COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS
WILLIAM BROWDER AND JONATHAN PAKIANATHAN
Abstract.In this paper we will study the cohomology of a family of p-gro*
*ups
associated to Fp-Lie algebras. More precisely we study a category BGrp of
p-groups which will be equivalent to the category of Fp-bracket algebras*
* (Lie
algebras minus the Jacobi identity). We then show that for a group G in *
*this
category, its Fp-cohomology is that of an elementary abelian p-group if *
*and
only if it is associated to a Lie algebra.
We then proceed to study the exponent of H*(G; Z) in the case that G i*
*s as-
sociated to a Lie algebra L. To do this, we use the Bockstein spectral s*
*equence
and derive a formula that gives B*2in terms of the Lie algebra cohomolog*
*ies of
L. We then expand some of these results to a wider category of p-groups.*
* In
particular we calculate the cohomology of the p-groups n;kwhich are defi*
*ned
to be the kernel of the mod p reduction GLn(Z=pk+1Z) mod-!GLn(Fp):
1991 Mathematics Subject Classification. Primary: 20J06, 17B50; Secondar*
*y:
17B56.
1.Introduction and Motivation
Throughout this paper, p will be an odd prime. First some definitions:
Definition 1.1.Given a p-group G, 1(G) = where the brackets
mean "smallest subgroup generated by". G is called p-central if 1(G) is central.
Definition 1.2.For any positive integer k, Gpk = .
Definition 1.3.F rat(G) = Gp[G; G]:
We will study a category BGrp of p-groups which is naturally equivalent to t*
*he
category of bracket algebras over Fp. This is exactly the category of p-central*
*, p-
groups G which have 1(G) = Gp = F rat(G). For such a group G, the associated
bracket algebra will be called Log(G).
In the case that one of these groups is associated to a Lie algebra, we will
show that it has the same Fp-cohomology as an elementary abelian p-group. More
precisely we will show: (The necessary definitions for the theorems quoted in t*
*his
introductory section can be found in the relevant parts of the paper.)
Theorem 1 ( 2.10).Let G 2 Obj(BGrp ) and n = dim(1(G)). Then
H*(G; Fp) = ^(x1; : :;:xn) Fp[s1; : :;:sn]
(where the xi have degree 1 and the si have degree 2) if and only if Log(G) is *
*a Lie
algebra. When this is the case, the polynomial algebra part restricts isomorphi*
*cally
to that of H*(1(G); Fp) and the exterior algebra part is induced isomorphically
from that of H*(G=1(G); Fp) via the projection homomorphism.
(This theorem was proven independently by T. Weigel for the case p 6= 3 in [W1 *
*],
[W2 ].)
1
2 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
However the integral cohomology of these groups is complicated and indeed
we will show that one can recover the Lie algebra associated to the group from
knowledge of the Bockstein on its Fp-cohomology. To do this we calculate the
full structure of the Fp-cohomology as a Steenrod-module. As an application, in
corollary 2.26, we completely determine the comodule algebra structure (see [W2*
* ])
of H*(G; Fp).
With the mild additional hypothesis that the associated Lie algebra lifts to *
*one
over Z=p2Z, we compute B*2of the Bockstein spectral sequence in terms of the Lie
algebra cohomologies of the corresponding Lie algebra:
Theorem 2 ( 2.36).Let G 2 Obj(BGrp ) with Log(G) = L, a Lie algebra, and
suppose that L lifts to a Lie algebra over Z=p2Z. Then B*2of the Bockstein spec*
*tral
sequence for G is given by
B*2= 1k=0H*-2k(L; Sk):
Here Sk is the L-module of symmetric k-forms on L with the usual action. (This
action is described in detail before the proof of the theorem.)
The integral cohomology of p-groups is a rich subject and there are many the-
orems and conjectures about the exponent of H* (P ; Z). (Where the bar denotes
reduced cohomology and we recall that the exponent of an abelian group G is the
smallest positive integer n such that ng = 0 for all g 2 G). We will obtain some
partial results on the exponent of the integral cohomology of the p-groups stud*
*ied
in this paper via the Bockstein spectral sequence.
In the last section of the paper, we extend the results mentioned to a bigger
family of groups: the uniform, p-central, p-groups. (Elementary abelian p-groups
are uniform and more generally one defineskinductively that a p-central, p-grou*
*p G
is uniform if and only if 1(G) = Gp for some nonnegative integer k and G=1(G)
is itself uniform. Thus a uniform p-group will give rise to a tower of uniform*
* p-
groups called a uniform tower where each group in the tower is the quotient of *
*the
previous group G by 1(G). In this paper such uniform towers will be indexed so
that G1 is always an elementary abelian p-group. When this is done, G2 will alw*
*ays
correspond to a group in the category BGrp .)
