution
Qo : 0 ! Qm ! . .!.Q0 ! Q-1 = Zp ! 0
of finite length m of the trivial Zp[[Gn]]-module Zp as given by Proposition 17*
* and
Proposition 21. The modules Qr in this resolution are finitely generated projec*
*tives
if r > n, and of the form Q0r Pr "GnNwith Q0rfinitely generated projective and*
* Pr
as in Proposition 21, if r n. We can realize these modules by spectra Zr which
are summands in a finite wedge of En's if r > n, and of the form Z0r_ Xr with Z*
*0r
a summand in a finite wedge of En's and Xr as in Theorem 25. Then the strategy
of the proof of Theorem 25 can be applied to this situation. The Zp[[Gn]]-line*
*ar
maps Qr ! Qr-1 can be uniquely realized by maps Zr-1 ! Zr of spectra and the
resulting sequence of spectra is a resolution of finite length in the sense of *
*3.3.1.
In fact, the factorizations Zr ! Cr ! Zr+1, 0 r n, can be constructed just
as in the proof of Theorem 25 by using again that Qo is F(Gn)-exact. For r > n
ordinary exactness of Qo implies, as in the proof of Theorem 4, that [Zo, Zr] is
exact. This allows to contruct the factorizations for r > n and therefore we ob*
*tain
the following result.
Theorem 26. Let p be odd and n = p - 1. Suppose Qo is an F(Gn)-resolution
of length m of the trivial Zp[[Gn]]-module Zp such that Qr is a finitely genera*
*ted
projective Zp[[Gn]]-module if r > n while Qr ~= Q0r Pr "GnN with Q0rfinitely
generated projective and Pr as in Theorem 25 resp. Proposition 21 if 0 r n.
Then there is a resolution of length m (in the sense of 3.3.1)
Zo : * ! LK(n)S0 := Z-1 ! Z0 ! . .!.Zn ! . .!.Zm ! *
such that the complex E*(Zo) is isomorphic as a complex of Morava modules to the
complex Hom cts(Qo, (En)*).
Furthermore, Zr is a direct summand in a finite wedge of En's if r > n while for
0 r n ` 2
Zr ' Z0r_ 2p n(i1+...+ir)EhFn
(i1,...,ir)
where the wedge is taken over all sequences (i1, . .,.ir) with n-1 i1 > i2 > *
*. .>.
ir 0 and Z0ris a direct summand in a finite wedge of En's.
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 39
3.6.6. Proof of Theorem 10 and Theorem 11.
We note that for p = 3 and n = 2 the group F of Theorem 26 (see also 3.6.4) is
equal to the group G24 of Theorem 10. Likewise, the character O(1) of Propositi*
*on
21 is equal to the character eOof G24. By Theorem 26 and the splitting on En
discussed in the appendix below it is therefore enough to prove Theorem 10 and
for this we only need to show that the kernel K of the augmentation Z3 "G2N! Z3
admits a projective resolution of the form
1 G1 G1 G1
0 ! Z3 "G2SD16! (Z3 ~2)S"2D16! (~2 O)S"2D16! O "S2D16! K ! 0 .
In fact, because of the isomorphism G2 ~=G12x Z3 (cf.1 [GHMR1]) it is enough
to show that the kernel K1 of the augmentation Z3 "G2N1! Z3 with N1 = NG12(E)
admits a projective resolution of the form
1 G1 G1
0 ! Z3 "G2SD16! ~2 "2SD16! O "S2D16! K1 ! 0 .
From [He] we know that ExtiZ3[[S12]](K1, F3) can be identified with the cokerne*
*l of
the map
Y1
H*(S12; F3) ! H*(!iCS12(E)!-i; F3)
i=0
given by the inclusions !iCS12(E)!-i ! S12, i = 0, 1. Furthermore, this map is
Z3[SD16]-linear and from the explicit description of this map in Theorem 4.4 of
[GHMR1] it is straightforward to see that, as Z3[SD16]-modules, we obtain
8
>>>O Z3 F3 ifi = 0
< ~2 F3 ifi = 1
ExtiZ3[[S12]](K1, F3) ~=> Z3
>>:Z3 Z3 F3 ifi = 2
0 ifi > 2 .
The resolution of K1 is then constructed as in the proof of Theorem 6.
Appendix: Splitting En with respect to Fn(q-1)
A.1. Let p be an odd prime, q = pn and F := Fn(q-1)= Cq-1 o Gal(Fq=Fp) be as
in section 2.3. The homotopy groups of En can be described as
ss*(En) ~=WFq[v1, . .,.vn-1][u 1]b ,
with vi:= uiu1-pi, i.e. ui= viupi-1, and where b means completion with respect
to the ideal (u1, . .,.un-1). The elements viare invariant with respect to the *
*action
of F and the action of a generator ! of the cyclic subgroup Cq-1 on u is given *
*by
multiplication with !, i.e. !*(u) = !u (cf. formula (8) in the proof of Lemma 2*
*2),
while the Galois group acts on the coefficients WFq only via Frobenius.
