THE NILPOTENT FILTRATION AND THE ACTION OF
AUTOMORPHISMS ON THE COHOMOLOGY OF FINITE
p-GROUPS
NICHOLAS J. KUHN
Abstract.We study H*(P), the mod p cohomology of a finite p-group P,
viewed as an Fp[Out(P)]-module. In particular, we study the conjecture, *
*first
considered by Martino and Priddy, that, if eS 2 Fp[Out(P)] is a primitive
idempotent associated to an irreducible Fp[Out(P)]-module S, then the Kr*
*ull
dimension of eSH*(P) equals the rank of P. The rank is an upper bound by
Quillen's work, and the conjecture can be viewed as the statement that e*
*very
irreducible Fp[Out(P)]-module occurs as a composition factor in H*(P) wi*
*th
similar frequency.
In summary, our results are as follows. A strong form of the conjectur*
*e is
true when p is odd. The situation is much more complex when p = 2, but is
reduced to a question about 2-central groups (groups in which all elemen*
*ts of
order 2 are central), making it easy to verify the conjecture for many f*
*inite
2-groups, including all groups of order 128, and all groups that can be *
*written
as the product of groups of order 64 or less.
The the odd prime theorem can be deduced using the approach to U, the
category of unstable modules over the Steenrod algebra, initiated by H.-*
*W.
Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 ma*
*ke
heavy use of the nilpotent filtration of U introduced in [S1], as applie*
*d to group
cohomology in [HLS2]. Also featured are unstable algebras of cohomology
primitives associated to central group extensions.
1.Introduction
Fix a prime p, and let H*(P ) denote the mod p cohomology ring of a finite
p-group P . Interpreted topologically, H*(P ) = H*(BP ; Fp), where BP is the
classifying space of P , and interpreted algebraically, H*(P ) = Ext*Fp[P](Fp, *
*Fp).
The automorphism group of P , Aut(P ), acts on P , and thus via ring homo-
morphisms on H*(P ). As the inner automorphism group, Inn(P ), acts trivially
on cohomology, H*(P ) becomes a graded Fp[Out(P )]-module, where Out(P ) =
Aut(P )=Inn(P ) is the outer automorphism group.
The ring H*(P ) is known to be Noetherian, and D. Quillen [Q1 ] computed its
Krull dimension: dim H*(P ) = rk(P ). Here rk(P ) denotes the maximal rank of an
elementary abelian p-subgroup of P .
Now let S be an irreducible Fp[Out(P )]-module and eS 2 Fp[Out(P )] an asso-
ciated primitive idempotent. Then eSH*(P ) is a finitely generated module over
the Noetherian ring H*(P )Out(P)(see x4.3), and thus has a Krull dimension with
evident upper bound rk(P ). In the early 1990's, J. Martino and S. Priddy [MP ]
____________
Date: May 3, 2006.
2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40, 20D*
*15.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
asked whether the following conjecture might be true.
Conjecture A dimeSH*(P ) = rk(P ) for all pairs (P, S).
As dimeSH*(P ) determines the growth of the Poincar'e series of eSH*(P ), and
this Poincar'e series gives a count of the occurrences of S as a composition fa*
*ctor in
H*(P ), the conjecture is roughly the statement that every irreducible Fp[Out(P*
* )]-
module occurs in H*(P ) with similar frequency.
The topological interpretation of H*(P ) makes it evident that H*(P ) is an o*
*bject
in K and U, the categories of unstable algebras and modules over the mod p Stee*
*nrod
algebra A, and in this paper, we study Conjecture A using modern `U-technology'.
In summary, our results are as follows. A strong form of Conjecture A is true
when p is odd. The situation is more complex when p = 2. We reduce the conjectu*
*re
to a different conjecture about 2-central groups, making it easy to verify Conj*
*ecture
A for many finite 2-groups, including all with order dividing 128, and all grou*
*ps
that can be written as the product of groups of order 64 or less.
The the odd prime theorem can be deduced using the functor category approach
to U initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1 ]. The reduct*
*ions
when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], *
*as
applied to group cohomology in [HLS2 ].
Featured here, and in a companion paper [K5 ], are algebras of cohomology pri*
*m-
itives associated to central group extensions, and p-central groups: groups in *
*which
every element of order p is central.
2.Main results
2.1. Notation. Before describing our results in more detail, we introduce some
notation. Throughout, V < P will denote an elementary abelian p-subgroup of P
and C = C(P ) < P will be the maximal central elementary abelian p-subgroup.
We let c(P ) be the rank of C, depthH*(P)H*(P ) be the depth of H*(P ) with
respect to the ideal of positive degree elements, and mrk(P ) be the minimal ra*
*nk
of a maximal V < P . These are related by
c(P ) depthH*(P)H*(P ) mrk(P ) rk(P ),
where the first inequality is due to J.Duflot [D ], and the second follows from
Quillen's work and standard commutative algebra (see x4.6).
Given V < P , we let Aut0(P, V ) C Aut(P, V ) < Aut(P ) be defined by
Aut(P, V ) = {ff 2 Aut(P ) | ff(V ) = V } and
Aut0(P, V ) = {ff 2 Aut(P ) | ff(v) = v for allv 2 V },
Then let Out0(P, V ) and Out(P, V ) be the projection of these groups in Out(P *
*).
Note that Out(P, V ) = Aut(P, V )=NP (V ) and Out0(P, V ) = Aut0(P, V )=CP (V ),
so that there is an extension of groups
WP (V ) ! Aut(P, V )=Aut0(P, V ) ! Out(P, V )=Out0(P, V ),
where WP (V ) = NP (V )=CP (V ).
Finally, if S is an irreducible Fp[G]-module, and M is an arbritrary Fp[G]-
module, we say that S occurs in M if it is a composition factor.
GROUP AUTOMORPHISMS 3
2.2. Quillen's approximation. Let A(P ) denote Quillen's category, having as
objects the elementary abelian p-subgroups V < P , and as morphisms the homo-
morphisms generated by inclusions and conjugation by elements in P . Let H*(P )
denote H*(P ) = lim H*(V ). It is not hard to see the same inverse limit is att*
*ained
A(P)
by using the smaller category AC (P ), where AC (P ) A(P ) is the full subcat*
*egory
consisting of elementary abelian V < P containing the maximal central elementary
abelian C.
The restriction maps associated to the inclusions V < P assemble to define a
natural map of unstable A-algebras
H*(P ) ! H*(P ),
which Quillen shows is an F -isomorphism. As a special case of Proposition 2.6
below, we have
dimeSH*(P ) dimeSH*(P ),
for all irreducible Fp[Out(P )]-modules S.
Theorem 2.1. Let S be an irreducible Fp[Out(P )]-module. Then eSH*(P ) 6= 0 if
and only if S occurs in Fp[Out(P )=Out0(P, V )] for some maximal V < P . In that
case, dimeSH*(P ) = max{rk(V ) | S occurs inFp[Out(P )=Out0(P, V )]}.
This implies an observation noted earlier by Martino and Priddy [MP , Prop.4.*
*2].