Specifically we prove:
Theorem 3 ( 3.14).Fix p 5. Let
Gk+1 ! Gk ! . .!.G2 ! G1 ! 1
be a uniform tower with k 2. Let L = Log(G2) and let ckijbe the structure
constants of L with respect to some basis. Then for suitable choices of degree*
* 1
elements x1; : :;:xn and degree 2 elements s1; : :;:sn, one has
H*(Gk; Fp) ~=^*(x1; : :;:xn) Fp[s1; : :;:sn]
with
Xn
fi(xt) =- ctijxixj
i : W x W ! V given by = ^x^y^x-1^y-1
where ^x; ^yare lifts of x; y to G, i.e., ss(^x) = x; ss(^y) = y.
We also define a p-power map function
(2) OE : W ! V given byOE(x) = ^xp:
It is routine to verify that <.; .> is a well-defined alternating, bilinear map*
* and that
OE is a well-defined linear map. (the linearity of OE uses that p is odd.)
We will now restrict ourselves to the case where the p-power map OE is an iso-
morphism. So V and W are isomorphic. In this case one can define
(3) [.; .] : W W ! W by [w1; w2] = 1_2OE-1():
Then it is easy to see this is still an alternating, bilinear map on W . Note W
is a vector space over Fp, the field on p elements. (The factor of 1_2is put i*
*n for
convenience in order to avoid messy expressions later on.)
Definition 2.1.A bracket algebra over Fp is a finite dimensional vector space W
over Fp equipped with an alternating, bilinear form [.; .] : W W ! W . The
dimension of a bracket algebra is the dimension of the underlying vector space.
Definition 2.2.Brak is the category whose objects consist of bracket algebras o*
*ver
Fp and with morphisms the linear maps which preserve the brackets.
Now we can define a category of p-groups which is essentially equivalent to t*
*he
category Brak . These groups will be our primary objects of study in the first *
*part
of this paper.
4 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
Definition 2.3.A p-power exact extension is a central short exact sequence:
1 ! V ! G ss!W ! 1
where V, W are elementary abelian and where the p-power map OE is an isomor-
phism.
Note in such an extension, V = 1(G) and W = G=1(G) (see definition 1.1).
So any homomorphism between two groups G1; G2 in the middle of p-power exact
extensions takes V1 = 1(G1) into V2 = 1(G2) and hence induces maps V1 ! V2
and W1 ! W2 where Wi = Gi=1(Gi). These will be called respectively the
induced maps on the V; (W ) level.
Definition 2.4.BGrp is the category whose objects consist of finite groups G
which fit in the middle of a p-power exact extension. These will be called brac*
*ket
groups. The morphisms in BGrp are just the usual group morphisms between these
groups, except we will identify two morphisms if they induce the same map on the
V and W levels.
One can show that the two categories Brak and BGrp are naturally equivalent
(see [Pak]). This follows from a general exponent-log correspondence (see [W1 *
*],
[W3 ] or [Laz]), suitably reworded for our purposes. We will give a short descr*
*iption
of the functors involved in this equivalence of categories.
Define a covariant functor Log : BGrp ! Brak as follows, to a bracket group G
we associate the bracket algebra Log(G) = (W; [.; .]) which is obtained as expl*
*ained
before. Notice that the underlying vector space of Log(G) is just G=1(G) so giv*
*en
2 Mor(G1; G2) we note induces a well-defined linear map Log( ) : Log(G1) !
Log(G2), and it is easy to check that this is a map of bracket algebras.
A description of the inverse functor Exp : Brak ! BGrp is given as follows.
Given (L; [.; .]) a bracket algebra, let K be a free Z=p2Z-module of rank equal*
* to
the dimension of L. Let OE : L ! K and ss : K ! L be injective/surjective maps
such that OE O ss = p (multiplication by p). Then Exp(L) = (K; O) where
(4) l O m = l + m + OE[ss(l); ss(m)]:
(Here + is the addition of K as a Z=p2Z-module. Notice in this notation, 0 is t*
*he
identity of Exp(L), l-1 corresponds to -l, lp corresponds to pl, and the p-power
map is indeed OE. )
For clarity we state the following proposition which summarizes these facts:
Proposition 2.5.The functors Log : BGrp ! Brak and Exp : Brak ! BGrp
give a natural equivalence between the categories BGrp and Brak . Thus to every
bracket algebra there naturally corresponds a unique bracket group.