40 HANS-WERNER HENN
In particular, the homotopy groups of the homotopy fixed points are given by the
appropriate completion
ss*(EhFn) ~=Zp[v1, . .,.vn-1][vn1]b ,
with vn = u-q. Furthermore we have an isomorphism of ss*(EhFn)[F ]-modules
q-1M
(15) ss*(En) ~= ss*(EhFn) WFqui
i=0
and the action on the modules ~i:= WFquiis as described above. We are interested
in the decomposition of En obtained from a splitting of the group algebra Zp[F *
*] as
a module over itself. The decomposition suggested by (15) is close but not quite
equal to the decomposition obtained from such a splitting.
A.2. From now on we will restrict attention to the case n = 2 and p odd which we
have used in section 2.3, section 3.3 and 3.3.6.
If i is divisible by p + 1 then !i belongs to Zp and therefore the action of ! *
*on
~i commutes with the action of Frobenius. Therefore, for such i, ~i splits into*
* 2
one-dimensional pieces ~i,+and ~i,-, with Frobenius acting trivially on ~i,+and
by multiplication by -1 on ~i,-.
We claim that there is a direct sum decomposition of Zp[F ] into a direct sum of
Zp[F ]-modules
M M
(16) Zp[F ] ~= ~i ~i,+ ~i,- .
i6 0(p+1) i 0(p+1)
We leave it to the reader to check that the modules on the right hand side of t*
*his
isomorphism are all indecomposable and that the only repetition in the decompo-
sition comes from the isomorphisms ~i~= ~pi, i 6 0 mod p + 1, which are induced
by Frobenius.
At the request of the referee we outline a direct construction of the decomposi*
*tion
given in (16): our identification of the roots of unity of WFq with Cq-1 specif*
*ies a
character O1 of Cq-1 defined over WFq. Let Oi= O1 i. Then the elements
X
ei= __1___(q - 1)Oi(g-1)g
g2Cq-1
belong to WFq[Cq-1] and they are easily checked to be orthogonal idempotents
which sum up to the element 1 2 WFq[Cq-1]. The elements ei form a basis of
WFq[Cq-1] as a WFq-module and each eigenerates a one-dimensional representation
over WFq on which Cq-1 acts via Oi.
For i 0 mod (p + 1) the element ei lives already in Zp[Cq-1], while for i 6
0 mod (p + 1) the elements ei+ epi and ! p+1_2(ei- epi) belong to Zp[Cq-1] and
together they form a basis of Zp[Cq-1] as a Zp-module. Furthermore, the Zp-modu*
*le
generated by ei, i 0 mod (p + 1), is a Zp[Cq-1]-module on which Cq-1 acts via
ON FINITE RESOLUTIONS OF K(n) - LOCAL SPHERES 41
Oi, and the Zp-submodule generated by ffii := ei+ epiand "i := ! p+1_2(ei- epi)*
* is
also a Zp[Cq-1]-module which is isomorphic to ~i restricted to Cq-1.
Now consider the isomorphism of Zp[Cq-1]-modules
Zp[F ] ~=Zp[Cq-1] Zp[Cq-1]oe
where oe is Frobenius considered as an element of F . Then there is a Zp-basis *
*of
Zp[F ] given by ei oeei, if i 0 mod (p+1), and ffii ffiioe, "i "ioe if i 6 0*
* mod (p+1).
In WFq[F ] we have oeei = epioe, in particular oeei = eioe if i 0 mod (p + 1)*
*, and
therefore oe(ei eioe) = (ei eioe) ifi 0 mod (p + 1) i.e. (ei eioe) genera*
*tes a
direct summand isomorphic to ~i, . Furthermore we have oeffii = ffiioe, oe"i = *
*-"ioe
which shows that the Zp-submodule generated by ffii+ ffiioe and "i+ "ioe is a Z*
*p[F ]-
module, and likewise the Zp-submodule generated by ffii- ffiioe and "i- "ioe is*
* a
Zp[F ]-module. Both of these modules are isomorphic to ~i (or ~pi).
A.3. Consequently, by the elementary theory of stable splittings we find that E2
splits, with respect to the F - action, into a direct sum of EhF2-module spectr*
*a whose
homotopy groups are given by
q-1M
Hom Zp[F](~i, , WFquj) ss*(EhF2) ~=(WFqui) ss*(EhF2)
j=0
resp.
q-1M
Hom Zp[F](~i, WFquj) ss*(EhF2) ~= Zpui Zpupi ss*(EhF2) .
j=0
where (WFqui) is the -eigenspace of the action of Frobenius on WFqui. In the
first case, the corresponding summand of E2 can be identified as EhF2-module sp*
*ec-
trum with 2iEhF2, in the second case with 2iEhF2_ 2piEhF2and the multiplicity
of this spectrum in a splitting of E2 (constructed via the action of F ) is 2.
References
[B] A.K. Bousfield, The localization of spectra with respect to homology, T*
*opology 18
(1979), 257-281.