Corollary 2.2. If eSH*(P ) 6= 0, then dimeSH*(P ) mrk(P ).
Given H < G, a standard argument shows that if H is a p-group, then every irr*
*e-
ducible Fp[G]-module occurs as a submodule of Fp[G=H]. Thus the theorem implies
that if Out0(P, V ) is a p-group, then dimeSH*(P ) rk(V ) for all irreducible*
*s S.
The converse will be true if P has a unique maximal elementary abelian subgroup,
for example, if P is p-central: see Corollary 6.3.
One can now make effective use of group theory results about p0-automorphisms
of p-groups, particularly the Thompson A x B lemma [Gor, Thm.5.3.4].
Theorem 2.3. Let P a finite p-group. If V < P is an elementary abelian subgroup,
then Out0(P, V ) will be a p-group if Out0(CP (V ), V ) is a p-group. If V is*
* also
maximal, and p is odd, this will always be the case. Thus if p is odd, dimeSH*(*
*P ) =
rk(P ) for all irreducible Fp[Out(P )]-modules S.
For 2-groups, one does not need to look far to see that story is quite differ*
*ent.
The quaternionic group of order 8, Q8, is 2-central with center C of rank 1. As
Aut(Q8) clearly must act trivially on the center, Out0(Q8, C) = Out(Q8) ' 3,
which is not a 2-group.
This example generalizes: it is the case t = 1 in the following.
Example 2.4. Let Gt be the 2-Sylow subgroup of SU3(F2t). This group is 2-
central with center Ct' F2t, and Gt=Ct' F22t. The group Out(Gt, Ct) contains a
cyclic subgroup of order 2t+ 1. Thus there is at least one irreducible F2[Out(G*
*t)]-
module S with eSH*(Gt) = 0.
The next proposition, an easy application of properties of Lannes TV -functor,
shows that Theorem 2.3 can be applied to 2-groups with reduced cohomology, i.e.,
groups P for which H*(P ) ! H*(P ) is monic. This includes groups built up by
iterated products and wreath products of Z=2's.
4 KUHN
Proposition 2.5. If H*(P ) is reduced, and V < P is a maximal elementary abelian
subgroup, then CP (V ) = V , and thus Out0(CP (V ), V ) is the trivial group.
To say more about dimeSH*(P ), we now begin to take a deeper look at H*(P )
using the nilpotent filtration.
2.3. A stratification of the problem. In x4, we study functors ~Rd: U ! U. In
brief, they are defined as follows. An unstable module M has a natural nilpotent
filtration
. . .nil2M nil1M nil0M = M,
and nildM=nild+1M = dRdM, where RdM is a reduced unstable module. Then
~RdM is defined as the nilclosure of RdM.
Henn [H ] has shown that the nilpotent filtration of a Noetherian unstable al*
*ge-
bra is finite, so only finitely many of the the modules ~RdH*(P ) will be nonze*
*ro.
Furthemore, ~R0H*(P ) = H*(P ), and each ~RdH*(P ) is a finitely generated H*(P*
* )-
module. The study of dimeSH*(P ) stratifies as follows.
Proposition 2.6. Let S be an irreducible F[Out(P )]-module. Then
dimeSH*(P ) = maxd{dim eSR~dH*(P )}.
This will be a special case of a more general statement about K: see Proposi-
tion 4.9.
Martino and Priddy observe [MP , Prop.4.1] that the Depth Conjecture, subse-
quently proved by D. Bourguiba and S. Zarati [BoZ ], implies that dimeSH*(P )
depthH*(P)H*(P ). We prove a stratified variant of this, and note that an analo*
*gue
of Duflot's theorem holds.
Proposition 2.7. If eSR~dH*(P ) 6= 0, then
dim eSR~dH*(P ) depthH*(P)~RdH*(P ) c(P ).
That dimeSR~dH*(P ) c(P ) is also a corollary of part (c) of Theorem 2.8 be*
*low.
2.4. Computing dim eSR~dH*(P ). In [K5 ], the work of [HLS2 ] will allow us to
write down useful formulae for ~RdH*(P ) analogous to
R~0H*(P ) = lim H*(V ).
V 2AC(P)
Using these, we can generalize Theorem 2.1 to a statement about dimeSR~dH*(P )
for all d.
To state this, we need to define the primitives associated to a central exten*
*sion.
The cohomology of an elementary abelian p-group is a Hopf algebra. If Q is a
finite group, and V < Q is a central elementary abelian p-subgroup, then multip*
*li-
cation m : V x Q ! Q is a homomorphism, and the induced map in cohomology,
m* : H*(Q) ! H*(V x Q) = H*(V ) H*(Q) makes H*(Q) into an H*(V )-
comodule. We then let PV H*(Q) denote the primitives:
PV H*(Q) = {x 2 H*(Q) | m*(x) = 1 x}
m* *
= Eq {H*(Q) -!-!H (V x Q)},
ss*
where ss : V x Q ! Q is the projection.
GROUP AUTOMORPHISMS 5
As equalizers of algebra maps are algebras, PV H*(Q) is again an unstable A-
algebra. Note also that PV H0(Q) = Fp. It is not hard to check that PV H1(Q) '
H1(Q=V ) = Hom (Q=V, Fp), via the inflation map. The reader may find it illumi-
nating to know that PV H*(Q) is again Noetherian, and has Krull dimension equal
to rk(Q) - rk(V ): see [K4 , K5].
Our theorem about dimeSR~dH*(P ) now goes as follows.
Theorem 2.8. Let S be an irreducible Fp[Out(P )]-module. For each d, there exis*
*ts
a set supd(S) obAC (P ), the d-support of S, with the following properties.
(a) supd(S) is a union of Aut(P )-orbits in ob AC (P ).
(b) Suppose that V1 < V2 < P , and PV1Hd(CP (V1)) ! PV1Hd(CP (V2)) is monic.
Then V1 2 supd(S) implies that V2 2 supd(S).
(c) eSR~dH*(P ) 6= 0 if and only if supd(S) 6= ;, and, in this case,
dimeSR~dH*(P ) = max{rk(V ) | V 2 supd(S)}.
(d) Let V 2 obAC (P ) be maximal. Then V 2 supd(S) if and only if S occurs in
IndOut(P)Out(P,V()[Fp[Aut(P, V )=Aut0(P, V )] PV Hd(CP (V ))]WP(V )).
When d = 0, we recover Theorem 2.1. In this case, the hypothesis in part
(b) always holds. Thus maximal elements in sup0(S) will always be maximal in
ob AC (P ). Thus part (d) implies that V will be a maximal element in sup0(S) if
and only if S occurs in
IndOut(P)Out(P,V()Fp[Aut(P, V )=Aut0(P, V )]WP(V ))
which rewrites as
Fp[Out(P, V )=Out0(P, V )].
Thus part (c) implies the calculation of dimeSH*(P ) in Theorem 2.1.
Theorem 2.8 now lets us recast Conjecture A as follows.
Corollary 2.9. Conjecture A is true for a pair (P, S) if and only if there exis*
*ts a
V < P of maximal rank such that S occurs in
IndOut(P)Out(P,V()[Fp[Aut(P, V )=Aut0(P, V )] PV H*(CP (V ))]WP(V )).