Example:
A direct computation shows that the group n;2which is the kernel of the map
GLn(Z=p3Z) mod-!GLn(Fp);
is a bracket group, and that Log(n;2) = gln, the Lie algebra of n x n matrices.
Now we will study the groups in BGrp from a cohomological viewpoint. Recall
if G 2 Obj(BGrp ) then G fits in a central short exact sequence
1 ! V ! G ss!W ! 1
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 5
where we can identify V and W via the p-power isomorphism OE. If we do not put
the restriction that OE is an isomorphism then such extensions as above are in *
*one
to one correspondence with H2(W ; V ). Now for W an elementary abelian p-group
of rank n (recall p is odd), we have that
H*(W ; Fp) ~=^*(x1; : :;:xn) Fp[fix1; : :;:fixn]
the tensor product of an exterior algebra on degree 1 generators and a polynomi*
*al
algebra on degree 2 generators which are the Bocksteins of the degree 1 generat*
*ors.
(Here fi : H1(W ; Fp) ! H2(W ; Fp) is the Bockstein. ) In basis free notation, *
*this
can be written
H*(W ; Fp) ~=^*(W *) Fp[fi(W *)]
where W *= H1(W ; Fp) is the dual vector space of W and fi(W *) is its image un*
*der
the Bockstein fi: Now,
H2(W ; V ) ~=H2(W ; Fp) V by the Universal Coefficient Theorem
(5) ~=(H2(W ; Fp))n using a choice of basisVof:
Note that a bracket on W gives us a map br : ^2(W ) ! W . If we take the dual
of this map we get a map br* : W *! ^2(W *). (Once we identify the dual of
^2(W ) with ^2(W *) in the usual way.) With this notation, it is easy to argue *
*by
comparisons (see [BC ]) and from the description of the Exp functor in equation*
* 4
that the extension elements are of the form fi(xi) + br*(xi) for i = 1; : :;:n.*
* (If we
use a suitable basis for V in equation 5. One chooses a basis E for W , the bas*
*is
OE(E) for V and the canonical dual basis for V *and W *so for example {x1; : :;*
*:xn}
are dual to E.) In this formulation br*(xi) are just the components of the brac*
*ket
of Log(G), with respect to the basis E. We will use this form of the extension
element from now on.
2.2. Lie Algebras and co-Lie algebras.
Definition 2.6.Given a bracket algebra (W; [.; .]) we define the Jacobi form of
(W; [.; .]) to be
J(x; y; z) = ([[x; y]; z] + [[y; z]; x] + [[z; x]; y]):
This is easily checked to be an alternating 3-form. Thus J : ^3(W ) ! W . Brack*
*et
algebras with J = 0 are called Lie algebras.
Given a bracket algebra, (W; [.; .]), the bracket defines a map br : ^2(W ) !*
* W .
Thus by taking duals and identifying the dual of ^*(W ) with ^*(W *) in the usu*
*al
way, we obtain a map br* : W *! ^2(W *), where W *is the dual vector space of
W . This motivates the following definition:
Definition 2.7.A co-bracket algebra is a vector space W *equipped with a linear
map br* : W *! ^2(W *). We will always extend br* as a degree 1 map on the
whole of the graded algebra ^*(W *) in the unique way that makes it a derivation
on that algebra. We call such a co-bracket algebra a co-Lie algebra if br* O br*
** = 0.
As was mentioned before, the dual of a bracket algebra, has a natural co-brac*
*ket
algebra structure and similarly, the dual of a co-bracket algebra has a natural
bracket algebra structure.
Given a co-bracket algebra W *, it is easy to check, as br* is a derivation, *
*that
br*O br* = 0 if and only if br*O br* : W *= ^1(W *) ! ^3(W *) is zero. The dual*
* of
this map is a map from ^3(W ) ! W (where we are calling W **= W ). It is easy
6 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
to check that this is just the Jacobi form defined before for the bracket algeb*
*ra W .
Thus we can conclude that the dual of a co-Lie algebra is a Lie algebra and vice
versa.
