PRIMITIVES AND CENTRAL DETECTION NUMBERS IN
GROUP COHOMOLOGY
NICHOLAS J. KUHN
Abstract.Fix a prime p. Given a finite group G, let H*(G) denote its mod
p cohomology. In the early 1990's, Henn, Lannes, and Schwartz introduced
two invariants d0(G) and d1(G) of H*(G) viewed as a module over the mod *
*p
Steenrod algebra. They showed that, in a precise sense, H*(G) is respect*
*ively
detected and determined by Hd(CG(V )) for d d0(G) and d d1(G), with
V running through the elementary abelian psubgroups of G.
The main goal of this paper is to study how to calculate these invaria*
*nts.
We find that a critical role is played by the image of the restriction o*
*f H*(G)
to H*(C), where C is the maximal central elementary abelian psubgroup of
G. A measure of this is the top degree e(G) of the finite dimensional Ho*
*pf
algebra H*(C) H*(G)Fp, a number that tends to be quite easy to calculat*
*e.
Our results are complete when G has a pSylow subgroup P in which every
element of order p is central. Using BensonCarlson duality, we show tha*
*t in
this case, d0(G) = d0(P) = e(P), and a similar exact formula holds for d*
*1.
As a bonus, we learn that He(G)(P) contains nontrivial essential cohomol*
*ogy,
reproving and sharpening a theorem of Adem and Karagueuzian.
In general, we are able to show that d0(G) max{e(CG(V ))  V < G}
if certain cases of Benson's Regularity Conjecture hold. In particular,*
* this
inequality holds for all groups such that the difference between the pr*
*ank of
G and the depth of H*(G) is at most 2. When we look at examples with p =*
* 2,
we learn that d0(G) 14 for all groups with 2Sylow subgroup of order u*
*p to
64, with equality realized when G = SU(3, 4).
Enroute we study two objects of independent interest. If C is any cent*
*ral
elementary abelian psubgroup of G, then H*(G) is a H*(C)comodule, and
we prove that the subalgebra of H*(C)primitives is always Noetherian of
Krull dimension equal to the prank of G minus the prank of C. If the
depth of H*(G) equals the rank of Z(G), we show that the depth essential
cohomology of G is nonzero (reproving and extending a theorem of Green),
and CohenMacauley in a certain sense, and prove related structural resu*
*lts.
1.Introduction
Fix a prime p, and let H*(G) denote the mod p cohomology ring of a finite gro*
*up
G. The pelementary abelian subgroups of G have had a featured role in the study
of group cohomology since D.Quillen's famous work [Q ] in the late 1960's. In p*
*ar
ticular, these subgroups become the objects in a category A(G) having morphisms
the homomorphisms generated by subgroup inclusion and conjugation by elements
in G. The inclusions V < G then induce a map
Y
H*(G) ~0! lim H*(V ) H*(V ),
V 2A(G) V
____________
Date: December 5, 2006.
2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
and ~0 is shown to have kernel and cokernel that are nilpotent in an appropriate
sense.
Viewing H*(G) as the mod p cohomology of the classifying space BG makes it
evident that H*(G) is an object in K and U, the categories of unstable algebras
and modules over the mod p Steenrod algebra A. The 1980's and 1990's saw a
revolution in our understanding of these categories, and the 1995 paper of H.
W.Henn, J.Lannes, and L.Schwartz [HLS1 ] revisited Quillen's approximation of
H*(G) from this new perspective.
For each d 0, the group homomorphisms V x CG (V ) ! G induce a map of
unstable algebras Y
H*(G) ! H*(V ) H d (CG (V )),
V
where M d denotes the quotient of a graded module M* by all elements of degree
more than d. The image of this map lands in an evident subalgebra of `compatibl*
*e'
elements which Henn, Lannes, and Schwartz show can be naturally identified with
LdH*(G), where Ld : U ! U is localization away from the localizing subcategory
generated by (d+1)fold suspensions of unstable modules. Thus Quillen's map can
be viewed as the just the bottom of a tower of localizations of H*(G) associate*
*d to
the nilpotent filtration of U:
..
.


fflffl
L2H*(G)99
ss
ssss p2
~2 ssss fflffl
ssss iL1H*(G)44
sss iiii
ssssii~1iiiii p1
siiiiiiss~0 fflffl
H*(G) ________________//_L0H*(G),
where we have
Y
H*(G) ~d!LdH*(G) H*(V ) H d (CG (V )).
V
This caused the authors of [HLS1 ] to introduce two new invariants of G: d0(G)
and d1(G) are the smallest d's such that H*(G) is respectively detected by, and
isomorphic to, LdH*(G). Alternatively, d0(G) is the smallest d such that H*(G)
contains no (d + 1)fold suspensions of a nontrivial unstable module, and d1(G)*
* is
the smallest d such that also Ext1A( d+1N, H*(G)) = 0 for all N 2 U.
These invariants satisfy a few easily verified nice properties: d0(GxH) = d0(*
*G)+
d0(H), d1(G x H) = max {d1(G) + d0(H), d0(G) + d1(H)}, and di(G) di(P ) if
P is a pSylow subgroup of G. However, they are not well behaved under taking
subgroups, quotient groups, and extensions; e.g., every G embeds in a symmetric
group n and d0( n) = 0. Rough upper bounds for d0(G) and d1(G) were found in
[HLS1 ]; e.g., d0(G) is bounded by n2 if a pSylow subgroup of G admits a faith*
*ful
n dimensional complex representation. However, in all but a few examples, these
bounds seem far from optimal. Up to now, what determines these group invariants
GROUP COHOMOLOGY PRIMITIVES 3
has remained mysterious, and they have not been connected to other work in group
cohomology.
A main goal of this paper is to present a way to calculate the number d0(G), *
*and,
in some cases, d1(G). Our finding is that these numbers seem to be controlled by
the restriction of cohomology to maximal central pelementary abelian subgroups.
Our results are complete when G has a Sylow subgroup that is pcentral, i.e. a
group in which every element of order p is central. For example, when p = 2, we
compute that d0(SU(3, 4)) = 14 and d1(SU(3, 4)) = 18, where, by constrast, the
estimates from [HLS1 ] yield only that d0(SU(3, 4)) 64 and d1(SU(3, 4)) 120.
Our method is to combine Utechnology, in the spirit of [HLS1 ], with duality
results as in the work of D.Benson and J.Carlson [BC1 ]. We ultimately connect a
conjectured upper bound for d0(G) to Benson's Regularity Conjecture [Be], known
to hold if the prank of G and the prank of Z(G) differ by at most 2. This is *
*the
case for all 2groups of order 64 or less, and, using cohomology calculations f*
*rom
[CTVZ ], we've been able to verify by hand that d0(G) 14 for all such groups.
A number of side results of independent interest come up in our investigation*
*s.
We are led to study carefully the cohomology of central extensions, in partic*
*ular
the structure of associated algebras of primitives. One outcome of this is a n*
*ew
proof of A.Adem and D.Karagueuzian's theorem [AK ] that pcentral pgroups have
nonzero essential cohomology. We show that in an explicit degree there is a non*
*zero
cohomology class that is simultaneously essential and annihilated by all Steenr*
*od
operations of positive degree.
Deriving our general estimate of d0(G) involves a careful study of the depth
essential cohomology of Carlson, et. al. [CTVZ ] in the important special case*
* that
the depth of H*(G) equals the rank of the center. We prove that then the depth
essential cohomology is both nonzero  reproving the main theorem of [G1 ] with*
*out
D.Green's hypothesis that G be a pgroup  and CohenMacauley.
In the next section we describe our results in more detail.
2.Main Results
2.1. The cohomology of central extensions. Suppose we have a central exten
sion of finite groups
C i!G q!Q,
where C is pelementary abelian of rank c.
We define various objects associated to this situation.
The extension corresponds to an element o 2 H2(Q; C). Since H2(Q; C) =
Hom (H2(Q), C), the extension can also be considered as corresponding to a homo
morphism o : H2(Q) ! C, or, equivalently, its dual o# : C# ! H2(Q).
Let {E*,*r} denote the Serre spectral sequence associated to the extension, c*
*on
verging to H*(G), and with E*.*2= H*(Q) H*(C). Under the identification
C# = H1(C), it is standard that o# corresponds to d2 : E0,12! E2,02.
Let Io H*(Q) be the ideal generated by A . im(o#G), so that H*(Q)=Io is an
unstable algebra. It is easy to see that Io is contained in the kernel of infl*
*ation
q* : H*(Q) ! H*(G).
Call a*subalgebra A of H*(G) a (G, C)Duflot subalgebra, if the composite A
H*(G) i!im(i*) is an isomorphism, where i* : H*(G) ! H*(C) is the restriction.
As we will describe more precisely in x2.5, as an algebra, the Hopf algebra im(*
*i*)
4 KUHN
H*(C) will necessarily be free graded commutative on c polynomial generators,
possibly tensored with an exterior algebra on some generators in degree 1, if p*
* is
odd. It follows that Duflot subalgebras exist and have the same form. Let QA H**
*(G)
denote the graded algebra1 of Aindecomposables H*(G) A Fp, or, equivalently,
the quotient of H*(G) by the ideal generated by the positive degree elements of*
* A.
H*(C) is a Hopf algebra, and the multiplication map m : C x G ! G induces a
map of unstable algebras
m* : H*(G) ! H*(C) H*(G)
making H*(G) into a H*(C)comodule. We define the associated algebra of prim
itives to be
PC H*(G) = {x 2 H*(G)  m*(x) = 1 x}
m* *
= Eq {H*(G) !!H (C x G)},
ss*
where ss : C x G ! G is the projection. It is easy to check that PC H*(G) is an
unstable algebra that contains the image of the inflation map. Thus q* : H*(Q) !
H*(G) refines to a map of unstable algebras
qo : H*(Q)=Io ! PC H*(G).
Theorem 2.1. With the notation as above, the following are true.
(a) H*(G) is a free Amodule. Moreover {E*,*r} is a spectral sequence of free E*
*0,*1
modules, and applying QE0,*1to the spectral sequence yields a spectral sequence*
* con
verging to QA H*(G) with E2term QE0,*1H*(C) H*(Q).
(b) The composite PC H*(G) ,! H*(G) i QA H*(G) is monic.
(c) Both PC H*(G) and QA H*(G) are finitely generated H*(Q)modules.
(d) The map qo : H*(Q)=Io ! PC H*(G) is an F isomorphism2, and the rings
H*(Q)=Io, im(q*), PC H*(G), and QA H*(G), are all Noetherian of Krull dimen
sion equal to (the prank of G)  (the rank of C).
Let C(G) < G be the pelementary abelian part of Z(G). If C = C(G), the first
part of statement (a) recovers J. Duflot's result [D ] that the depth of H*(G) *
*is at
least as great as the rank of C(G).3 We will call a (G, C(G))Duflot subalgebra*
* of
H*(G) simply a Duflot subalgebra.
The prank of G equals the rank of C exactly when C = C(G) and G is pcentral,
and we have the following corollary.
Corollary 2.2. If G is pcentral, and C = C(G), then the rings H*(Q)=Io, im(q*),
PC H*(G), and QA H*(G) all have Krull dimension zero and so are finite dimen
sional Fpalgebras.
____________
1QAH*(G) will not necessarily be an unstable algebra, as A need not be close*
*d under Steenrod
operations.
2In the sense of Quillen [Q ]: ker(~q*) is nilpotent, and for all x 2 PCH*(G*
*), there exists a k so
that xpk2 im(~q*).
3We are claiming no originality in the proof of this, which is similar to al*
*l proofs of Duflot's
theorem following [BrH]. The spectral sequence refinement seems to be a new obs*
*ervation.
GROUP COHOMOLOGY PRIMITIVES 5
2.2. Quillen's category and functors involving primitives. Our algebras of
primitives arise in two formulae associated to H*(G), viewed as an object in K.*
* To
describe these, we need to introduce some notation.
Given a small category C, we let C# denote the associated twisted arrow categ*
*ory:
the objects of C# are the morphisms of C, and a morphism ff _ fi from ff : A1 !*
* A2
to fi : B1 ! B2 is a commutative diagram in C
A1 __ff_//A2OO
 
 
fflfflfi
B1 _____//B2.