Starting from this, it is not hard to show (see Lemma 8.2) that Conjecture A *
*for
a 2-group P would be implied by
Conjecture B Let P be a finite 2-group. If V < P is a maximal elemen-
tary abelian subgroup, then every irreducible F2[Out0(P, V )]-module occurs in
PV H*(CP (V ))WP(V ).
Note that the subgroups CP (V ) that arise here are 2-central. Our general th*
*eory
shows
Proposition 2.10. If Q is 2-central with maximal elementary abelian subgroup
C, then Conjecture B is true for Q, i.e. every irreducible F2[Out0(Q, C)]-module
occurs as a composition factor in PC H*(Q).
6 KUHN
We conjecture a subtle strengthening of this.
Conjecture C If Q is 2-central with maximal elementary abelian subgroup C,
then every irreducible F2[Out0(Q, C)]-module occurs as a submodule of PC H*(Q).
Again using the Thompson A x B lemma, we have
Theorem 2.11. Conjecture B is true for the pair (P, V ) if the 2-central group
CP (V ) satisfies Conjecture C. Thus if P is a 2-group such that CP (V ) satisf*
*ies
Conjecture C for some V < P of maximal rank, then dimeSH*(P ) = rk(P ) for all
irreducible F2[Out(P )]-modules S.
Though Out0(Q, C) need not be a 2-group if Q is 2-central, its 20part tends to
be very small, even in cases when Out(Q) is quite complicated. This makes it not
hard to check the following theorem, using information about 2-groups of order
dividing 64 available at the website [Ca ], or, in book form, [CTVZ ].
Theorem 2.12. Conjecture C is true for all 2-central groups that can be written
as the product of groups of order dividing 64.
A counterexample to Conjecture A would have to be a 2-group P that is not
2-central, and having the property that the proper subgroups CP (V ), with V of
maximal rank, are all counterexamples to Conjecture C. Thus the last theorem
implies
Corollary 2.13. Conjecture A is true for all 2-groups of order dividing 128, and
all 2-groups that can be written as the product of groups of order 64 or less.
We end this section with an example1 that illustrates the sorts of patterns t*
*hat
the numbers dimeSR~dH*(P ) can take, when P is not p-central.
Example 2.14. Let P be the group of order 64 number #108 on the Carlson
group cohomology website. From the information there one learns that Out(P ) has
order 3 . 28, c(P ) = depthH*(P)H*(P ) = 2, while mrk(P ) = rk(P ) = 3. One can
also deduce that F2[Out(P )] has precisely two irreducibles - the one dimension*
*al
trivial module `1' and a two dimensional module `S'. We compute the following
table of nonzero dimensions, where ; denotes that the corresponding summand of
~Rd(H*(P )) is 0.
__________________________________
||_d||dime1R~dH*(P|)d|imeSR~dH*(P_|) |
|_0_|_____3_______|_______;_______|
|_1_|_____2_______|_______3_______|
|_2_|_____2_______|_______3_______|
|_3_|_____3_______|_______2_______|
|_4_|_____2_______|_______2_______|
|_5_|_____2_______|_______2_______|
|_6_|_____2_______|_______2_______|
|_7_|_____2_______|_______;_______|
____________
1Checking Conjecture A for this group led us to the formulation and proof of*
* Theorem 2.8.
GROUP AUTOMORPHISMS 7
Note that this example shows that, when d > 0, eSR~dH*(P ) 6= 0 does not imply
that dim sSR~dH*(P ) mrk (P ), in constrast to Corollary 2.2. Indeed, for th*
*is
group, dimR~dH*(P ) = 2 < 3 = mrk(P ) for d = 4, 5, 6, 7.
2.5. Organization of the paper. The rest of the paper is organized as follows.
In x3, we quickly survey previous results related to Conjecture A. The nilpotent
filtration is reviewed in x4, which has various relevant general results about *
*U and
K concerning both dimension and depth. In x5, we give a first proof of Theorem *
*2.1
using the functor category description of U=Nil introduced in [HLS1 ]. In x6, *
*we
then collect various results about p0-automorphisms of p-groups, and prove Theo-
rem 2.3 and Proposition 2.5. In this section we also discuss the family of 2-ce*
*ntral
groups given in Example 2.4. In x7, we use a convenient formula for R~dH*(P )
from [K5 ] to prove Theorem 2.8. A short proof of Proposition 2.10 also appears
there. That Conjecture C implies Conjecture A is shown in the short x8, and in *
*x9
various 2-central groups are shown to satisfy Conjecture C, including those lis*
*ted
in Theorem 2.12. Finally, in x10, we will discuss Example 2.14 in detail.
The author wishes to thank the Cambridge University Pure Mathematics De-
partment for its hospitality during a visit during which a good part of this re*
*search
was done. Information compiled by Ryan Higginbottom has been very useful, and
was used by him in [Hi] to verify Conjecture A for all groups of order dividing
64 except for group 64#108 . Example 2.4 arose from a conversation with David
Green.
3.Previous results
In the 1984 paper [DS ], T. Diethelm and U. Stammbach gave a group theoretic
proof that, if P is a finite p-group, every irreducible Fp[Out(P )]-module occu*
*rs
as a composition factor in H*(P ). Independently and concurrently, it was noted
explicitly in [HK ], and implicitly in [N ], that this same result was a conseq*
*uence of
the proof of the Segal Conjecture in stable homotopy theory. From both [DS ] and
[HK ], one can conclude that every irreducible occurs an infinite number of tim*
*es,
so that dimeSH*(P ) 1 for all irreducibles S.
The 1999 paper of P. Symonds [Sy] gives a second group theoretic proof that
every irreducible Fp[Out(P )]-module occurs in H*(P ), and inspection of his pr*
*oof
also yields Martino and Priddy's lower bound dimeSH*(P ) c(P ).2
A proof of Conjecture A for an elementary abelian p-group V serves as a start*
*ing
point for work on the general question, and goes as follows.
Let S*(V ) H2*(V ) denote the symmetric algebra generated by fi(H1(V ))
H2(V ), where fi is the Bockstein. Thus S*(V ) is a polynomial algebra on r = r*
*k(V )
generators in degree 2. It is a classic result of Dickson [D , Cr] that the inv*
*ariant
ring S*(V )GL(V )is again polynomial on homogeneous generators cV (1), . .,.cV *
*(r),
where cV (i) has degree 2(pr - pr-i-1).
For any idempotent e 2 Fp[GL(V )], eH*(V ) is an S*(V )GL(V )-module via
left multiplication. Now one observes that S*(V ) (and thus H*(V )) is a free
____________
2[Sy] was written with knowledge of [HK ], but apparently not of [DS].
8 KUHN
S*(V )GL(V )-module, and that every finite Fp[GL(V )]-module occurs as a sub-
module of S*(V ) [Alp, p.45]. These facts imply that, for all irreducible Fp[GL*
*(V )]-
modules S, eSH*(V ) is a nonzero finitely generated free S*(V )GL(V )-module, a*
*nd
thus dimeSH*(V ) = rk(V ).