[DH1] E. Devinatz and M. Hopkins, The action of the Morava stabilizer group o*
*n the Lubin-
Tate moduli space of lifts, Amer. J. Math. 117 (1995), 669-710.
[DH2] E. Devinatz and M. Hopkins, Homotopy fixed point spectra for closed sub*
*groups of the
Morava stabilizer groups, Topology 43 (2004), 1-48.
[DW] W.G. Dwyer and C.W. Wilkerson, Smith theory and the functor T, Comment.*
* Math.
Helvetici 66 (1991), 1-17.
[EM] S. Eilenberg and J.C. Moore, Foundations of relative homological algebr*
*a, Memoirs of
the Amer. Math. Soc. 55 (1965).
[EKMM] A.D. Elmendorf, I. Kriz, M.A. Mandell and J.P. May, Rings, modules and *
*algebras in
stable homotopy theory, Amer. Math. Soc. Surveys and Monographs 47 (199*
*6).
[GHM] P. Goerss, H.-W. Henn and M. Mahowald, The homotopy of L2V (1) for the *
*prime 3,
Categorical Decomposition Techniques in Algebraic Topology. Progress in*
* Mathematics
215 (2003).
42 HANS-WERNER HENN
[GHMR1] P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, A resolution of the K(2*
*)-local
sphere, to appear in Annals of Mathematics, Preprint available at http:*
*//hopf.math.
purdue.edu/.
[GHMR2] P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, in preparation.
[Ha] M. Hazewinkel, Formal groups and applications, Academic Press, 1978.
[He] H.-W. Henn, Centralizers of elementary abelian p-subgroups and mod-p co*
*homology of
profinite groups, Duke Math. J. 91 (1998), 561-585.
[HMS] M. Hopkins, M. Mahowald and H. Sadofsky, Constructions of elements in P*
*icard
groups, Topology and Representation Theory, Contemp. Math. 158 (1994), *
*89-126.
[Ho] M. Hovey, Bousfield localization functors and Hopkins' chromatic splitt*
*ing conjecture,
The ~Cech centennial (Boston, MA, 1993), Contemp. Math. 181 (1995), 225*
*-250.
[HS] M. Hovey and N. Strickland, Morava K-theories and Localisation, vol. 66*
*6, Memoirs
of the American Mathematical Society, 1999.
[La] M. Lazard, Groupes p-adiques analytiques, Inst. Hautes Etudes Sci. Publ*
*. Math. 26
(1965), 389-603.
[LT] J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups*
*, Bull. Soc.
Math. France 94 (1966), 49-60.
[Mi] H. Miller, On relations between Adams spectral sequences, with an appli*
*cation to the
stable homotopy of a Moore space, Journal of Pure and Applied Algebra 2*
*0 (1981),
287-312.
[MS] Ph. A. Minh and P. Symonds, Cohomology and finite subgroups of profinit*
*e groups,
Proc. Amer. Math. Soc. 132 (2004), 1581-1588.
[Mo] J. Morava, Noetherian localizations of categories of cobordism comodule*
*s, Annals of
Mathematics 121 (1985), 1-39.
[N] L. Nave, On the nonexistence of Smith-Toda complexes, preprint (http://*
*hopf.math.
purdue.edu).
[Q] D. Quillen, The spectrum of an equivariant cohomology ring I,II, Annals*
* of Mathemat-
ics 94 (1974), 549-572, 573-602.
[Ra1] D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Ac*
*ademic
Press, 1986.
[Ra2] D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. *
*of Math.
Studies, Princeton University Press, 1992.
[RZ] L. Ribes and P. Zaleskii, Profinite Groups, Ergebnisse der Mathematik u*
*nd ihrer Gren-
zgebiete, vol. 40, Springer Verlag, 2000.
[Re] C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algeb*
*raic geom-
etry and group representations. Contemp. Math. 220 (1998), 313-366.
[Se] J.P. Serre, Cohomologie des groupes discrets, Annals of Math. Studies 7*
*0 (1971), 77-
169.
[Sh] K. Shimomura, The homotopy groups of the L2-localized Toda-Smith comple*
*x V (1) at
the prime 3, Trans. Amer. Math. Soc. 349 (1997), 1821-1850.
[ShW] K. Shimomura and X. Wang, The homotopy groups ss*(L2S0) at the prime 3,*
* Topology
41 (2002), 1183-1198.
[SyW] P. Symonds and T. Weigel, Cohomology of p-adic analytic groups, New Hor*
*izons in
pro-p groups, Progress in Math. 184 (2000), 349-410.
[Y] A. Yamaguchi, The structure of the cohomology of Morava stabilizer alge*
*bra S(3),
Osaka Journal Math. 29 (1992), 347-359.
Institut de Recherche Math'ematique Avanc'ee, C.N.R.S. - Universit'e Louis Past*
*eur,
7 rue Ren'e Descartes, F-67084 Strasbourg, France
~~