The functor assigning H*(V ) to V 2 A(G) is contravariant, while the assignme*
*nt
of H*(CG (V )) is covariant. Now observe that the assignment of Pff(V1)H*(CG (V*
*2))
to ff : V1 ! V2 can be viewed as defining a contravariant functor of A(G)# .
Let AC (G) denote the full subcategory of A(G) having as objects the V con
taining C(G). If G is pcentral, then AC (G) has a single object and morphism.
2.3. A formula for the locally finite part of H*(G). If M is an unstable A
module, we define MLF , the locally finite part of M, by
MLF = {x 2 M Ax M is finite}.
This is again an unstable module, and is an unstable algebra if M is.
Theorem 2.3. There is a natural isomorphism of unstable algebras
H*(G)LF ' limffPff(V1)H*(CG (V2)),
V1!V2
where the limit is over AC (G)# .
Corollary 2.4. If G is pcentral, then H*(G)LF = PC(G)H*(G).
2.4. A formula for ~RdH*(G). An unstable module M 2 U has a canonical `nilpo
tent' filtration [S1, K1, HLS1 ]:
. . .nil2M nil1M nil0M = M.
In general, nildM=nild+1M = dRdM, where RdM is reduced, i.e. has no nontriv
ial submodules that are suspensions. We let ~RdM denote the nilclosure L0RdM of
RdM.
The module nildM identifies with the kernel of ~d : M ! Ld1M, and a bit
of diagram chasing will show that dR~dM is isomorphic to the kernel of LdM !
Ld1M: see Proposition 3.1. Thus d0(G) is the length of the filtration of H*(G),
and also is the biggest d such that ~RdH*(G) 6= 0.
Theorem 2.5. There is a natural isomorphism of unstable modules
~RdH*(G) ' lim H*(V1) Pff(V1)Hd(CG (V2)),
V1ff!V2
where the limit is over AC (G)# .
Corollary 2.6. If G is pcentral, then there is an isomorphism of unstable modu*
*les
R~dH*(G) ' H*(C(G)) PC(G)Hd(G).
6 KUHN
2.5. Invariants of restriction to C(G). If i : C < G is a central pelementary
abelian of rank c, then
(
H*(C) ' F2[x1, . .,.xc] if p = 2
(x1, . .,.xc) Fp[y1, . .,.yc]if p,is odd
where xi = 1 and yi= fi(xi), and is a Hopf algebra in the usual way.
In x6, we will see that, after a change of basis for H1(C), the image of the
restriction homomorphism i* : H*(G) ! H*(C) will be a sub Hopf algebra of
H*(C) of the form
( j1 jc
F2[x21 , . .,.x2c ] if p = 2
im (i*) = pj1 pjb
Fp[y1 , . .,.yb , yb+1, . .,.yc] (xb+1,i.f.,.xc)p,is odd
with the ji forming a sequence of nonincreasing nonnegative integers4.
Now suppose that C = C(G). We will say that G has type [a1, . .,.ac] where
( j
1, . .,.2jc) if p = 2
(a1, . .,.ac) = (2
(2pj1, . .,.2pjb, 1, .i.,.1)f p.is odd
The type of G has the form [1, . .,.1] if and only if G = C x H, where Z(H) h*
*as
order prime to p. In all other cases, a1 = 2pk for some k 0.
Define e(G) and h(G) by
Xc
e(G) = (ai 1),
i=1
and 8
><2pk1 ifa1 = 2pk withk 1
h(G) ' >1 ifa1 = 2
:0 ifa1 = 1.
For example, Q8x Z=4 has type [4, 2] when p = 2, so that e(Q8x Z=4) = 4, and
h(Q8 x Z=4) = 2.
Remark 2.7. The careful reader will observe that the type of G is just the list*
* of the
degrees of the unstable Aalgebra generators of im(i*), listed in decreasing or*
*der,
e(G) is the top nonzero degree of the finite dimensional Hopf algebra
H*(C) H*(G)Fp = H*(C)=(im(i*>0)),
and h(G) is the top nonzero degree of the module A . H1(C) projected into this
Hopf algebra.
2.6. PC(G)H*(G), d0(G), and d1(G) when G is pcentral. If G is pcentral with
C = C(G), then PC H*(G) is a finite dimensional unstable algebra. We identify i*
*ts
top degree and more.
Theorem 2.8. Let G be pcentral, C = C(G), and A be a Duflot subalgebra of
H*(G). Then both PC H*(G) and QA H*(G) are zero in degrees greater than e(G),
and one dimensional in degree e(G). Furthermore, PC He(G)(G) is annihilated by
____________
4In the odd prime case, c  b will be the rank of the largest subgroup of C *
*splitting off G as a
direct summand.
GROUP COHOMOLOGY PRIMITIVES 7
all positive degree elements of the Steenrod algebra, and, if G is a pgroup, c*
*onsists
of essential5 cohomology classes.
The last statement implies the main result of [AK ]: pcentral pgroups have
nonzero essential cohomology.
This theorem, combined with Corollary 2.6 and related results, leads to the
following calculation.
Theorem 2.9. Let G be pcentral. Then d0(G) = e(G) and d1(G) = e(G) + h(G).
Furthermore, if G is a finite group with pSylow subgroup P , and P is pcentra*
*l,
then d0(G) = d0(P ) and d1(G) = d1(P ).
Corollary 2.10. If G is pcentral, and H < G, then d0(H) d0(G) and d1(H)
d1(G).
Examples 2.11. (a) As Q8 is 2central of type [4], d0(Q8) = 3 and d1(Q8) =
3 + 2 = 5, in agreement with [HLS1 , (II.4.6)].
(b) The hypotheses of pcentrality is needed in the last part of the theorem: as
observed in [HLS1 , II.4.7], if G = GL2(F3) and P = SD16, then P is the 2
Sylow subgroup of G, but d0(G) = 0 < 2 = d0(P ), and d1(G) = 2 < 4 = d1(P ).
Similarly, G needs to be pcentral in the corollary: if H = Z=4 < D8 = G,
then d0(H) = 1 > 0 = d0(G) and d1(H) = 2 > 0 = d1(G). The example
H = Z=4 < Z=8 = G shows that the inequalities of the corollary can be equal
ities, even when H is a proper subgroup of a pcentral pgroup G.
(c) The 2Sylow subgroup P of the simple group SU(3, 4) is 2central of type [8*
*, 8].
Thus d0(SU(3, 4)) = d0(P ) = 14 and d1(SU(3, 4)) = d1(P ) = 18. Similarly, the
2Sylow subgroup Q of the simple group Sz(8) is 2central of type [4, 4, 4]. Th*
*us
d0(Sz(8)) = d0(Q) = 9 and d1(Sz(8)) = d1(Q) = 11. We will see that P and
Q have the largest d0 of all 2groups of order dividing 64. For more about the
SU(3, 4) example, see x9.
(d) In [AKM ], the authors associate a 2central Galois group GF to every field
F of characteristic different from 2 that is not formally real. (They call thi*
*s the
W group of F because of its connections to the Witt ring W F [MiS ].) From the*
*ir
construction it is easy to deduce that GF has type [2, . .,.2]. Thus d0(GF) = r*
* and
d1(GF) = r + 1, where GF has rank r. In particular, the universal group W group
W (n) has d0(W (n)) = n+12 and d1(W (n)) = n+12+ 1. For more about this
example, see x9.
2.7. Central essential cohomology. Our calculation of d0(G) when G is p
central relies on Corollary 2.6. To understand d0(G) for general G, one needs
to use the more complicated formula given in Theorem 2.5. Using some analysis of
this already done by us in our companion paper [K3 ], we are led to a formula6 *
*for
d0(G) that makes use of the following variant of essential cohomology.
____________
5Recall that x 2 H*(G) is essential if it restricts to zero on all proper su*
*bgroups.
6Thus far, we have not found an analogous formula for d1(G).
8 KUHN
We define Cess*(G), the central essential cohomology of G, to be the kernel of
the restriction map Y
H*(G) ! H*(CG (U)),
C(G) 0, nildM = ker~d1.
An unstable module M is called reduced if nil1M = 0. As observed in [K1 ,
Prop.2.2], nildM=nild+1M = dRdM, where RdM is a reduced unstable module.
(See also [S2, Lemma 6.1.4].) Then ~RdM is defined to be the N il1closure of R*
*dM.
Thus RdM L0RdM = ~RdM.
____________
8This section necessarily overlaps with the presentation in our recent prepr*
*int [K3].
9What we are calling Ld here was called Ld+1in [HLS1].
GROUP COHOMOLOGY PRIMITIVES 11
We have the following useful alternative definition of R~dM. (Compare with
[HLS1 , I(3.8.1)].)
Proposition 3.1. There is a natural isomorphism
dR~dM ' ker{LdM ! Ld1M}.
The functors Ld and Ld1 are left exact, as they are localizations, and thus *
*we
conclude
Corollary 3.2. R~d: U ! U is left exact.
Proof of Proposition 3.1.Let cd+1M = coker{~d : M ! LdM}. Then cd+1M 2
N ild+1, and there is an exact sequence
0 ! nild+1M ! M ! LdM ! cd+1M ! 0.
Diagram chasing then shows that there is a natural short exact sequence
0 ! nildM=nild+1M ! ker{LdM ! Ld1M} ! ker{cd+1M ! cdM} ! 0.
As the middle module here is N ild+1closed, and the right module is in N ild+1*
*, we
see that the left map identifies with ~d. Recalling that nildM=nild+1M = dRdM,
this says that there is a natural isomorphism
Ld( dRdM) ' ker{LdM ! Ld1M}.
The proof of the proposition is then completed by observing that Ld( dRdM) '
dR~dM, a consequence of the next proposition.
Proposition 3.3. There is a natural isomorphism Lc+d( dM) ' dLcM, for all
M 2 U.
Proof.We need to check that the map d~c : dM ! dLcM satisfies the two
properties characterizing localization away from N ilc+d+1.
That ker( d~c) and coker( d~c) are both in N ilc+d+1 is clear, as ker(~c) and
coker(~c) are both N ilc+1, and the dfold suspension of a module in N ilc+1 wi*
*ll
be in N ilc+d+1.
To see that the range of d~c is N ilc+d+1closed, we check that if M 2 U
is N ilc+1closed then dM is N ilc+d+1closed. This follows from the following
characterization of N ilc+1closed modules: M 2 U is N ilc+1closed if and only*
* if
it fits into an exact sequence of the form
Y Y
0 ! M ! H*(Vff) Mff! H*(Wfi) Nfi,
ff fi
with all the modules Mffand Nficoncentrated in degrees between 0 and c. See
[BrZ2 , Prop.1.15].
3.2. Further properties of Ld, Rd, and ~Rd. We need to recall some notation
and terminology. If V is an elementary pgroup, TV : U ! U is defined to be
the left adjoint to H*(V ) ___, as famously studied by Lannes [L1, L3]. Given*
* a
Noetherian unstable algebra K 2 K, Kf.g. U is defined to be the category studi*
*ed
in [HLS1 , I.4] whose objects are finitely generated Kmodules M whose Kmodule
structure map K M ! M is in U, and morphisms are Kmodule maps in U.
12 KUHN
Proposition 3.4. The functor Ld : U ! U satisfies the following properties.
(a) There are natural isomorphisms L0(M N) ' L0M L0N.
(b) There are natural isomorphisms TV LdM ' LdTV M.
(c) If K 2 K, then LdK 2 K, and K ! LdK is a map of unstable algebras. If
K is also Noetherian, and M 2 Kf.g. U, then LdK 2 Kf.g. U, and thus is
Noetherian, and LdM 2 LdKf.g. U.
Property (b) can be deduced from properties of TV as follows. First, to see t*
*hat
TV LdM is N ild+1closed, we compute, for s = 0, 1 and N 2 N ild+1:
ExtsU(N, TV LdM) = ExtsU(H*(V ) N, LdM) = 0,
since H*(V ) N will be in N ild+1 if N is. Second, TV ~d : TV M ! TV LdM is a
N ild+1isomorphism, as the kernel and cokernel are in N ild+1, since TV is exa*
*ct
and sends N ild+1 to itself.