Using [HK ], this result immediately extends to all abelian p-groups.
In [MP ], Martino and Priddy use the analogous action of H*(P )Out(P)on eH*(P*
* )
to show that, for all nonzero idempotents e, dim eH*(P ) depthH*(P)H*(P ).
Their proof is critically dependent on an unpublished paper that was later with-
drawn. However, as we note in the next section, their argument can be salvaged *
*by
using the later work of D. Bourguiba and S. Zarati [BoZ ] utilizing deep proper*
*ties
of U and K.
Remark 3.1. Readers of [MP ] will know that much of Martino and Priddy's paper
concerns a refined version of Conjecture A. The double Burnside ring A(P, P ) a*
*cts
on H*(P ), and one can conjecture that dim eH*(P ) = rk(P ) for all idempotents
e 2 A(P, P ) Fp that project to a nonzero element under the retraction of alg*
*ebras
A(P, P ) Fp ! Fp[Out(P )]. The paper [HK ] notes that one can deduce that
eH*(P ) 6= 0, and thus that dim eH*(P ) 1, from the Segal conjecture; there is
currently no `group theoretic' proof of this fact. Partly for this reason, we *
*have
focused on the Out(P ) conjecture, though some of our general theory evidently
applies to the A(P, P ) version.3
4. Krull dimension and the nilpotent filtration of U
4.1. The nilclosure functor. Let N il1 U be the localizing subcategory gener-
ated by suspensions of unstable A-modules, i.e. N il1 is the smallest full subc*
*ate-
gory containing all suspensions of unstable modules that is closed under extens*
*ions
and filtered colimits.
An unstable module M is called nilreduced (or just reduced) if it contains no*
* such
nilpotent submodules, or, equivalently, if Hom U(N, M) = 0 for all N 2 N il1. M*
* is
called nilclosed if also Ext1U(N, M) = 0 for all N 2 N il1.
Let L0 : U ! U be localization away from N il1. Thus L0M is nilclosed, and
there is a natural transformation M ! L0M with both kernel and cokernel in N il*
*1.
Recall that TV : U ! U is defined to be left adjoint to H*(V ) ___. The var*
*ious
marvelous properties of TV are reflected in similar properties of L0.
Proposition 4.1. The functor L0 : U ! U satisfies the following properties.
(a) There are natural isomorphisms L0(M N) ' L0M L0N.
(b) There are natural isomorphisms TV L0M ' L0TV M.
(c) If K 2 K, then L0K 2 K, and K ! L0K is a map of unstable algebras.
One approach to properties (a) and (b) is to use Proposition 5.1. See [HLS2 ,
I.4.2] and [BrZ1 ] for more detail about property (c).
Given a Noetherian unstable algebra K 2 K, we recall the category Kf.g.- U
as studied in [HLS2 , I.4]. The objects are finitely generated K-modules M whose
____________
3Our results do imply the refined conjecture for p-central groups at odd pri*
*mes. See Re-
mark 6.5.
GROUP AUTOMORPHISMS 9
K-module structure map K M ! M is in U, and morphisms are K-module
maps in U.
Proposition 4.2. Let K 2 K be Noetherian, and M 2 Kf.g.- U. Then L0K 2
Kf.g.- U, and thus is Noetherian, and L0M 2 L0Kf.g.- U.
See [HLS2 , I(4.10)].
4.2. The nilpotent filtration. For d 0, let N ild U be the localizing subca*
*te-
gory generated by d-fold suspensions of unstable A-modules. An unstable module
M admits a natural filtration
. . .nil2M nil1M nil0M = M,
where nildM is the largest submodule in N ild.
As observed in [K3 , Prop.2.2], nildM=nild+1M = dRdM, where RdM is a
reduced unstable module. (See also [S2, Lemma 6.1.4].)
Proposition 4.3. The functors Rd : U ! U satisfy the following properties.
(a) There are a natural isomorphisms R*(M N) ' R*M R*N of graded objects
in U.
(b) There are natural isomorphism TV RdM ' RdTV M.
(c) Let K 2 K be Noetherian, and M 2 Kf.g.- U. Then R0K is also a Noetherian
unstable algebra, and RdM 2 R0Kf.g.- U, for all d.
For the first two properties, see [K3 , x3], and the last follows easily from*
* the
first.
Now let ~RdM denote the nilclosure of Rd(M). Thus RdM L0RdM = ~RdM.
We also note that L0M = ~R0M.
The previous propositions combine to prove the following.
Proposition 4.4. The functors ~Rd: U ! U satisfy the following properties.
(a) There are natural isomorphisms ~R*(M N) ' ~R*M ~R*N of graded objects
in U.
(b) There are natural isomorphisms TV ~RdM ' ~RdTV M.
(c) Let K 2 K be Noetherian, and M 2 Kf.g.- U. Then ~R0K is also a Noetherian
unstable algebra, and ~RdM 2 ~R0Kf.g.- U, for all d.
Henn [H ] proved the following important finiteness result.
Proposition 4.5. Let K 2 K be Noetherian, and M 2 Kf.g.- U. Then the
nilpotent filtration of M has finite length. Equivalently, ~RdM = 0 for d >> 0.
4.3. Invariant rings as Noetherian unstable algebras.
Proposition 4.6. Let K 2 K be Noetherian. Given any subgroup G < AutK (K),
the invariant ring KG is again a Noetherian unstable algebra, and K is a finite*
*ly
generated KG -module.
10 KUHN
Proof.Let D(d) 2 K denote the dth Dickson algebra: in the notation of x3, D(d)
is the polynomial algebra S*((Fp)d)GLd(Fp). Then let D(d, j) 2 K denote the
subalgebra consisting of pj powers of the elements of D(d). In [BoZ , Thm.A.1],
Lannes shows that, if d = dimK, then there is an embedding of unstable algebras
D(d, j) ! K that is unique in the sense that any two such embeddings will agree
after restriction to D(d, k) with k large enough. As G is necessarily finite, i*
*t follows
that for large enough j, there is an embedding D(d, j) ! KG such that K is a
finitely generated D(d, j)-module. The proposition follows.
Our various propositions apply to the case when K = H*(P ).
Corollary 4.7. Let P be a finite p-group.
(a) Both H*(P )Out(P)and H*(P )Out(P)are Noetherian.
(b) eSH*(P ) 2 H*(P )Out(P)f.g.- U and eSR~dH*(P ) 2 H*(P )Out(P)f.g.- U, for *
*all
irreducible Fp[Out(P )]-modules S and all d.
4.4. How to deal with odd primes. When p is odd, K 2 K is not necessarily
commutative unless K is concentrated in even degrees. To use standard definitio*
*ns
and results from the commutative algebra literature, and for other technical re*
*asons,
it is useful to have a systematic way of ridding ourselves of this problem.