See [HLS1 , I.4] and [BrZ1 ] for more detail about properties (a) and (c).
Proposition 3.5. 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) If K 2 K, then R0K 2 K, and K ! R0K is a map of unstable algebras. If
K is also 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 [K1 , x3], and the last follows easily from*
* the
first.
Proposition 3.6. 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) If K 2 K, then ~R0K 2 K, and K ! ~R0K is a map of unstable algebras. If
K is also Noetherian, and M 2 Kf.g. U, then ~R0K is also a Noetherian unstable
algebra, and ~RdM 2 ~R0Kf.g. U, for all d.
This, of course, follows from the previous two propositions.
A Noetherian unstable algebra K has a finite Krull dimension dimK. We have
an addendum to Proposition 3.4.
Proposition 3.7 ([K3 , Prop.4.10]). If an unstable algebra K is Noetherian, then
dimK = dimL0K.
Another special property of L0 that we will need goes as follows.
GROUP COHOMOLOGY PRIMITIVES 13
Proposition 3.8 ([L1, Lem.4.3.3]). Let f : M ! N be a map in K. Then
L0f : L0M ! L0N
is an isomorphism if and only if, for all pelementary abelian groups V , the i*
*nduced
map
f* : Hom K(N, H*(V )) ! Hom K(M, H*(V ))
is a bijection.
As in the introduction, given M 2 U, MLF denotes the submodule of locally
finite elements: x 2 M such that Ax M is finite.
Proposition 3.9. There is a natural isomorphism (RdM)0 = (R~dM)0 ' (MLF )d.
See [K1 , x3] for a proof.
Finally, Henn [H ] proved the following important finiteness result.
Proposition 3.10. Let K 2 K be Noetherian, and M 2 Kf.g. U. Then the M is
N ildlocal for d >> 0. In particular, the nilpotent filtration of M has finite*
* length.
3.3. Properties of d0M and d1M. The authors of [HLS1 ] define d0M and d1M
as follows.
Definition 3.11. Let M be an unstable module.
(a) Let d0M be the smallest d such that ~d is monic, or 1 if no such d exists.
Equivalently, d0M is the smallest d such that Hom U(N, M) = 0 for all N 2 N ild*
*+1,
or the smallest d such that nild+1M = 0. If M is nonzero, d0M is also the large*
*st
d such that RdM is nonzero, or the largest d such that ~RdM is nonzero.
(b) Let d1M be the smallest d such that ~d is an isomorphism, or 1 if no such d
exists. Equivalently, d1M is the smallest d such that ExtsU(N, M) = 0 for s = 0*
*, 1
and all N 2 N ild+1.
As fundamental examples, we have that d0H*(V ) = d1H*(V ) = 0 for all ele
mentary abelian pgroups V .
Proposition 3.12. Let M and N be unstable modules.
(a) For s = 0, 1, ds(M N) = max{dsM, dsN}.
(b) If M and N are nonzero, d0(M N) = d0M + d0N and d1(M N) =
max{d1M + d0N, d0M + d1N}.
(c) For s = 0, 1, dsTV M = dsM.
(d) If M is nonzero, for s = 0, 1, ds( nM) = dsM + n.
For properties (a) and (b) see [HLS1 , Prop.I.3.6]. Using the exactness of T*
*V ,
property (c) follows from Proposition 3.4(b). Property (d) follows from Proposi
tion 3.3.
14 KUHN
Proposition 3.13. Let 0 ! M1 ! M2 ! M3 ! 0 be a short exact sequence in U.
(a) For s = 0, 1, dsM2 max{dsM1, dsM3}. Furthermore, if dsM3 < dsM1, then
dsM2 = dsM1.
(b) d0M1 d0M2 and d1M1 max{d1M2, d0M3}. Furthermore, if d1M2 < d0M3,
then d1M1 = d0M3.
This is proved with straightforward use of the long exact Ext* sequence assoc*
*i
ated to a short exact sequence. Compare with [HLS1 , Prop.I.3.6].
Corollary 3.14. If M 2 U is reduced, then d1M = d0(L0M=M).
This follows by applying Proposition 3.13(b) to 0 ! M ! L0M ! L0M=M ! 0.
3.4. Basic properties of d0(G) and d1(G). By abuse of notation, if G is a finite
group, for s = 0, 1, we write ds(G) for dsH*(G). For example, d0(V ) = d1(V ) =*
* 0
for all elementary abelian pgroups V .
The properties of d0M and d1M presented above have the following immediate
consequences for d0(G) and d1(G)
Proposition 3.15. Let G and H be finite groups.
(a) d0(G x H) = d0(G) + d0(H).
(b) d1(G x H) = max{d1(G) + d0(H), d0(G) + d1(H)}.
(c) If P is a pSylow subgroup of G, then ds(G) ds(P ) for s = 0, 1.
(d) If V is a pelementary abelian subgroup of G, then ds(CG (V )) ds(G) for
s = 0, 1.
Properties (a) and (b) follow from Proposition 3.12 (b). As the unstable modu*
*le
H*(G) is a direct summand of H*(P ) if P is a pSylow subgroup, property (c) fo*
*l
lows from Proposition 3.12 (a). Similarly property (d) follows from Proposition*
* 3.12
(c), as H*(CG (V )) is a direct summand of TV H*(G) [L2]10.
4.Formulae for H*(G)LF and ~Rd(H*(G))
In this section we prove the formulae for H*(G)LF and ~RdH*(G) given in x2.
4.1. A formula for LdH*(G). The starting point for all of these are the followi*
*ng
constructions. Given a morphism ff : V1 ! V2 in A(G), there are maps
ff* : H*(V2) ! H*(V1),
ff* : H*(CG (V1)) ! H*(CG (V2)), and
m*ff: H*(CG (V2)) ! H*(V1) H*(CG (V2)).
____________
10It is unfortunate that this much referenced elegant 1986 preprint has neve*
*r been published.
GROUP COHOMOLOGY PRIMITIVES 15
Here ff* is induced by conjugation by g1 where g 2 G is any element11chosen so
that conjugation by g induces ff, and
mff: V1 x CG (V2) ! CG (V2)
is the homomorphism sending (x, y) to ff(x)y. We also let mV : V x V ! V denote
multiplication in an elementary abelian group V .
To state one of the formulae from [HLS1 ], we recall two other bits of notati*
*on
from x2. Given an unstable module M, we let M d denote M modulo degrees
greater than d. Given a category C, we let C# denote the associated twisted arr*
*ow
category: the objects of C# are the morphisms of C, and a morphism ff _ fi from
ff : A1 ! A2 to fi : B1 ! B2 is a commutative diagram in C
A1 __ff_//A2OO
 
 
fflfflfi
B1 _____//B2.
[HLS1 , Formula I(5.5.1)] now reads
Theorem 4.1. The homomorphisms V1 x CG (V2) mff!CG (V2) G induce an
isomorphism of unstable algebras from LdH*(G) to
~(ff)
limffEq { H*(V1) H d (CG (V2))___////_H*(V1) (H*(V1) H*(CG (V2)))}d,
V1!V2 (ff)
where ~(ff) is induced by 1 m*ff, (ff) is induced by m*V1 1, and the limit *
*is over
A(G)# .
4.2. A formula for ~RdH*(G). Recall our notation from x2: if W is a central ele
mentary abelian psubgroup of Q, then PW H*(Q) denotes the algebra of primitives
in the H*(W )comodule H*(Q).
Proposition 4.2. As unstable modules, ~RdH*(G) is naturally isomorphic to
limffH*(V1) Pff(V1)Hd(CG (V2)),
V1!V2
where the limit is over A(G)# .
Proof.Recall that dR~dM is the kernel of LdM ! Ld1M. As kernels commute
with limits and equalizers, it follows from the previous theorem that ~RdH*(G) *
*is
naturally isomorphic to
~(ff)
limffEq{ H*(V1) Hd(CG (V2))____////_H*(V1) (H*(V1) H*(CG (V2)))d},
V1!V2 (ff)
where ~(ff) is induced by 1 m*ffand (ff) is induced by m*V1 1. But now we
observe that the equalizer in this formula is precisely H*(V1) Pff(V1)Hd(CG (*
*V2)).
For (ff) is the composite
m*V1 1
H*(V1) Hd(CG (V2))!H*(V1) H*(V1) Hd(CG (V2))
truncate!H*(V 0 d
1) H (V1) H (CG (V2)),
____________
11This is well defined as any two choices will differ by an element of CG(V1*
*), and so will agree
on cohomology.
16 KUHN
and this identifies with
*
H*(V1) Hd(CG (V2)) 1ss!H*(V1) (H*(V1) H*(CG (V2)))d,
where ss : V1 x CG (V2) ! CG (V2) is the projection.
4.3. A formula for H*(G)LF .
Proposition 4.3. As unstable algebras, H*(G)LF is naturally isomorphic to
limffPff(V1)H*(CG (V2)),
V1!V2
where the limit is over A(G)# .
Proof.As there are no nonzero locally finite elements in "H*(V1) H*(CG (V2)), t*
*he
composite H*(G)LF H*(G) ! H*(CG (V2)) has image in Pff(V1)H*(CG (V2)) for
any ff : V1 ! V2 in A(G). Thus one gets a natural map of unstable algebras
H*(G)LF ! limffPff(V1)H*(CG (V2)).
V1!V2
That this is an isomorphism follows from Proposition 4.2, recalling that Propos*
*i
tion 3.9 said that there is a natural isomorphism (R~dM)0 ' (MLF )d.
4.4. Replacing A(G) with AC (G). Recall that C(G) denotes the maximal central
pelementary abelian subgroup of G, and AC (G) denotes the full subcategory of
A(G) with objects C(G) V < G.
Theorem 4.4. One can take the limit over AC (G)# , rather than A(G)# in The
orem 4.1, Proposition 4.2, and Proposition 4.3.
This will follow quite formally from the following simple observations. Let C*
* =
C(G). Given V < G, let CV < G be the subgroup generated by C and V . This
induces an evident functor C : A(P ) ! AC (G). Furthermore, the natural inclusi*
*on
V ! CV induces an identification CG (CV ) = CG (V ).
Given ff : V1 ! V2, let ffC : V1 ! CV2 be the evident map, and then let
ffofffoffC__gff//_Cff,
morphisms in A(G)# , correspond to the diagram in A(G)
V1_______V1_____//_CV1
ff ffC Cff
fflffl fflffl fflffl
V2_____//CV2_____CV2.
Lemma 4.5. Let F : A(G)# ! Fpvector spacesbe a contravariant functor such
that for all ff : V1 ! V2, F (fff) : F (ff) ! F (ffC ) is an isomorphism. Then*
* the
natural map
: lim F (ff) ! lim F (ff)
ff2A(G)# ff2AC(G)#
is an isomorphism.
Note that both F (V1 ff!V2) = H*(V1) and F (V1 ff!V2) = H*(CG (V2)) sat
isfy the hypothesis of the lemma. Theorem 4.4 then follows from the lemma, as
the relevant F 's are built from these two examples by constructions that prese*
*rve
isomorphisms.
GROUP COHOMOLOGY PRIMITIVES 17
Proof of Lemma 4.5.We define : lim F (ff) ! lim F (ff), an inverse
ff2AC(G)# ff2A(P)#
to , as follows. Given x = (xfi) 2 lim F (fi), let (x) = ( (x)ff) 2
Q fi2AC(G)#
ff2A(G)#F (ff), where (x)ff= F (fff)1F (gff)(xCff). One then checks that (*
*x) 2
lim F (ff), O = 1, and O = 1.
ff2A(G)#
4.5. Rewriting the formulae. If C is a small category, and
F : C# ! Fpvector spaces
is a contravariant functor, there is a canonical isomorphism
( )
Y ~ Y
limF = Eq F (1C ) !! F (ff),
C# C2obC ff2morC
where, given ff : C1 ! C2, the ffcomponent of ~ and are induced by applying F
to the canonical morphisms in C# from ff to 1C1 and 1C2 respectively.
Thus, for example, ~RdH*(G) will be naturally isomorphic to
( )
Y ~ Y
Eq H*(V ) PV Hd(CG (V )) !! H*(V1) Pff(V1)Hd(CG (V2)),
V ff:V1!V2
where ~ and are induced by
1 ff* : H*(V1) PV1Hd(CG (V1)) ! H*(V1) Pff(V1)Hd(CG (V2))
and
ff* i : H*(V2) PV2Hd(CG (V2)) ! H*(V1) Pff(V1)Hd(CG (V2))
for each ff : V1 ! V2. (i is the evident inclusion.)
Morphisms in A(G) factor as inclusions followed by isomorphisms induced by
the inner automorphism group Inn(G), so this last formula rewrites as follows.
Proposition 4.6. R~dH*(G) is naturally isomorphic to
8" # 9
< Y Inn(G) ~ Y =
Eq : H*(V ) PV Hd(CG (V )) !! H*(V1) PV1Hd(CG (V2)),
V V1> 0 [E ].
Recall that the extension corresponds to an element o 2 H2(Q; C), or equiva
lently a homomorphism o : H2(Q) ! C. Under the identification C# = H1(C), its
dual o# : C# ! H2(Q) corresponds to d2 : E0,12! E2,02.
5.1. H*(C)comodule structure of the spectral sequence. As C is central,
multiplication m : C x G ! G is a group homomorphism. The induced algebra
map
m* : H*(G) ! H*(C) H*(G),
makes H*(G) into a H*(C)comodule. The restriction i* : H*(G) ! H*(C) is
both an algebra and comodule map, and it follows that E0,*1= im(i*) is a subHopf
algebra of E0,*2= H*(C).
One can strengthen these last observations to statements about the whole spec
tral sequence. A good functorial model for BG, say the reduced bar construction,
shows that BC is an abelian topological group, BG is a BCspace equipped with
proper free action via Bm : BC x BG ! BG, and BG ! BQ is the associated
principal BCbundle. The Serre spectral sequence arises from the pullback to BG
of the skeletal filtration of BQ. This will be a filtration of BG by BCsubspac*
*es,
and we conclude the following.
Lemma 5.1. For all k and r, Ek,*ris an H*(C)comodule, such that the maps
dr : Ek,*r! Ek+r,*r
and
Ei,*r Ej,*r! Ei+j,*r
are maps of H*(C)comodules. In particular, E0,*ris a subHopf algebra of E0,*2=
H*(C).
GROUP COHOMOLOGY PRIMITIVES 19
5.2. A handy Hopf algebra lemma. We now digress to state and prove a handy
statement about (connected graded) Hopf algebras that we can apply to the situa
tion of the previous subsection.
We need some notation. Let H be a graded connected Hopf algebra over a field
F. There is a canonical splitting of vector spaces H = F I(H), where I(H) is *
*the
augmentation ideal. If M is a right Hmodule, let the module of indecomposables
be defined by QH M = M H F = M=MI(H). Dually, if M is a right Hcomodule,
let the module of primitives be defined by
PH M = Eq {M !!M H}
i
= ker{ ~: M ! M I(H)},
where : M ! M H is the comodule structure, i is the inclusion induced by the
unit F ! H, and ~ is the composite M ! M H ! M I(H).
Lemma 5.2. Let K be a subHopf algebra of a Hopf algebra H. Suppose M is si
multaneously an Hcomodule and Kmodule such that the Kmodule structure map
M K ! M is a map of Hcomodules. Then
(a) M is a free Kmodule, and
(b) the composite PH M ,! M i QK M is monic.
Remark 5.3. To put this in perspective, the lemma has long been known if K = H,
and, in this case, PH M ' QH M [Sw , Thm.4.1.1]. Our proof is very similar to
the proofs of Proposition 1.7 and Theorem 4.4 of Milnor and Moore's classic pap*
*er
[MM ]. Compare also to Green's lemma [G3 , Lem2.1].
Before proving the lemma, we note the following consequence. Given H and K as
in the lemma, let K H Mod be the category of M as in the lemma: an object is
a vector space M that is simultaneously an Hcomodule and Kmodule such that
the Kmodule structure map M K ! M is a map of Hcomodules. Morphisms
are linear maps that are both Kmodule and Hcomodule maps. K  H  Mod
is an abelian category in the obvious way.
Corollary 5.4. (a) Every short exact sequence 0 ! M1 ! M ! M2 ! 0 in
K  H  Mod is split as a sequence of Kmodules.
(b) The functor sending M to QK (M) is exact on K  H  Mod.
Proof of Lemma 5.2.Choose a section s : QK M ! M of the quotient ss : M !
QK M, and let ms : QK M I(K) ! MI(K) be the epimorphism given by
ms(x, k) = s(x)k. Statement (a) is asserting that ms is an isomorphism.
Let K : MI(K) ! M I(H) be the composite
~
MI(K) M ! M I(H),
Statement (b) asserts that PH M \ MI(K) = {0}, i.e. that K is monic.
Thus both statements will follow from the following claim:
K O ms : QK M I(K) ! M I(H)
is monic.
20 KUHN
To prove this claim, let FnM be the Ksubmodule of M generated byPelements
of degree upPto n. Given x 2 (QK M)n, and k 2 I(K), let (s(x)) = y0 h0,
and (k) = k0 k00. Then
K (ms(x, k))= ~(s(x)k)
s(x) k
modulo terms of the form y0k0 h0k00with either y0 < s(x) = n, or k0 2 I(K).
Otherwise said,
K (ms(x, k)) s(x) k mod (Fn1M + I(K)M) I(H).
Thus
ss( K (ms(x, k))) x k mod (QK M)<2pk1 ifa1 = 2pk withk 1
and we let e(G) = (ai 1) and h(G) = 1 ifa1 = 2
i=1 >:0 ifa1 = 1.
We have the following lemma about products.
Lemma 7.1. Suppose G0 and G1 have maximal central pelementary abelian sub
groups C0 and C1, and Duflot subalgebras A0 and A1. Then the following hold.
(a) C0 x C1 = C(G0 x G1), and
PC0xC1H*(G0 x G1) = PC0H*(G1) PC1H*(G1).
(b) A0 A1 will be a Duflot subalgebra for G0 x G1, and
QA0 A1H*(G0 x G1) = QA0H*(G1) QA1H*(G1).
(c) e(G0 x G1) = e(G0) + e(G1), and h(G0 x G1) = max{h(G0), h(G1)}.
Note that a subgroup H of a pcentral group G is again pcentral, and C(H) =
C(G) \ H. The next lemma is easily deduced.
Lemma 7.2. Let G be pcentral, and let A H*(G) be a Duflot subalgebra. If
j : H < G is a subgroup, then e(H) e(G), h(H) h(G), and j*(A) will be a
Duflot subalgebra of H*(H).
Thanks to this lemma, Corollary 2.10 immediately follows from Theorem 2.9.
Remark 7.3. The example H = Z=4 < Z=8 = G shows that the inequalities of the
lemma can be equalities, even when H is a proper subgroup of a pgroup G.
7.1. BensonCarlson duality. If G is pcentral, then QA H*(G) will be a finite
dimensional Fpalgebra if A is any Duflot subalgebra. Benson and Carlson tell us
much more:
Theorem 7.4. If G is pcentral and A is a Duflot subalgebra of H*(G), then
QA H*(G) is a Poincar'e duality algebra with top class in degree e(G).
GROUP COHOMOLOGY PRIMITIVES 29
Under the assumption that A is a polynomial algebra (always true if p = 2), t*
*his
is an immediate application of the main theorem in [BC1 ]. The general case red*
*uces
to this one: G and A will admit decompositions G = C0 x G1 and A = H*(C0)
A1, with C0 pelementary, G1 having no Z=p summands, and A1 a (necessarily
polynomial) Duflot subalgebra of H*(G1). Then QA H*(G) = QA1H*(G1), and
e(G) = e(G1).
7.2. Proof of Theorem 2.8. Let G be pcentral, C = C(G), and A H*(G) a
Duflot subalgebra. We now prove the various parts of Theorem 2.8.
Firstly, Theorem 7.4 implies that QA H*(G) is zero in degrees greater than e(*
*G),
and one dimensional in degree e(G).
Now consider the Serre spectral sequence for C ! G ! G=C, as studied in
Theorem 2.1. The bigraded algebra QE0,*1E*,*1is the graded object associated to
a decreasing filtration of the Poincar'e duality algebra QA H*(G) with top degr*
*ee
e(G). This forces the following to be true: there is a largest s, s(G), such th*
*at Es,*1
is nonzero, QE0,*1Es(G),*1will be one dimensional and concentrated in total deg*
*ree
e(G), and nonzero classes in Es(G),e(G)s(G)1 He(G)(G) will be Poincar'e duali*
*ty
classes.
These classes will also be H*(C)comodule primitives, as Es(G),*1is a sub
H*(C)comodule of H*(G), and everything in lowest degree must be primitive.
As PC H*(G) is contained in QA H*(G), we conclude that PC H*(G) is also zero in
degrees greater than e(G), and one dimensional in degree e(G).
By Corollary 2.4, PC He(G)H*(G) is also be the top nonzero degree of H*(G)LF ,
and so consists of classes annihilated by all positive degree Steenrod operatio*
*ns.
It remains to show that, under the additional assumption that G is a pgroup,
PC He(G)H*(G) is essential cohomology. This we prove in the next subsection.
7.3. pcentral pgroups and essential cohomology. Let P be a pcentral p
group. We have shown that He(P)(P )LF = PC(P)He(P)(P ) is a one dimensional
subspace of He(P)(P ).
Proposition 7.5. He(P)(P )LF is essential.
Proof.As P is a pgroup, maximal proper subgroups have the form j : Q < P ,
where Q is the kernel of a nonzero homomorphism x : P ! Z=p. We need to show
that j*(i) = 0 2 H*(Q) if i 2 He(P)(P )LF is nonzero.
The map j* : H*(P ) ! H*(Q) will take He(P)(P )LF to He(P)(Q)LF . If e(Q) <
e(P ), we are done: j*(i) will be an element of a zero group.
If e(Q) = e(P ), we reason as follows. Let A be a Duflot subalgebra of H*(G),
so that j*(A) is a Duflot subalgebra of H*(Q). If j*(i) 6= 0, it will project *
*to
a nonzero element in Qj*(A)H*(Q). We show that this is impossible. Regard
x as an nonzero element in H1(P ). By construction, j*(x) = 0 2 H1(Q). By
Poincar'e duality, there exists y 2 H*(P ) such that i = xy 2 QA H*(P ). But th*
*en
j*(i) = j*(x)j*(y) = 0 2 Qj*(A)H*(Q).
Let A(P, P ) be the two sided Burnside ring over Fp: the Fpalgebra with basis
given by equivalence classes of diagrams P Q ff!P , and multiplication defin*
*ed
using the double coset formula14. If J is the ideal generated by all such diagr*
*ams
____________
14There are more elegant descriptions, but this is better for our purposes.
30 KUHN
with ff not an isomorphism, then A(P, P )=J ' Fp[Out(P )], the group ring of the
outer automorphism group.
Using transfers (a.k.a. induction), A(P, P ) acts on H*(P ), with a basis ele*
*ment
* TrPQ
[P Q ff!P ] inducing H*(P ) ff!H*(Q) ! H*(P ). As these are unstable A
module maps, it follows that He(P)(P )LF is a one dimensional A(P, P )submodul*
*e.
Corollary 7.6. The ideal J acts trivially on He(P)(P )LF .
Proof.The previous proposition shows that if a homomorphism ff : Q ! P is not
onto, then ff*(He(P)(P )LF ) = 0.
It follows that the A(P, P )module He(P)(P )LF is the pullback of a one dime*
*n
sional representation of Out(P ) over the prime field Fp. We let !(P ) denote t*
*his
representation. Clearly !(P ) will be trivial if p = 2, but this need not be th*
*e case
when p is odd.
Example 7.7. Let p = 3. Then !(Z=9) = H1(Z=9) is nontrivial, as 1 : Z=9 !
Z=9 induces multiplication by 1 on H1(Z=9).
7.4. d0(G) when G has a pcentral pSylow subgroup. We prove the parts of
Theorem 2.9 involving d0.