A standard thing to do, done many times before, and going back at least to
[LZ ], goes as follows. Let U0 denote the full subcategory of U consisting of m*
*odules
concentrated in even degrees. Given M 2 U, we let M0 2 U0 denote the image of M
under the right adjoint of the inclusion U0 U: in more down-to-earth terms, M0
is the largest submodule of M contained in even degrees. It is easy to see that*
* if K
is an unstable algebra, then K0 is a subalgebra. As an example, H*(V )0= S*(V ).
Statements about Kf.g.- U become statements about K0f.g.- U using the next
result.
Proposition 4.8. If K 2 K is Noetherian, so is K0, and K is a finitely generated
K0-module.
This is [BrZ1 , Lemma 5.2].
4.5. Krull dimension. We will generally work with the standard definition of
Krull dimension. Given a commutative graded Noetherian Fp-algebra K, dimK =
d if
"0 . . ."d
is a chain of prime ideals in K of maximal length. If M is a finitely generated
K-module, dim M is then defined to be dim(K=Ann(M)), where Ann(M) is the
annilator ideal of M.
Given a Noetherian K 2 K and M 2 Kf.g.- U, the proposition of the last
subsection shows that M 2 K0f.g.- U, and thus dimM will be a well defined finite
natural number. P
In this situation, the Poincar'e series of M, 1i=0(dimFp Mi)ti, will be a r*
*ational
function, and dimM equals the order of the pole at t = 1. There is a third way *
*of
calculating dimK: it is the number d such that there exist algebraically indepe*
*ndent
elements k1, . .,.kd 2 K with K finitely generated over Fp[k1, . .,.kd]. See [C*
*TVZ ,
x10.2] for a nice discussion of these and related facts.
Proposition 2.6 of the introduction is a special case of the following.
GROUP AUTOMORPHISMS 11
Proposition 4.9. Let K 2 K be Noetherian. Given M 2 Kf.g.- U,
dim M = maxd{dim ~RdM}.
Proof.As the modules dRdM are the composition factors associated to a finite
filtration of M, standard properties of Krull dimension [Mat , (12.D)] imply th*
*at
dimM = maxd{dim RdM}. Recalling that ~Rd= L0Rd, the next proposition finishes
the proof.
Proposition 4.10. Given K and M as above, if M is reduced then
dim M = dimL0M.
This follows from the next two lemmas.
Lemma 4.11. Ann(M) is a sub A-module of K.
Proof.We are claiming that, if kx = 0 for all x 2 M, then, for all a 2 A and
x 2 M, we have (ak)x = 0. Fixing a 2 A, assume byPinduction that (a0k)y = 0 for
all y 2 M and a02 A with |a0| < |a|. With a = a0 a00, we then have
0= a(kx)
X
= (ak)x + (a0k)(a00x)
|a0|<|a|
= (ak)x,
where the last equality uses the inductive hypothesis.
Lemma 4.12. With M reduced, Ann(M) = Ann(L0M).
Proof.
Ann(M) = {k 2 K | Ak M ! M is 0}
= {k 2 K | Ak M ! L0M is 0}
= {k 2 K | Ak L0M ! L0M is 0}
= Ann(L0M).
Here the first and last equalities are consequences of the last lemma. The seco*
*nd is
immediate since M is reduced, i.e. M ! L0M is monic. Finally, the third equality
follows from the universal property of nilclosed modules, as Ak M ! Ak L0M
is a N il1-isomorphism.
4.6. Depth. If M is a K-module, and I K is an ideal, the depth of M with
respect to I is defined to be the maximal length l of an M-regular sequence in
I: r1, . .,.rl 2 I such that for each i between 1 and l, ri is not a zero divis*
*or on
M=(r1, . .,.ri-1)M.
If K is a Noetherian unstable algebra, and M 2 Kf.g.- U, we let depthK M
denote the depth of M with respect to the ideal of positive degree elements in *
*K.
It is standard that depthK M dimN, where N M is any nonzero submod-
ule. This is usually stated in the following equivalent formulation [Mat , Thm.*
*29]:
depthKM dim K=" where " is any associated prime ideal, i.e. a prime ideal
arising as the annihilator of an element of M. The associated primes include all
the minimal primes in the support of M [Mat , Thm.9].
12 KUHN
Quillen [Q2 , Prop.11.2] shows that the minimal primes correspond to the maxi-
mal elementary abelian subgroups, and so depthH*(P)H*(P ) mrk(P ), as asserted
in the introduction.
The work of Bourguiba, Lannes, and Zarati [BoZ ] shows
Proposition 4.13. Let K 2 K be Noetherian, G < AutK (K), and M 2 Kf.g.- U.
Then depthKG M = depthKM.
The point here is that the main theorem of [BoZ ] says that depthK M can be
calculated by using a very specific regular sequence of `generalized Dickson in*
*vari-
ants', and, as discussed in the proof of Proposition 4.6, Lannes' theorem in the
appendix of [BoZ ] says that these elements will be in KG .
Corollary 4.14. Let K 2 K be Noetherian, G < AutK (K), M 2 Kf.g.- U, and
N be a nonzero direct summand of M, viewed as a KG -module. Then dim N
depthKM.
Proof.dim N depthKG N depthKG M = depthKM.
Specializing further, we have
Corollary 4.15. Let P be a finite p-group, and S an irreducible Fp[Out(P )]-
module. If eSH*(P ) 6= 0, then dimeSH*(P ) depthH*(P)H*(P ). If eSR~dH*(P ) 6=
0, then dimeSR~dH*(P ) depthH*(P)~Rd(P ).
Duflot showed [D ] that depthH*(P)H*(P ) rk(C) = c(P ). Subsequent proofs,
beginning with [BrH ], have emphasized that this theorem is a reflection of the
H*(C)-comodule structure on H*(P ) induced by the multiplication homomorphism
C x P ! P . This same homomorphism also induces an H*(C)-comodule structure
on ~R*H*(P ), and the proof of Duflot's theorem given in [CTVZ , Thm.12.3.3] g*
*oes
through without change to prove
Proposition 4.16. depthH*(P)~R*(P ) c(P ).
This proposition and the preceeding corollary imply Proposition 2.7.
5. Proof of Theorem 2.1
In this section, we give a proof of Theorem 2.1 based on the identification of
U=N il as a certain functor category.
Following [HLS1 ], let F be the category of functors from finite dimensional *
*Fp-
vector spaces to Fp-vector spaces. This is an abelian category in the obvious w*
*ay:
F ! G ! H is exact if F (W ) ! G(W ) ! H(W ) is exact for all W .
Let l : U ! F be defined by l(M)(W ) = (TW M)0 = Hom U(M, H*(W ))0. (Here
M0 denotes the continuous dual of a profinite vector space M.) This has right
adjoint r : F ! U given by r(F )d = Hom F(Hd, F ), where H*(W ) is the mod p
homology of the group W .
Proposition 5.1. The functors l and r satisfy the following properties.
(a) l is exact.
(b) The natural transformation M ! r(l(M)) identifies with M ! L0M. In par-
ticular l(M) = 0 if and only if M 2 N il1.
GROUP AUTOMORPHISMS 13
(c) Both l and r commute with tensor products.
For proofs of these properties see [HLS1 , K2, S2].