Firstly, if G is pcentral, then Corollary 2.6 says that
R~dH*(G) ' H*(C(G)) PC(G)Hd(G).
Since d0(G) is the largest d such that ~RdH*(G) 6= 0, it follows that d0(G) wil*
*l equal
the top nonzero degree of PC(G)H*(G), which we have computed to be e(G).
Now suppose that G is not necessarily pcentral, but has a pcentral pSylow
subgroup P . We show that then d0(G) = d0(P ).
We need to show that the largest d such that ~RdH*(G) 6= 0 is d = d0(P ) = e(*
*P ).
Let e1 2 A(P, P ) be an idempotent chosen so that A(P, P )e1 is the projective *
*cover
of ffl, the trivial Fp[Out(P )]module, pulled back to A(P, P ). Standard argum*
*ents
show that there are inclusions
e1R~dH*(P ) ~RdH*(G) ~RdH*(P ).
Thus it suffices to show that e1R~e(P)H*(P ) 6= 0. Otherwise said, it suffices *
*to show
that ffl is a composition factor in the A(P, P )modules ~Re(P)H*(P ).
If p = 2, we are done: by Corollary 7.6, ~Re(P)He(P)(P ) = He(P)(P )LF ' ffl.*
* As
a bonus, we learn that H*(G)LF is one dimensional in degree e(P ).
When p is odd, more care (and maybe luck) is needed. Recall that ~Re(P)H*(P )*
* =
H*(C(P )) He(P)(P )LF . The fact that J acts as 0 on He(P)(P )LF implies that
same is true for H*(C(P )) He(P)(P )LF . Thus we just need to show that the
trivial Out(P )module occurs as a composition factor in H*(C(P )) He(P)(P )LF =
H*(C(P )) !(P ), or, equivalently, that !(P )1 occurs as an Out(P )composit*
*ion
factor H*(C(P )). We are done with the following lemma15.
Lemma 7.8. If P is a pcentral pgroup with p an odd prime, then every irreduci*
*ble
Fp[Out(P )]module occurs as a composition factor of H*(C(P )).
The lemma, in a stronger form than stated, follows by combining [K3 , Prop.5.7
and Cor.6.8]. The key point is that, since C(P ) = 1(P ), the kernel of Aut(P *
*) !
Aut(C(P )) will be a pgroup if p is odd [Gor, Thm.5.3.10].
____________
15This lemma is false if p = 2, as the example P = Q8 illustrates.
GROUP COHOMOLOGY PRIMITIVES 31
Example 7.9. Let p = 3 and G be the semidirect product Z=9 o Z=2. Then
d0(G) = d0(Z=9) = 1, but "H*(G)LF = 0.
7.5. d1(G) when G is pcentral. In this subsection, let G be pcentral. We show
that d1(G) = e(G) + h(G).
We get control of d1(G) by working directly with the desuspended composition
factors RdH*(G) of H*(G), rather than their N il1localizations ~RdH*(G), as was
done in our calculation of d0(G).
To simplify notation, write Rd for RdH*(G), ~Rdfor ~RdH*(G), and C for C(G).
We have that ~R0= H*(C), and R0 = im(i*), where i : C ,! G is the inclusion.
In the nilpotent filtration of H*(G), the last nonzero submodule, nile(G)H*(G*
*) =
e(G)Re(G), has been shown to be isomorphic to e(G)R0 as an unstable module.
Thus d1 of this submodule of H*(G) equals e(G) + d1(R0). By Lemma 3.13, the
next lemma implies that if d < e(G), d1( dRd) is strictly smaller than this.
Lemma 7.10. Each Rd with d < e(G) admits a filtration by unstable modules with
subquotients all of the form kR0 with d + k < e(G).
Again appealing to Lemma 3.13, we then have the next corollary.
Corollary 7.11. d1(G) = d1( e(G)Re(G)) = e(G) + d1(R0).
Thus we will have proved that, when G is pcentral, d1(G) = e(G) + h(G), once
we have proved Lemma 7.10, and calculated that d1(R0) = h(G). We begin the
proof of Lemma 7.10 here, and then both finish it, and calculate d1(R0), in the*
* two
subsections that follow, which correspond to the cases p = 2 and p odd.
Proof of Lemma 7.10.For all d, we have inclusions
R0 PC Hd(G) Rd ~R0 PC Hd(G),
where PC Hd(G) is regarded as an unstable module concentrated in degree 0. These
are inclusions of unstable modules, enriched with compatible R0module structur*
*es
and ~R0comodule structures. Call the category of such objects R0  ~R0 U.
Say that M 2 R0  ~R0 U admits a nice filtration if it admits a filtration in
R0  ~R0 U with subquotients all of the form kR0. We will show that each Rd
admits a nice filtration.
(That the composition factors will then also satisfy d + k < e(G) follows imm*
*e
diately from the fact that QR0( dR*) is a graded object associated to QA H*(G),
which we know is one dimensional in degree e(G) and zero above that.)
We claim that, if N admits a nice filtration, and M N, then M also admits
a nice filtration. To see this, suppose F0N F1N . .i.s a filtration of N wi*
*th
FjN=Fj1N = kjR0. Let FjM = M \FjN. Then FjM=Fj1M FjN=Fj1N =
kjR0 will be an inclusion of objects in R0R~0U that will be split as R0modu*
*les,
thanks to Corollary 5.4. We conclude that FjM=Fj1M is either 0 or kjR0.
Thus to prove Rd has a nice filtration, it suffices to prove that ~R0 PC Hd(*
*G)
has a nice filtration, or just that ~R0has a nice filtration. We show this in t*
*he next
two subsections, which separately deal with the cases p = 2 and p is odd.
7.6. A calculation of d1(R0), and a nice filtration of R~0, when p = 2.
Suppose that p = 2, and that
j1 2jc
R0 = F2[x21 , . .,.x1 ] F2[x1, . .,.xc] = ~R0,
with j1 . . .jc. We show the following.
32 KUHN
Lemma 7.12. R~0has a good filtration as an object in R0  ~R0 U.
Lemma 7.13. If j1 > 0, d1(R0) = 2j11.
Proof of Lemma 7.12.For all 1 b c and 1 ib jb, the module
i1 2ic
R(i1, . .,.ic) = F2[x21, . .,.xc ]
will be an object in R0  ~R0 U in the evident way.
Clearly R(j1, . .,.jc) = R0 admits a nice filtration. The short exact sequenc*
*es
ib
0 ! R(i1, . .,.ic) ! R(i1, . .,.ib 1, . .,.ic) ! 2 R(i1, . .,.ic) ! 0
then shows that if R(i1, . .,.ic) admits a nice filtration, so does R(i1, . .,.*
*ib 
1, . .,.ic). By downward induction, we conclude that R(0, . .,.0) = R~0 admits
a nice filtration.
Proof of Lemma 7.13.By Proposition 3.12(c), it suffices to prove that, if j > 0,
j j1
d1(F2[x2 ]) = 2 .
In the short exact sequence
j 2j1 2j1 2j
0 ! F2[x2 ] ! F2[x ] ! F2[x ] ! 0,
d1 of the middle term is strictly less than d0( 2j1F2[x2j]) = 2j1: this is cl*
*ear
when j = 1, and for larger j this follows byjan inductive hypothesis. Thus Prop*
*o
sition 3.13(b) applies to say that d1(F2[x2 ]) = 2j1.
7.7. A calculation of d1(R0), and a nice filtration of R~0, when p is odd.
Suppose that p is odd. We can assume that
j1 pjc *
R0 = Fp[yp1 , . .,.y1 ] (x1, . .,.xc) Fp[y1, . .,.yc] = ~R0,
with j1 . . .jc, and we show the following.
Lemma 7.14. R~0has a good filtration as an object in R0  ~R0 U.
Lemma 7.15. If j1 = 0, d1(R0) = 1. If j1 > 0, d1(R0) = 2pj11.
Proof of Lemma 7.14.As a first step, we note that the filtration of ~R0given by*
* let
ting FkR~0= k(x1, . .,.xc) Fp[y1, . .,.yc] is a filtration in the category R0*
*R~0
U, and the associated subquotients are direct sums of suspensions of Fp[y1, . .*
*,.yc].
It follows that it suffices to prove the lemma with ~R0replaced by Fp[y1, . .,.*
*yc].
Our next reduction will allow us to reduce to the case when c = 1.
If K is a subHopf algebra of a Hopf algebra H, and both objects and all struc*
*ture
maps are in U, one has a category K  H  U, analogous to R0  ~R0 U. One
can then say that M 2 K  H  U has a good filtration if it has a filtration wi*
*th
subquotients that are all suspensions of K. It is easy to see that if M1 2 K1H*
*1U
and M2 2 K2  H2  U has a good filtration, then so does M1 M2, viewed as an
object in K1 K2  H1 H2  U.
Applying this observation to the evident tensor decompositions of K = R0 and
H = Fp[y1, . .,.yc], we are left just needing to show that Fp[y] has a nice fil*
*tration,
when viewed as an object in Fp[ypj]  Fp[y]  U.
By downwards induction on i, we show that, for 0 i j, Fp[ypi] has a nice
filtration, when viewed as an object in Fp[ypj]  Fp[y]  U. The case i = j is *
*clear.
GROUP COHOMOLOGY PRIMITIVES 33
For the inductive step, we filter Fp[ypi]. For 0 r p  1, define M(r) to
be the span of {ypim  m s modPp, for some0 s r}. Using the formulae
Pkyn = nkyn+k(p1)and (yn) = k nkyk ynk, one easily checks that each
M(r) is an object in Fp[ypj]  Fp[y]  U: if nk6 0 mod p, and n has the form
pi(pa + s) with 0 s r p  1, then both n + k(p  1) and k also have this *
*form.
Thus we have a filtration in Fp[ypj]  Fp[y]  U:
i+1 pi
Fp[yp ] = M(0) M(1) . . .M(p  1) = Fp[y ],
and we are assuming by induction that Fp[ypi+1] has a good filtration. Now one
checks that M(r)=M(r  1) ' 2pirM(0) as objects in Fp[ypj]  Fp[y]  U, so by
upwards induction on r we conclude that each M(r) has a good filtration.
Proof of Lemma 7.15.By Proposition 3.12(c), it suffices to prove that d1(Fp[y])*
* =
1, and, if j > 0, d1(Fp[ypj]) = 2pj1.
Corollary 3.14 (or Proposition 3.13(b)), applied to the short exact sequence
0 ! Fp[y] ! *(x) Fp[y] ! Fp[y] ! 0,
shows that d1(Fp[y]) = d0( Fp[y]) = 1.
If j > 1, we consider the short exact sequence:
j pj1 pj1 pj
0 ! Fp[yp ] ! Fp[y ] ! Fp[y ]=Fp[y ] ! 0.
We claim that d0 of the last term is 2pj1, which, by induction, will be strict*
*ly
more than d1 of the middle term. Thus Proposition 3.13(b) applies to say that
d1(Fp[ypj]) = 2pj1.
To verify the claim, one checks that the map Fp[ypj1] ! 2pj1Fp[ypj1] send*
*ing
ypj1nto the 2pj1th suspension of nypj1(n1)isjamap1ofjunstablejAmodules,1*
*j1
and thus induces an embedding Fp[yp ]=Fp[yp ] ,! 2p Fp[yp ] in U. Since
the range of this embedding is the 2pj1th suspension of a reduced module, the
same is true of the domain, which thus has d0 = 2pj1.
7.8. d1(G) when G has a pcentral pSylow subgroup. Now suppose that G
is not necessarily pcentral, but has a pcentral pSylow subgroup P . Here we *
*show
that then d1(G) = d1(P ). As d1(G) d1(P ) is always true, the point is to show
that d1(G) is as big as it could be.
Let e! 2 Fp[Out(P )] be an idempotent chosen so that Fp[Out(P )]e! is the
projective cover of the one dimensional module !(P )1.
Lemma 7.16. d1(G) = d1(P ) if and only if d1(Re(P)H*(G)) = h(P ), and either
of these equalities are implied by d1(e!R0H*(P )) = h(P ).