The proposition implies that a nilclosed module, like H*(P ), is completely d*
*e-
termined as an object in U by its associated functor. It is well known that Qui*
*llen's
work allows for the identification of l(H*(P )) = l(H*(P )). Let Rep (W, P ) =
Hom (W, P )=Inn(P ).
Proposition 5.2. l(H*(P ))(W ) = FRep(W,P)p.
For a simple proof, see [K1 ].
Proof of the first statement of TheoremB2.1.y the discussion above, an irreduci*
*ble
Fp[Out(P )]-module S occurs in H*(P ) if and only if it appears in the permutat*
*ion
module FRep(W,P)pfor some W .
Let ff : W ! P represent an element in Rep(W, P ). The stablizer of the Out(P*
* )-
orbit of this element will be Out0(P, V ) where V = ff(W ). Thus, as an Out(P )-
module, FRep(W,P)pis a direct sum of modules of the form FOut(P)=Out0(P,Vp), and
every such module appears as a summand of FRep(W,P)pif rkW rkV .
Finally, we note that if V1 < V2 < P , then Out0(P, V2) Out0(P, V1), and th*
*us
FOut(P)=Out0(P,V1)pis a quotient of FOut(P)=Out0(P,V2)p. It follows that if S *
*occurs
in FOut(P)=Out0(P,Vp)for some V < P , then S occurs in FOut(P)=Out0(P,Vp)for so*
*me
maximal V < P .
We now finish the proof of Theorem 2.1 by showing that if S occurs in the
permutation module FOut(P)=Out0(P,Vp)then dim eSH*(P ) rkV . We do this by
looking more carefully at FRep(W,P)pas an object in F with Out(P ) action.
We begin by giving an increasing filtration of the set valued bifunctor Rep(W*
*, P ).
Let Repk(W, P ) Rep(W, P ) be the set of elements represented by homomorphisms
with image of rank at most k. The inclusions
{0} = Rep0(W, P ) Rep1(W, P ) . . .Reprk(P)(W, P ) = Rep(W, P )
induces epimorphisms of objects in F with Out(P ) action
Fp = FRep0(W,P)p FRep1(W,P)p . . .FReprk(P)(W,P)p= FRep(W,P)p.
An appeal to Proposition 4.10 thus shows
Lemma 5.3. dim eSH*(P ) = maxk{dim eSr(ker{FRepk(W,P)p! FRepk-1(W,P)p})}.
To more usefully describe ker{FRepk(W,P)p! FRepk-1(W,P)p}, we need to introdu*
*ce
some notation. Given V < P , let GLP (V ) = Out(P, V )=Out0(P, V ). Note that
GLP (V ) is naturally a subgroup of GL(V ) and also acts on the right of the Ou*
*t(P )-
set Out(P )=Out0(P, V ).
Lemma 5.4. There is an isomorphism of objects in F with Out(P ) action,
Y Epi(W,V )xGL (V )Out(P)=Out0(P,V )
ker{FRepk(W,P)p! FRepk-1(W,P)p} ' Fp P ,
[V ]
where the product is over Aut(P )-orbits of elementary abelian subgroups of ran*
*k k.
14 KUHN
Now we need to identify the nilclosed unstable module with Out(P ) action as-
sociated to FEpi(W,Vp)xGLP(V )Out(P)=Out0(P,V.)
Let cV 2 S*(V )GL(V )be the `top' Dickson invariant: cV is the product of all*
* the
elements fi(x) 2 H2(V ), with 0 6= x 2 H1(V ). The key property we need is that
j*(cV ) = 0 for all proper inclusions j : U < V .
Lemma 5.5 (Compare with [HLS1 , proof of Thm.II.6.4]). l(cV S*(V ))(W ) =
FEpi(W,Vp).
Corollary 5.6. The functor l assigns to the reduced module
[cV S*(V ) FOut(P)=Out0(P,Vp)]GLP(V ),
the object in F which sends W to FEpi(W,Vp)xGLP(V )Out(P)=Out0(P,V.)
To finish the proof of Theorem 2.1, we are left needing to prove
Proposition 5.7. If an irreducible Fp[Out(P )]-module S occurs in the permutati*
*on
module Fp[Out(P )=Out0(P, V )], then
dimeS[cV S*(V ) FOut(P)=Out0(P,Vp)]GLP(V )= rkV.
Proof.The proof is similar to the proof of Conjecture A when P = V given
in x3. Our first observation is that eS[cV S*(V ) FOut(P)=Out0(P,Vp)]GLP(V )*
*is a
S*(V )GL(V )-submodule of the free S*(V )GL(V )-module S*(V ) FOut(P)=Out0(P,Vp*
*).
As dimS*(V )GL(V )= rkV , it suffices to show that if S occurs in the permutati*
*on
module Fp[Out(P )=Out0(P, V )], then S occurs in
[cV S*(V ) FOut(P)=Out0(P,Vp)]GLP(V ).
The group ring Fp[GLP (V )] occurs as a submodule of S*(V ) [Alp, p.45], and th*
*us as
a submodule of cV S*(V ). Thus S will occur in [cV S*(V ) FOut(P)=Out0(P,Vp)]GL*
*P(V )
if it occurs in [Fp[GLP (V )] FOut(P)=Out0(P,Vp)]GLP(V ). But this last Fp[Ou*
*t(P )]-
module rewrites as FOut(P)=Out0(P,Vp).
6. Proof of Theorem 2.3 and related results
Theorem 2.1 tells us that calculating dim eSH*(P ) amounts to understanding
the Fp[Out(P )]-module composition factors of the permutation modules
Fp[Out(P )=Out0(P, V )],
when V < P is a maximal elementary abelian subgroup.
In this section, we use group theory to find various conditions ensuring that*
* all
irreducibles occur in this way.
6.1. Composition factors of permutation modules. We begin with an ele-
mentary, but useful, lemma about permutation modules.
Lemma 6.1. Let H be a subgroup of a finite group G. Then (a) ) (b) ) (c),
and, if H is also normal, (c) ) (a).
(a) H is a p-group.
(b) Every irreducible Fp[G]-module occurs as a submodule of Fp[G=H].
GROUP AUTOMORPHISMS 15
(c) Every irreducible Fp[G]-module occurs as a composition factor of Fp[G=H].
Proof.We begin by reminding the reader that permutation modules are self dual,
so an irreducible S occurs as a submodule of Fp[G=H] if and only if S occurs as*
* a
quotient module. This happens if and only if there exists a nonzero homomorphism
Fp[G=H] ! S, which is equivalent to SH 6= 0.
If H is a p-group, then MH 6= 0 for all nonzero Fp[H]-modules M. Thus (a)
implies (b), and (b) implies (c) is obvious.
Now suppose that H is normal. Then the composition factors of Fp[G=H] are just
the irreducible Fp[G=H]-modules pulled back to G. But if H is not a p-group, th*
*en
G=H has fewer p0 conjugacy classes than G, and thus fewer irreducible modules.
Thus (c) implies (a) under the normality assumption.
Remark 6.2. It is not hard to see that, if H is not normal, then neither implic*
*ation
(b) ) (a) nor (c) ) (b) need hold. For the former, let p = 3, G = SL2(F3), and
H = Z=2 permuting a basis for (F3)2. For the latter, let p = 3, G = 3, and
H = Z=2.
Corollary 6.3. If a p-group P has a unique maximal elementary abelian subgroup
V , then every irreducible occurs in Fp[Out(P )=Out0(P, V )] if and only if Out*
*0(V )
is a p-group.
Proof.The hypothesis implies that Out0(P, V ) arises as the kernel of an evident
homomorphism Out(P ) ! Aut(V ), and thus is normal in Out(P ). Now the lemma
applies.
6.2. p0-automorphisms of p-groups. We will use various well known results
about detecting automorphisms of a p-group P of order prime to p.
The most classic is due to Burnside and Hall, and goes as follows. Let (P ) *
*be
the Frattini subgroup of P , so that P= (P ) = H1(P ; Fp).
Proposition 6.4 ([Gor, Thm.5.1.4]). The kernel of the homomorphism Aut(P ) !
Aut(P= (P )) is a p-group.
With groups A and B as indicated, the Thompson A x B lemma ([Asch, (24.2)],
[Gor, Thm.5.3.4]) immediately applies to prove the next proposition.
Proposition 6.5. Let B be an abelian subgroup of a finite p-group P , and let
A = {ff 2 Aut(P ) | ff(g) = g for allg 2 CP (B)}. Then A is a p-group.
Corollary 6.6. For all elementary abelian V < P , the kernel of the homomorphism
Aut(P, V ) ! Aut(CP (V )) is a p-group. Thus if Aut(CP (V )) is a p-group so is
Aut(P, V ), and if Aut0(CP (V ), V ) is a p-group so is Aut0(P, V ).
The next result requires that p be odd. Let 1(P ) be the subgroup of P gener*
*ated
by the elements of order p.
Proposition 6.7 ([Gor, Thm.5.3.10]). Let P be a p-group, with p odd. Then the
kernel of the homomorphism Aut(P ) ! Aut( 1(P )) is a p-group.
Corollary 6.8. Let Q is p-central, with p odd, and with maximal central element*
*ary
abelian subgroup C. Then Aut0(Q, C) is a p-group.
This follows from the proposition, noting that C = 1(Q), since Q is p-centra*
*l.
16 KUHN
Proof of Theorem 2.3.The first statement follows immediately from Corollary 6.6.
Now we note that, if V < P is a maximal elementary abelian subgroup, then
CP (V ) is p-central with V as its maximal central elementary abelian subgroup.
Thus Corollary 6.8 applies to prove the second statement.
6.3. 2-central group examples. We elaborate on Example 2.4.
Example 6.9. Let Gt be the 2-Sylow subgroup of SU3(F2t). This is a 2-central
group of rank t, and if Ctis its maximal elementary abelian subgroup, Out0(Gt, *
*Ct)
contains a cyclic subgroup of order 2t+ 1, and thus is not a 2-group.
This will be a consequence of a few facts4 about Gt.
Let q = 2t. Given a 2 Fq2, let ~a= aq, so that Fq identifies with the set of *
*b 2 Fq2
such that b + ~b= 0. Then Gt SU3(Fq) GL3(Fq2) can be described as
8 0 1 9
< 1 a b =
Gt= :A(a, b) = @ 0 1 ~aA : a, b 2 Fq2with b + ~b=.a~a
0 0 1 ;
The center Ct of Gt is the set of matrices of the form A(0, b): note that then
b 2 Fq. All other elements have order 4, and thus Gt is 2-central sitting in t*
*he
central extension
Ct! Gt! Gt=Ct,
with Ct' Fq and Gt=Ct' Fq2.
The normalizer of Gt in SU3(F2t) is the semidirect product Gt >CTt, where Tt
is the set of matrices
8 0 1 9
< c 0 0 =
@ 0 ~cc-1 0 A : c 2 Fx2.
: D(c) = 0 0 ~c-1 q ;
Direct computation shows that
D(c)A(a, b)D(c-1) = A(~c-1c2a, ~ccb) = A(c2-qa, cq+1b).
From this, one deduces that Tt ! Aut(Gt=Ct), and thus Tt ! Aut(Gt), is monic
if gcd(q - 2, q2 - 1) = 1. This is the case unless q = 2, i.e., t = 1. Meanwh*
*ile,
the kernel of the homomorphism Tt ! Aut(Ct) identifies with the kernel of the
multiplicative norm Fxq2! Fxq, and thus is cyclic of order q + 1. Thus, if t *
*2,
Aut0(Gt, Ct) = ker{Aut(Gt) ! Aut(Ct)} contains a cyclic group of order (q + 1).
(When t = 1, Gt = Q8, and so Aut0(Gt, Ct) also contains a group of order
(q + 1) = 3.)
Remark 6.10. The group G2 has been of interest to those studying group cohomol-
ogy. Even though it is 2-central, a presentation of its cohomology ring is rema*
*rkably
nasty to write down: see the calculations for group number # 187 of order 64 in
[Ca , CTVZ ]. It is the smallest group with nontrivial products in its essentia*
*l coho-
mology [Gr ]. The calculation of the nilpotent length of p-central groups in [K*
*4 , K5]
shows that its nilpotent length is 14, which seems likely to be maximal among a*
*ll
groups of order 64. Presumably, the groups Gtfor larger t are similarly interes*
*ting.
____________
4The authors are very grateful to David Green for showing us his unpublished*
* `Appendix B'
to [Gr], which is our source of information about these groups.
GROUP AUTOMORPHISMS 17
6.4. p-groups with reduced cohomology. Recall that P has reduced cohomol-
ogy if and only if H*(P ) is detected by restriction to the various V < P , i.*
*e.,
H*(P ) ! H*(P ) is monic.
We restate Proposition 2.5.
Proposition 6.11. If H*(P ) is reduced, and V < P is a maximal elementary
abelian subgroup, then CP (V ) = V .
Recalling that CP (V ) is p-central if V < P is maximal, the proposition is a
consequence of the next two lemmas.
Lemma 6.12. If H*(P ) is reduced, so is H*(CP (V )) for any elementary abelian
V < P .
Lemma 6.13. If Q is a p-central p-group with reduced cohomology, then Q is
elementary abelian.
Proof of Lemma 6.12.Properties of the functor TV imply that if M 2 U is re-
duced so is TV M: recalling that TV preserves monomorphisms, property (b) of
Proposition 4.1 impies this. Now one uses that, given V < P , H*(CP (V )) is a
(canonical) direct summand in TV H*(P ). (See [HLS2 , I(5.2)] for a calculation*
* of
TV H*(P ).)
Proof of Lemma 6.13.Let C be the center of Q. The hypothesis is that H*(Q) !
H*(C) is monic, which implies that the inflation map H*(Q=C) ! H*(Q) is zero.
But H1(Q=C) ! H1(Q) is always monic, so we conclude that H1(Q=C) = 0. But,
since Q=C is a p-group, this means that Q=C is trivial, i.e., C = Q.