Proof.As we proved that d1(P ) = e(P )+h(P ), we showed that d1(nile(P)H*(P )) =
e(P ) + h(P ) and d1(H*(P )=nile(P)H*(P )) < e(P ) + h(P ). This second fact i*
*m
plies that d1(H*(G)=nile(P)H*(G)) < e(P ) + h(P ) also holds, as H*(G) is a di
rect summand of H*(P ) in U. We conclude that d1(G) = d1(P ) if and only if
d1(nile(P)H*(G)) = e(P ) + h(P ). As d1(nile(P)H*(G)) = e(P ) + d1(Re(P)H*(G)),
we deduce that d1(G) = d1(P ) if and only if d1(Re(P)H*(G)) = h(P ).
Now reasoning as in x7.4, this last equality would follow if one could show t*
*hat
d1(e!R0H*(P )) = h(P ).
34 KUHN
We now sketch a proof that d1(e!R0H*(P )) = h(P ). This involves redoing the
calculation that d1(R0H*(P )) = h(P ) in a way that allows one to keep track of*
* the
Out(P )action. Let C = C(P ).
The 0line of the spectral sequence associated to C ! P ! P=C is natural with
respect to the action of Out(P ). Thus the filtration studied in x6,
H*(C) = E0,*2 E0,*3 E0,*2p+1 . . .E0,*2pk1+1 E0,*2pk+1= E0,*1= R0H*(P ),
is a filtration by unstable modules with an Out(P )action.
Our work above shows that
8
><2pj1 ifr = 2pj+ 1 withj 1
d1(E0,*r) = >1 ifr = 3
:0 ifr = 2.
We now suppose that p > 2 and k 1: the cases when p = 2 or when E0,*1
equals E0,*2or E0,*3are similar and easier. Recall that then h(P ) = 2pk1. Usi*
*ng
Proposition 3.13 in the usual way, we conclude that d1(e!R0H*(P )) = h(P ) if a*
*nd
only if d0(e!B) = 2pk1, where B = E0,*2pk1+1=E0,*2pk+1.
From x6, we see that
(7.1) B = *(C#0) S*(fi(C#1) + fi(C#2) + . .+. k1fi(C#k))
S*( k1fi(C# =C#k))=S*( kfi(C# =C#k)),
where C#0 C#1 . . .C#k C# is a filtration of C# as an Out(P )module.
As an unstable module, B thus has the form
M (S*( k1fi(V ))=S*( kfi(V ))),
where M is reduced. Now one observes that S*(fi(V ))=S*( fi(V )) = 2N where
N is reduced. Thus
S*( k1fi(V ))=S*( kfi(V=)) k1(S*(fi(V ))=S*( fi(V )))
= k1( 2N)
k1 k1
= 2p ( N),
which is the 2pk1st suspension of a reduced module.
We conclude that d0(e!B) = 2pk1 if and only if e!B is nonzero.
The image of Out(P ) ! GL(C) lands in the parabolic subgroup GL(C, P ) re
specting the filtration of C, and the idempotent e! will project to a nonzero i*
*dem
potent in Fp[GL(C, P )].
We claim that if e 2 Fp[GL(C, P )] is any nonzero idempotent, then e acts non
trivially on B as described in (7.1). Equivalently, we claim that all irreduci*
*ble
Fp[GL(C, P )]modules occur as composition factors in B.
To prove the claim, we note that all irreducible GL(C, P ) modules will be pu*
*ll
backs from the associated Levi factor (i.e. the product of `block diagonal' GL(*
*Vj)'s),
as the projection from the one to the other has kernel which is a pgroup. This
reduces us quickly to verifying the following lemma.
Lemma 7.17. Every irreducible Fp[GL(V )]module occurs as a composition factor
in S*(fi(V ))=S*( fi(V )).
GROUP COHOMOLOGY PRIMITIVES 35
Proof.It is well known that every such irreducible S occurs in S*(fi(V )). Choo*
*sing
an occurrence of lowest polynomial degree, it is clear that it will remain nonz*
*ero in
the quotient S*(fi(V ))=S*( fi(V )).
8.Central essential cohomology
Recall that Cess*(G) is defined to be the kernel of the restriction map
Y
H*(G) ! H*(CG (U)).
C(G) r  c), and let ae be the regular repres*
*en
tation of G=C. For 1 i r  c, let ~~i2 H2(pnpni)(G=C) be the (pn  pni)th
Chern class of ae, and then let ~i = InfGG=C(~~i) 2 H*(G). It is easy to check
that ,1, . .,.,c, ~1, . .,.~rc is a polarized system of parameters in the sens*
*e of [G1 ,
Def.2.2]. It follows that the element , = ~1 satisfies the conclusion of the le*
*mma.
Proposition 8.9. Suppose c < r and A = Fp[,1, . .,.,c] is a Duflot algebra of
H*(G). The following are equivalent for a fixed integer e 0.
(a) e0(G) < e.
(b) With , as in the lemma, the kernel of multiplication by ,,
ker{,. : Hd(G)=(,1, . .,.c) ! Hd+,(G)=(,1, . .,.c)},
is zero for all d e.
"
(c) ker{,. : Hd(G)=(,1, . .,.c) ! Hd+,(G)=(,1, . .,.c)}
,2H"*(G)
is zero for all d e.
Proof.For each d and , 2 H*(G), we have a commutative diagram
f(d) L d
(8.1) Hd(G)=(,1, . .,.c)_____// C d e. Given 0 6= ~ 2 Hd(G)=(,1, . .,.c), we need to show *
*that
f(d)(~) 6= 0. By (c), there exists , 2 "H*(G) such that , . ~ 6= 0. As f(d + ,*
*) is
monic by inductive assumption, f(d + ,)(, . ~) 6= 0. But this equals , . f(d)*
*(~),
and so f(d)(~) 6= 0.
Proof of Theorem 2.13.With notation as in the proposition just proved, we wish
to prove that Cess*(G) = 0 if and only if the depth of H*(G) is greater than c.
Thanks to Theorem 2.12, Cess*(G) = 0 if and only if statement (a) of the last
proposition holds when e = 0. But then statement (b) is true with e = 0, and th*
*us
the depth of H*(G) is at least c + 1.
Conversely, if the depth of H*(G) is at least c + 1, there exists a , such th*
*at
,1, . .,.,c, , is a regular sequence on H*(G), and so statement (c) certainly h*
*olds
with e = 0. Thus statement (a) does as well.
Now we study statements (b) and (c) of the last proposition, using work by
Carlson and Benson.
Proposition 8.10. If r  c = 1, then QA Cess*(G) satisfies Poincar'e dual
ity with duality degree equal to e(G). In other words, the Poincar'e polynomial
pQACess*(G)(t) satisfies
pQACess*(G)(t) = te(G)pQACess*(G)(1=t).
Proof.The conclusion of the proposition is obvious if Cess*(G) = 0, so we can
assume that the depth of H*(G) is precisely c. Let ,1, . .,.,c be as in Propos*
*i
tion 8.9, and choose , as in the lemma. Replacing , by a large power of itself,*
* if
necessary, we can assume that, in diagram (8.1), f(d + ,) is monic for all d.*
* Thus
QA Cess*(G), the kernel of the top map in (8.1), identifies with the kernel of *
*multi
plication by ,, the left map in (8.1). But a careful reading of [BC2 , Lemma 3.*
*2 and
its proof] reveals that the Poincar'e series of this kernel is precisely the po*
*lynomial
called `pr(t)' there, and then [BC2 , Theorem 3.9] says that the functional equ*
*ation
of the proposition holds.
Corollary 8.11. If r  c = 1, then
e0(G) = e(G)  min{d  Cessd(G) 6= 0} < e(G).
To state what we know about the situation when r c > 1, we need to introduce
local cohomology. If I is a homogeneous ideal in a graded ring R, and M is a gr*
*aded
Rmodule, H0,*I(M) is defined to be the Itorsion in M, i.e. the set of x 2 M s*
*uch
that Ikx = 0 for some k. This is a left exact functor of M, and Hd,*I(M) is def*
*ined
to be the associated dth right derived functor.
In [Be], Benson conjectured
Conjecture 8.12 (Strong Regularity Conjecture).
(
Hi,j"H*(G)(H*(G)) = 0 for j i ifc i < r
j > i ifi = r.
Proof of Proposition 2.19.This proposition asserted that, for a fixed finite gr*
*oup
G, Conjecture 2.18 is implied by the Strong Regularity Conjecture.
GROUP COHOMOLOGY PRIMITIVES 41
q 

2 
1  u, vxv
_0_1__x,_y_________
0 1 2 p
Figure 1. Ep,q3= Ep,q1modulo (a2, b2, c2)
In the terminology of [Be], if the Strong Regularity Conjecture holds, then, *
*by
[Be, Thm.4.5], every filter regular sequence is of type beginning with the sequ*
*ence
(1, 2, . .,.(c + 1)). In particular, with , as in statement (b) of Propositi*
*on 8.9,
the sequence ,1, . .,.,c, , is the beginning of such a sequence. From the defin*
*ition
of filter regular, we see that statement (b) of Proposition 8.9 thus holds with*
* e =
e(G).
As mentioned in the introduction, in [Be], Benson shows that his conjecture is
true if r  c 2. I have my own `heuristic' proof of statement (c) with e = e(*
*G)
under the same condition, and the failure of the method to go beyond r  c 2
makes one wonder if a counterexample to both of our conjectures is lurking among
the groups of order 128 or 256.
Remark 8.13. Slightly milder than the Strong Regularity Conjecture is Benson's
Regularity Conjecture, which asserts that the CastelnuovoMumford regularity of
H*(G) is precisely 0. In terms of local cohomology, this is the statement that
Hi,j"H*(G)(H*(G)) = 0 for j > i. For our purposes, this is enough to deduce th*
*at
e0(G) e(G).
9.Examples
Example 9.1. Let W (2) be the universal 2central group whose quotient by its
center C is V2 = (Z=2)2. Thus there is a central extension
H2(V2; F2) i!W (2) q!V2,
where C = H2(V2; F2) ' (Z=2)3. In terms of HallSenior numbering, and thus also
the numbering in [CTVZ ], W (2) is 32#18.
In the associated spectral sequence, one has that
E*,*2= F2[x, y, a, b, c],
with a, b, c 2 E0,12and x, y 2 E1,02, and d2(a) = x2, d2(b) = xy, and d2(c) = y*
*2.
As E*,03= F2[x, y]=(x2, xy, y2), it follows that a2, b2, and c2 must be perma
nent cycles. We conclude that W (2) will have type [2, 2, 2] so that d0(W (2))*
* =
e(W (2)) = 3 and d1(W (2)) = 4.
With a bit more work, one can show that E*,*3=(a2, b2, c2) is six dimensional*
* with
generators as indicated in Figure 1, where u and v are respectively represented*
* by
bx + ay and cx + by. This is a Poincar'e duality algebra with relations x2 = xy*
* =
y2 = u2 = v2 = uv = xu = yv = xv + yu = 0.
It follows that E*,*3= E*,*1, and then that
H*(W (2)) ' F2[ff, fi, fl, x, y, u, v]=(x2, xy, y2, u2, uv, v2, xu, yv, xv*
* + yu),
42 KUHN
with x, y 2 H1 and ff, fi, fl, u, v 2 H2. Here ff, fi, and fl are represented b*
*y a2, b2,
and c2 in the spectral sequence.
The polynomial subalgebra A = F2[ff, fi, fl] is a Duflot subalgebra. With res*
*pect
to the H*(C) = F2[a, b, c] comodule structure, the elements 1, x, y are in the *
*image
of the inflation map q* and so are primitive. The top class xv is not in the im*
*age
of inflation, but is primitive, by our general theory. The elements u and v are*
* not
primitive, as m*(u) = 1 u + b x + a y and m*(v) = 1 v + c x + b y in
H*(C) E*,*1. Thus each of the inclusions
im(q*) ,! PC H*(W (2)) ,! QA H*(W (2))
is proper.
The nilpotent filtration works as follows.
R0 = H*(W (2))=(x, y, u, v) ' F2[ff, fi, fl] and ~R0= F2[a, b, c]. The embedd*
*ing
R0 ~R0sends ff to a2, fi to b2, and fl to c2.
R1 is the free F2[ff, fi, fl]module on generators ~x, ~y, ~u, ~vof respectiv*
*e degrees
0,0,1,1, and R~1is the free F2[a, b, c]module on ~x, ~y. The embedding R1 R*
*~1
sends ~uto b~x+ a~yand ~vto c~x+ b~y. Thus Sq1(~u) = fi~x+ ff~yand Sq1(~v) = fl*
*~x+ fi~y.
R2 and ~R2are both 0, as PC H2(W (2)) = 0.
R3 is the free F2[ff, fi, fl]module on a single generator ~x~vof degree 0, a*
*nd ~R3is
the free F2[a, b, c]module on this same element.
Finally, H*(W (2))LF H*(W (2)) is the algebra spanned by 1, x, y, xv. All
nontrivial products and Steenrod operations are zero.
Example 9.2. Let G be the group of order 64 with HallSenior number #108.
Using information from [CTVZ ], we analyzed H*(G) in detail for other purposes*
* in
[K3 ]. Here we summarize relevant bits to illustrate how one can calculate H*(G*
*)LF
and ~RdH*(G) by using Proposition 4.7 and Proposition 4.6.
The commutator subgroup Z = [G, G] has order 2. The center C is elementary
abelian of rank 2, and C = (G), so Z < C and G=C is elementary abelian of
rank 4. There is a unique maximal elementary abelian group V of rank 3, and its
centralizer K has order 32, so that NG (V )=CG (V ) = G=K ' Z=2. More precisely,
K is isomorphic to (Z=2)2 x Q8, with Q8 embedded so that V \ Q8 = Z.
We have the following picture of AC (G):
__________________________________________*
*_____________
C _____//VffZ=2_____________________________________*
*___________________________
and from this it is already clear that ~R0H*(G) = H*(V )Z=2.
We have maps of unstable algebras equipped with Aut(G) action:
*
PV H*(K) ,! PC H*(K) j PC H*(G),
where j : K ! G is the inclusion. It is easily checked that j* is onto in degre*
*e 1.
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. B* has
dimension 1,2,2,1 in degrees 0,1,2,3.
GROUP COHOMOLOGY PRIMITIVES 43
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*(G) ! PC H*(K) *
*is
onto, and then that Inn(G) acts trivially on both PV H*(K) and PC H*(K).
Proposition 4.7 then tells us that there is a pullback diagram of unstable al*
*gebras:
H*(G)LF _____//_PC H*(G)
 *
 j
fflffl fflffl
PV Hd(K) _____//PC H*(K).
Similarly, Proposition 4.6 tells us that, for all d, there is a pullback diag*
*ram of
unstable modules:
R~dH*(G) __________//_H*(C) PC Hd(G)
  *
 1 j
fflffl fflffl
H*(V )Z=2 PV Hd(K)_____//H*(C) PC Hd(K).
Note that the kernel of j* : H*(G) ! H*(K) is precisely Cess*(G), which is
described in [CTVZ ]. In our terminology, we learn that a Duflot subalgebra A *
*is
polynomial on classes of degree 2 and 8 (so G has type [8, 2]), and QA Cess*(G)*
* is
a graded vector space of dimension 1,3,5,6,5,3,1 in degrees 1,2,3,4,5,6,7. Note*
* that
this evident Poincar'e duality is predicted by Proposition 8.10.
QA Cess*(G) has a basis in which every element is a product of 1 dimensional
classes, and thus PC Cess*(G) = QA Cess*(G). In [K3 , Prop.10.2], we further
showed that
PC Cess*(G) ' B*[y]=(y4)
as unstable modules.
It follows that there are short exact sequences in U:
0 ! B*[y]=(y4) ! H*(G)LF ! B* ! 0,
and
0 ! H*(C) [ B*[y]=(y4)]d ! ~RdH*(G) ! H*(V )Z=2 Bd ! 0.
Furthermore, d0(G) = e00(G) = e0(G) = 7, and e(G) = 8.
Example 9.3. One can often determine e(G) using minimal information about the
extension class o* : C* ! H2(G=C) (where C = C(G)), and in situations where
H*(G) has yet to be calculated.
For example, suppose that p = 2 and G has no Z=2 direct summands (so that
o* is monic). If the image of o* has a basis consisting of products of 1 dimens*
*ional
classes, then G has type [2, . .,.2] and so e(G) equals the rank of C. To see t*
*his,
we note that, if d2(a) = xy, then
d3(a2) = d3(Sq1a) = Sq1(d2(a)) = Sq1(xy) = x2y + xy2 0 mod (xy).
This criterion holds for the important family of groups studied in [AKM ]. T*
*here
the authors associate a 2central Galois group GF to every field F of character*
*istic
different from 2 that is not formally real. They call this group a W group due
to its connections to the Witt ring W F [MiS ]. Thus d0(GF) = e(GF) = r and
44 KUHN
q 

8  2
7 
6  6 8 8 8 4 1
5 
4  8 12 8 7 4
3 
2  4 7 8 8 8
1 
_0_1__4__8__10__8_6_______________
0 1 2 3 4 5 6 7 8 p
Figure 2. The dimension of Ep,q1modulo (a8, b8)
d1(GF) = r+1, where GF has rank r. Included among these groups are the universal
W groups W (n), the 2central group with extension sequence
H2((Z=2)n; F2) ! W (n) ! (Z=2)n.
n+1 n+1
Thus d0(W (n)) = 2 and d1(W (n)) = 2 + 1.
At odd primes p, analogous criteria exist, ensuring that G is pcentral of ty*
*pe
[2, . .,.2]. Interesting families of such groups were studied by BrowderPakian*
*athan
[BP ] and AdemPakianathan [AP ]. Included among these are the universal groups
W (n, p), with extension sequence
H2((Z=p)n; Fp) ! W (n, p) ! (Z=p)n.
n+1 n+1
Thus d0(W (n, p)) = 2 and d1(W (n, p)) = 2 + 1.
Example 9.4. Compared to the families in the last example, at the other extreme
among 2central 2groups is the 2Sylow subgroup P of the simple group SU(3, 4).
This group has order 64 and HallSenior number #187. Its center C is elementary
abelian of rank 2. In [G2 ], Green analyzed the associated spectral sequence17.*
* In
particular, P has type [8, 8], so that d0(P ) = e(P ) = 14 and d1(P ) = 18, and*
* the
analogue of Figure 1 is the impressively complex Figure 2 (reproduced from [G2 *
*]).
In spite of this complexity, it is interesting to note that one can get the b*
*ound
e(P ) 14 quite easily, by using representation theory and characteristic clas*
*ses.
We thank David Green for the following description of some complex represen
tations of P . Let H*(C) = F2[a, b], and then let aea and aeb be the 1dimensio*
*nal
complex representations of C with respective total StiefelWhitney classes w(ae*
*a) =
1 + a2, w(aeb) = 1 + b2. These representations extend to 1dimensional represen
tations "aeaand "aebof subgroups Qa and Qb of index 4 in P . Let !a and !b be t*
*he
4 dimensional representations one gets by inducing "aeaand "aebup to P : these *
*turn
out to be irreducible.
By construction ResPC(!a) = 4aea and ResPC(!b) = 4aeb. It follows that the to*
*tal
StiefelWhitney classes of !a and !b restrict to (1 + a2)4 = 1 + a8 and (1 + b2*
*)4 =
1 + b8 in H*(C). Thus imResPCcontains F2[a8, b8] and so e(P ) 14 must hold.
____________
17A key simplification comes by computing with F4 coefficients rather than F*
*2 coefficients.
GROUP COHOMOLOGY PRIMITIVES 45
Alternatively, one can just use the single 8 dimensional representation !a *
*!b.
This is faithful, as it is faithful when restricted to C, the subgroup of all e*
*lements of
order 2. It has characteristic classes that restrict to a8+b8 and a8b8 in H*(C)*
*. From
this, one can formally deduce that the special Hopf algebra imResPCmust contain
F2[a8, b8], so that d0(P ) 14 and d1(P ) 18. By contrast, the estimate of H*
*enn,
Lannes, and Schwartz in [HLS1 ] just lets one conclude that d0(P ) 64 and d1(*
*P )
120 if one knows that P has a faithful 8 dimensional complex representation. Th*
*is
suggests that there might be some general bounds for ds(G) for an arbitrary gro*
*up
G, determined by the dimensions of its faithful representations, that are much
better than those in [HLS1 ].
46 KUHN
Appendix A. Tables of group invariants
Here are various tables of some of our invariants for 2groups of order divid*
*ing
64. The tables were compiled by hand using the calculations in [CTVZ ] and the
website version [Ca2 ]. The type of a group G, and thus e(G) and h(G), can be
deduced by inspecting the description of restriction to maximal elementary abel*
*ian
subgroups; this is particularly easy when G is 2central. If G is not 2central*
*, one
can immediately determine if Cess*(G) 6= 0, since both the rank of Z(G) and the
depth of H*(G) are given, and then read off the number e0(G) from the descripti*
*on
of depth essential cohomology. The website source allows one to identify centra*
*lizers
of elementary abelian subgroups as needed.
We say a group is indecomposable if it cannot be written as a nontrivial dire*
*ct
product of two subgroups. The numbering of groups is as in [CTVZ ] which follo*
*ws
the HallSenior numbering [HS ].
In Tables 1 and 2, recall that, since G is 2central, d0(G) = e(G) = e0(G) =
e00(G), and d1(G) = e(G) + h(G).
In Table 3, `2' means Z=2, etc. To compute d0(G), we needed to observe that,
in all cases covered by this table, e00(G) = e0(G). Except when G is 32#41, this
can be checked by noticing that elements in the top degree in QA Cess*(G) are
represented by classes in the image of InfGG=C, and so are primitive. When G is
32#41, elements in the QA Cess5(G) are represented by essential classes of lowe*
*st
degree, and so are primitive.