6.5. A remark about odd prime p-central groups. Using a result of Henn
and Priddy, Theorem 2.3 has an addendum. For odd prime p-central groups P , the
refinement of Conjecture A described in Remark 3.1 holds: indeed, dimeH*(P ) =
rk(P ) for all idempotents e 2 A(P, P ) Fp that project to a nonzero element un*
*der
the retraction of algebras A(P, P ) Fp ! Fp[Out(P )]. Here dimension is defined*
* via
the Poincar'e series, and we remind the reader that A(P, P ) is the double Burn*
*side
ring.
The argument goes as follows. Given an irreducible Fp[Out]-module S, let eS 2
Fp[Out(P )] and "eS2 A(P, P ) Fp be the associated primitive idempotents5. Th*
*en
eSH*(P ) = "eSH*(P ) M* where M* is a finite direct sum of modules of the form
"eTH * (Q) for appropriate pairs (Q, T ). By [HP , Prop.1.6.1 and Lem.2.1], if*
* P
is an odd prime p-central group, the only Q's that can occur here will be proper
retracts of P . But such groups will necessarily have strictly smaller ranks. T*
*hus
dimM* < rk(P ), and so dime"SH*(P ) = dimeSH*(P ) = rk(P ).
7. Proof of Theorem 2.8
7.1. A formula for ~RdH*(P ). Recall that ~R0H*(P ) = limV 2AC(P)H*(V ). Using
the fact that morphisms in A(V ) factor as inner automorphisms composed with
inclusions, this rewrites as follows: there is a natural isomophism
8 " # 9
< Y Inn(P) ~ Y =
~R0H*(P ) = Eq H*(V ) - ! H*(V1) ,
: V - ! V1, C = , V = ,
Q8 = , and K = . K is maximal subgroup #11.
10See automorphism #7 on the Carlson website.
GROUP AUTOMORPHISMS 27
10.4. PV H*(K) and PC H*(K). We have maps of unstable algebras equipped with
with Aut(P ) action:
*
(10.2) PV H*(K) ,! PC H*(K) -j PC H*(P ),
where j : K ! P is the inclusion. In degree 1, this reads
Hom (K=V, F2) ,! Hom (K=C, F2) Hom (P=C, F2),
and so we see that j* is onto in degree 1, and that PV H1(K) is a copy of S.
The maps of pairs (Q8, Z) ! ((Z=2)2 x Q8, (Z=2)2 x Z) = (K, V ) induces an
isomorphism of algebras:
PV H*(K) ' PZH*(Q8).
The algebra PZH*(Q8) is familiar: the calculation of H*(Q8) using the Serre
spectral sequence associated to Z ! Q8 ! Q8=Z reveals that PZH*(Q8) =
Im {H*(Q8=Z) ! H*(Q8)} = B*, where B* is the Poincar'e duality algebra
F2[x, w]=(x2 + xw + w2, x2w + xw2), where x and w both have degree 1. As an
F2[Aut(P )]-module, PV H*(K) = B* is given by
8
>>>1 ifd = 3
~~
>>:S ifd = 1
1 ifd = 0.
From this we learn that PC H*(K) ' B*[y] where y is also in degree 1, and thus
is generated by elements in degree 1. It follows that j* : PC H*(P ) ! PC H*(K)*
* is
onto, and then that Inn(P ) acts trivially on both PV H*(K) and PC H*(K).
10.5. R~dH*(P ). Applying the formula for R~dH*(P ) in Proposition 7.1 to our
group, we see that, for all d, there is a pullback diagram of graded F2[Out(P )*
*]-
modules:
R~dH*(P )____________//_H*(C) PC Hd(P )
| | *
| |1 j
fflffl| fflffl|
(H*(V ) PV Hd(K))Inn(P)___//H*(C) PC Hd(K).
By our comments above, this simplifies: for all d, there is a pullback diagra*
*m of
unstable F2[Out(P )]-modules:
R~dH*(P )__________//_H*(C) PC H*(P )
| | *
| |1 j
fflffl| fflffl|
H*(V )Z=2 PV H*(K)_____//H*(C) PC H*(K).
As j* is onto, we conclude
Proposition 10.1. For all d, there is a short exact sequence of unstable F2[Out*
*(P )]-
modules 0 ! H*(C) kerjd! ~RdH*(P ) ! H*(V )Z=2 PV Hd(K) ! 0, where
H*(C) and H*(V )Z=2have only trivial composition factors, while PV H*(K) ' B*.
28 KUHN
We note that in the notation of x7, Ess*(C) = kerj*and Ess*(V ) = PV H*(K).
From Proposition 10.1, one can already place the 3's occurring in the table in
Example 2.14. The 2's will be determined by knowing the composition factors of
the ideal kerj* PC H*(P ). This we discuss next.
10.6. PC H*(P ) and the ideal kerj*. Obviously, kerj0 = 0, and we also know
that kerj1 is one dimensional. Using more detailed information about H*(P ), we
outline how all of the composition factors of kerj* can be determined.
H*(P ) is generated by classes z, y, x, w, v, u, t of degrees 1,1,1,1,2,5,8 r*
*espec-
tively, and the kernel of the restriction H*(P ) ! H*(K) is identified as the i*
*deal
(z).11 Furthermore, this ideal is a free module on F2[v, t] on an explicit basi*
*s of 24
monomials in the generators of degree 1.
All the degree 1 generators are H*(C)-primitive, while v and t restrict to al*
*ge-
braically independent elements of H*(C). It follows that kerj*= PC H*(P ) \ (z)
is precisely the F2-span of this basis of mononomials.
This allows us to conclude the following.
Firstly PC H*(P ) H*(P ) is precisely the subalgebra generated by the class*
*es
in degree 1. Explicitly, PC H*(P ) is
A* = F2[z, y, x, w]=(z2, zy + x2 + xw + w2, zy2 + x2w + xw2, y2w3 + yw4 + w5).
For our purposes, this can be simplified as follows. From the second and third
relations, one can deduce that x3 = z(yx + y2) and w3 = z(yw + y2). Using these
and the first relation, one deduces that y2w3 + yw4 + w5 = zy4. So we have
A* = F2[z, y, x, w]=(z2, zy4, zy + x2 + xw + w2, zy2 + x2w + xw2).
Secondly, kerj*= (A*=Ann(z)), as unstable F2[Out(P )]-modules. Since z2 =
zy4 = 0, A*=Ann (z) is a quotient of A*=(z, y4) = F2[y, x, w]=(y4, x2+xw+w2, x2*
*w+
xw2) = B*[y]=(y4). Inspection of the degrees of the monomial basis shows that
A*=Ann(z) has dimension 1,3,5,6,5,3,1 in degrees 0,1,2,3,4,5,6. This agrees wi*
*th
B*[y]=(y4), and we can conclude that Ann(z) = (z, y4), and then
Proposition 10.2. kerj*' (B*[y]=(y4)) as unstable F2[Out(P )]-modules.
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Department of Mathematics, University of Virginia, Charlottesville, VA 22904
E-mail address: njk4x@virginia.edu
~~