Table 1: Indecomposable, 2central, 2groups of order 32
_______________________________________
_Order#Type_d0(G)d1(G)Notes_
___2__1_[1]__0____0___Z=2__
___4__2_[2]__1____2___Z=4__
___8__3_[2]__1____2___Z=8__
______5_[4]__3____5___Q8___
___16_5[2]___1____2___Z=16_
______14[4]__3____5___Q16__
___32_18[2,2,2]3__4________
______19[2,2]_2___3________
______21[2,2]_2___3________
______28[4,2]_4___6________
______29[2,2]_2___3________
______30[2,2]_2___3________
______35[4,2]_4___6________
______40[4,4]_6___8________
______51[4]__3____5___Q32__
GROUP COHOMOLOGY PRIMITIVES 47
Table 2: Indecomposable, 2central, groups of order 64
___________________________________________
__#_Typed0(G)d1(G)____Notes______
__11_[2]__1___2_______Z=64_______
__30_[2,2,2]_3____4___________________
__37_[2,2,2]_3____4___________________
__38__[2,2]__2____3___________________
__39__[2,2]__2____3___________________
__41__[2,2]__2____3___________________
__59_[2,2,2]_3____4___________________
__63__[4,2]__4____6___________________
__64__[2,2]__2____3___________________
__65__[2,2]__2____3___________________
__82_[2,2,2]_3____4___________________
__87_[4,2,2]_5____7___________________
__88_[2,2,2]_3____4___________________
__90_[2,2,2]_3____4___________________
__92_[4,2,2]_5____7___________________
__93_[2,2,2]_3____4___________________
_101_[4,4]__6_____8___________________
_119_[4,2]__4_____6___________________
_139_[4,2]__4_____6___________________
_140_[2,2]__2_____3___________________
_141_[2,2]__2_____3___________________
_145_[4,2,2]_5____7___________________
_149_[4,2,2]_5____7___________________
_152_[4,2,2]_5____7___________________
_153[4,4,4]9_11___2Sylow_of_Sz(8)_ 
_162_[4,4]__6_____8___________________
_187[8,8]14__18___2Sylow_of_U3(F4)_
_190_[4,2]__4_____6___________________
_191_[4,4]__6_____8___________________
_192_[4,2]__4_____6___________________
_194_[4,4]__6_____8___________________
_199_[4,4]__6_____8___________________
_210_[4,4]__6_____8___________________
_211_[4,2]__4_____6___________________
_212_[4,4]__6_____8___________________
_222_[4,4]__6_____8___________________
_227_[4,4]__6_____8___________________
_233_[4,4]__6_____8___________________
_235_[4,2]__4_____6___________________
_236_[4,2]__4_____6___________________
_240_[4,4]__6_____8___________________
_267[4]__3____5________Q64_______
48 KUHN
Table 3: Indecomposable, non 2central, 2groups of order 32
________________________________________________________________________
_Order#TypeDepthRank_e(G)e0(G)d0(G)_CG_(V_)'s_Notes_
___8__4_[2]_2____2___1___1___0_______22______D8____
___16_8[4]__1____2___3___1___1_____4_x_2____AES16__
______9[2,2]_2___3___2___1____1_______22____________
______11[4]_1____2___3___2____2_____4_x_2___________
______12[2]_1____2___1___1___0_______22______D16___
______13[4]_1____2___3___2____2_______22_____SD16___
___32_16[4,2]2___3___4___3____3_____4_x_22__________
______17[4]_2____2___3___1___1_____8_x_2___________
______20[2,2]2___3___2___1____1_____4_x_22__________
______22[4]_1____2___3___2____2_____8_x_2___________
______26[4]_2____2___3___1___1___8_x_2,_4_x_2______
______27[2,2]2___3___2___1____1_______23____________
______31[4]_2____2___3___1___2___4_x_4,_4_x_2______
______32[4]_1____2___3___2____2_____8_x_2___________
______33[2,2]3___4___2___1___0_____24,_23__________
______34[2,2]3___3___2___1___0_______23____________
______36[2,2]3___3___2___1___1___4_x_22,_23________
______37[4,2]2___3___4___3____3_____4_x_22__________
______38[4,2]2___3___4___2____2___4_x_22,_23________
______39[4,2]2___3___4___3____3_______23____________
______41[4,4]2___3___6___5____5_______23____________
______42[4]_3____3___3___1___0_______23_____D8_*_D8_
______43[8]_2____2___7___1___3_____Q8_x_2___D8_*_Q8_
______44[4]_2____3___3___1___1____4_x_2,_23________
______45[8]_1____2___7___4____4___Q8_x_2,_4_x2______
______46[4]_2____3___3___1___3___Q8_x_2,_23________
______47[4]_1____3___3___1____1___D8_x_2,_23________
______48[8]_1____2___7___6____6_____Q8_x_2__________
______49[2]_2____2___1___1___0_______22______D32___
______50[4]_1____2___3___2____2_______22_____SD32___
GROUP COHOMOLOGY PRIMITIVES 49
Table 4: Indecomposable, non 2central, order 64, with Cess*(G) 6= 0
__________________________________________________________________________
__#_TypeRanke0(G)#TypeRank_e0(G)#TypeRanke0(G)_
__42__[4]___2____2___173_[4,4]__4___3___193_[4,2]__3____3___
__67__[4]___2____2___175_[4,4]__4___3___196_[4,2]__3____2___
_143__[4]___2____2___183_[4,4]__4___5___197_[4,2]__3____3___
_182__[4]___2____2___202_[4,2]__4___2___198_[4,2]__3____3___
_245__[8]___2____4______________________200_[4,4]__3____5___
_246__[4]___2____2____32_[4,2]__3___3___204_[4,2]__3____3___
_249__[8]___2____6____33_[4,2]__3___3___206_[4,2]__3____3___
_255__[8]___2____6____40_[2,2]__3___1___207_[4,2]__3____3___
_258__[8]___2____4____54_[4,2]__3___3___208_[4,2]__3____3___
_266__[4]___2____2____60_[4,2]__3___3___209_[4,2]__3____3___
______________________61_[4,2]__3___3___213_[4,2]__3____2___
_121__[8]___3____3____62_[2,2]__3___1___214_[4,2]__3____2___
_130__[8]___3____3____79_[4,4]__3___5___215_[4,4]__3____5___
_133__[8]___3____4____80_[4,4]__3___4___216_[4,4]__3____5___
_180__[8]___3____3____95_[4,2]__3___3___218_[4,2]__3____3___
_181__[8]___3____3____97_[4,2]__3___3___219_[4,2]__3____2___
_247__[4]___3____1____98_[4,2]__3___3___220_[4,2]__3____3___
_251__[8]___3____4____99_[4,2]__3___2___221_[4,4]__3____5___
_253__[8]___3____3___100_[4,2]__3___3___223_[4,4]__3____5___
_254__[8]___3____4___102_[4,4]__3___5___224_[4,4]__3____5___
_257__[8]___3____3___108_[8,2]__3___7___225_[4,2]__3____2___
_262__[8]___3____3___115_[8,2]__3___7___226_[4,2]__3____3___
_____________________116_[4,2]__3___3___228_[4,2]__3____2___
__81_[2,2,2]_5___1___118_[4,2]__3___3___229_[4,2]__3____3___
_____________________129_[4,2]__3___3___230_[4,2]__3____3___
__83_[2,2,2]_4___2___132_[4,2]__3___3___231_[4,4]__3____5___
__85_[2,2,2]_4___2___138_[2,2]__3___1___232_[4,4]__3____5___
__86_[2,2,2]_4___2___161_[4,4]__3___4___234_[2,2]__3____1___
__89_[4,2,2]_4___4___165_[4,4]__3___4___238_[4,4]__3____5___
__91_[2,2,2]_4___2___166_[4,4]__3___4___239_[4,2]__3____3___
_146_[2,2,2]_4___2___167_[4,4]__3___4_______________________
_147_[4,2,2]_4___4___168_[4,4]__3___5_______________________
_148_[2,2,2]_4___2___172_[8,2]__3___5_______________________
_150_[2,2,2]_4___2___174_[4,4]__3___5_______________________
_151_[2,2,2]_4___2___177_[4,4]__3___3_______________________
_____________________178_[4,4]__3___4_______________________
__94__[4,2]__4___2___179_[4,4]__3___5_______________________
_113_[4,2]__4____2___185_[4,4]__3___3_______________________
_131_[4,2]__4____2___186_[4,4]__3___4_______________________
_163_[4,4]__4____3___189_[4,2]__3___3_______________________
50 KUHN
References
[AK]A.Adem, and D.Karagueuzian, Essential cohomology of finite groups, Comment.*
* Math.
Helv. 72 (1997), 101109.
[AKM] A.Adem, D.B.Karagueuzian, and J.Min'a~c, On the cohomology of Galois grou*
*ps deter
mined by Witt rings, Adv. Math. 148 (1999), 105160.
[AP]A.Adem and J.Pakianathan, On the cohomology of central Frattini extensions,*
* J. Pure Appl.
Algebra 159 (2001), 114.
[Alp]J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced *
*Math. 11,
Cambridge U. Press, 1986.
[BC1]D.J.Benson, J.F.Carlson, Projective resolutions and Poincar'e duality comp*
*lexes, Trans.
Amer. Math. Soc. 342 (1994), 447488.
[BC2]D.J.Benson, J.F.Carlson, Functional equations for Poincar'e series in grou*
*p cohomology,
Bull. London Math. Soc. 26 (1994), 438448.
[Be]D.Benson, Dickson invariants, regularity and computation in group cohomolog*
*y, Illinois J.
Math. 48 (2004), 171197.
[BrH]C.Broto and H.W. Henn, Some remarks on central elementary abelian psubgr*
*oups and
cohomology of classifying spaces, Quart. J. Math. 44 (1993), 155163.
[BrZ1]C.Broto and S.Zarati, Nillocalization of unstable algebras over the Stee*
*nrod algebra, Math.
Zeit. 199 (1988), 525537.
[BrZ2]C.Broto and S.Zarati, On subA*palgebras of H*(V ), Springer L. N. Math.*
* 1509 (1992),
3549.
[BP]W.Browder and J. Pakianathan, Cohomology of uniformly powerful pgroups, Tr*
*ans. Amer.
Math. Soc. 352 (2000), 26592688.
[Ca1]J.F.Carlson, Depth and transfer maps in the cohomology of groups, Math. Ze*
*it. 218 (1995),
461468.
[Ca2]Jon Carlson, Mod 2 cohomology of 2 groups, MAGMA computer computations on *
*the
website http://www.math.uga.edu/~lvalero/cohointro.html.
[CTVZ]J.F.Carlson, L.Townsley, L.ValeriElizondo,and M. Zhang, Cohomology rings*
* of finite
groups. With an appendix, Calculations of cohomology rings of groups of orde*
*r dividing 64,
by Carlson, ValeriElizondo and Zhang. Algebras and Applications 3, Kluwer, *
*Dordrecht,
2003.
[Car]P.Cartier, Une nouvelle op'eration sure les formes diffe'rentielles, C.R.A*
*cad.Sci.Paris 244
(1957), 426428.
[D]J. Duflot, Depth and equivariant cohomology, Comm. Math. Helv. 56 (1981), 62*
*7637.
[E]L. Evens, The spectral sequence of a finite group extension stops, Trans. A.*
* M. S. 212 (1975),
269277.
[FLS]V. Franjou, J. Lannes, and L. Schwartz, Autour de la cohomologie de MacLan*
*e des corps
finis, Invent. Math. 115(1994), 513538.
[Gor]D. Gorenstein, Finite Groups, 2nd edition, Chelsea Publishing, New York, 1*
*980.
[G1]D.J.Green, On Carlson's depth conjecture in group cohomology, Math. Zeit. 2*
*44 (2003),
711723.
[G2]D.J.Green, The essential ideal in group cohomology does not square to zero,*
* J. Pure Appl.
Algebra 193 (2004), 129139.
[G3]D.J.Green, The essential ideal is a CohenMacauley module, Proc. A. M. S. 1*
*33 (2005),
31913197.
[HS]M.Hall and J.K.Senior, The groups of order 2n, n 6, Macmillan, New York, *
*1964.
[H]H.W. Henn, Finiteness properties of injective resolutions of certain unstab*
*le modules over
the Steenrod algebra and applications, Math. Ann. 291 (1991), 191203.
[HLS1]H.W.Henn, J.Lannes, and L.Schwartz, Localizations of unstable Amodules *
*and equivari
ant mod p cohomology, Math. Ann. 301 (1995), 2368.
[K1]N.J.Kuhn, On topologically realizing modules over the Steenrod algebra,, An*
*nals of Math
141(1995), 321347.
[K2]N.J.Kuhn, Cohomology primitives associated to central extensions, Oberwolfa*
*ch Reports 2
(2005), 23832386.
[K3]N.J.Kuhn, The nilpotent filtration and the action of automorphisms on the c*
*ohomology of
finite pgroups, preprint, May, 2006.
GROUP COHOMOLOGY PRIMITIVES 51
[L1]J.Lannes, Sur la cohomologie modulo p des pgroupes ab'eliens 'el'ementaire*
*s, Homotopy theory
(Durham, 1985), L.M.S. Lect. Note Ser. 117 (1987), 97116.
[L2]J.Lannes, Cohomology of groups and function spaces, 1986 preprint based on *
*a talk given at
the University of Chicago.
[L3]J.Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'*
*un pgroupe ab'elien
'el'ementaire, I.H.E.S. Pub. Math. 75 (1992), 135244.
[MM] J.W.Milnor and J.C.Moore, On the structure of Hopf algebras, Ann. Math. 81*
* (1965),
211264.
[MiS]J.Min'a~c and M.Spira, Witt rings and Galois groups, Ann. Math. 144 (1996)*
*, 3560.
[Q]D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 *
*(1971), 549
572.
[S1]L. Schwartz, La filtration nilpotente de la cat'egorie U et la cohomologie *
*des espaces de lacets
, Algebraic Topology  Rational Homotopy (Louvain la Neuve 1986), S. L. N. M*
*. 1318 (1988),
208218.
[S2]L. Schwartz, Modules over the Steenrod algebra and Sullivan's fixed point c*
*onjecture, Chicago
Lectures in Math., U. Chicago Press, 1994.
[Sw]M.E.Sweedler, Hopf algebras, Math. Lect. Note. Series, Benjamin, 1969.
Department of Mathematics, University of Virginia, Charlottesville, VA 22904
Email address: njk4x@virginia.edu