Let ss : ^2(W ) ! W W be the map defined by ss(a ^ b) = a b - b a. Thus
the usual definition of a module over a bracket (Lie) algebra can be worded in *
*the
following more categorical way:
Definition 2.8.We say V is a module over the bracket algebra (W; br) if we have
a map : W V ! V such that the following diagram commutes:
W ^ W V ----! W V
? br1 ?
(1)O(ss1)?y ?y
W V ----! V
Taking the dual of this definition, and noting that the dual of ss, ss* : W **
*W *!
^2(W *) is just the canonical quotient map, gives us the following definition:
Definition 2.9.We say V *is a comodule over the co-bracket algebra (W *; br*) if
we have a map * : V *! W * V *such that the following diagram commutes:
W *^ W * V * ---- W * V *
x br*1 x
1^* ?? ??*
*
W * V * ---- V *
It is easy to see that the dual of a comodule over the co-bracket algebra W **
*is
a module over the bracket algebra W = W **and vice-versa.
2.3. The Fp-cohomology of a group in BGrp . In this section we will show
that a group in BGrp has the Fp-cohomology of an elementary abelian p-group if
and only if it is associated to a Lie algebra. More precisely we will prove:
Theorem 2.10. Let G 2 Obj(BGrp ) and n = dim(1(G)). Then
H*(G; Fp) = ^(x1; : :;:xn) Fp[s1; : :;:sn]
(where the xi have degree 1 and the si have degree 2) if and only if Log(G) is *
*a Lie
algebra. When this is the case, the polynomial algebra part restricts isomorphi*
*cally
to that of H*(1(G); Fp) and the exterior algebra part is induced isomorphically
from that of H*(G=1(G); Fp) via the projection homomorphism.
Fix G 2 Obj(BGrp ) we now study the Lyndon-Hochschild-Serre (L.H.S.) spec-
tral sequence (with Fp-coefficients) associated to the extension
1 ! 1(G) = V !i G ss!W ! 1:
Recall that this is a spectral sequence with
Ep;q2= Hp(W; Hq(V; Fp))
and it abuts to H*(G; Fp). Since this is a central extension, if we use Fp coef*
*ficients
throughout then
Ep;q2= Hp(W; Hq(V; Fp)) = Hq(V; Fp) Hp(W; Fp):
So using that p is odd one gets explicitly:
(6) E*;*2~=^*(V *) Fp[fi(V *)] ^*(W *) Fp[fi(W *)]:
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 7
As before, we are using basis free notation, for example V *= H1(V ; Fp) and fi*
* is
the Bockstein.
The dual of the p-power map OE gives us an isomorphism OE* : V *! W *so given
a basis E = {e1; : :e:n} for V *, one can use OE*(E) = {x1; : :;:xn} as a basis*
* for W .
By standard comparisons it is easy to see that d2|W* = d2|fi(W*)= d2|fi(V *)=*
* 0
while
d2(ei) = (the ith component of the extension element) = fi(xi) + br*(xi)
for all i. Here we are using the form of the extension element as presented in *
*the
end of section 2.1. As OE*(ei) = xi, one has
d2|V *= fi O OE* + br* O OE*:
Let us calculate E*;*3.
Let
A = ^*(X1; : :;:Xn) Fp[Y1; : :;:Yn] ^*(T1; : :;:Tn) Fp[S1; : :;:Sn]
be an abstract free graded-commutative algebra where the polynomial generators
are degree 2 and the exterior generators are degree 1. Since this is a free obj*
*ect, the
assignment Xi! xi; Ti! ei; Si! fi(ei); Yi! fi(xi) + br*(xi) defines a map of
graded-algebras from A to E*;*2. As fi(xi)+br*(xi) differs form fi(xi) by a nil*
*potent
element it is easy to see by induction on the grading that is an isomorphism. *
*The
induced differential D2 on A of d2 under this isomorphism has D2(Xi) = D2(Yi) =
D2(Si) = 0 and D2(Ti) = Yi. Applying K"unneth's Theorem, we then see that
the cohomology of (A; D2) is isomorphic to ^*(X1; : :;:Xn) Fp[S1; : :;:Sn]. Us*
*ing
this, we see that E*;*3is given by
(7) E*;*3~=^*(W *) Fp[fi(V *)]
We have obviously that d3|W* = 0 and we also have (see for example page 155 of
[Be]) :
d3 O fi|V=*{fi O d2|V *} 2 E3;03= ^3(W *)
(8) = {fi O (fi O OE* + br* O OE*)|V *}
= {fi O br* O OE*|V *} sincefi O fi = 0:
Now note that fi is a derivation and that fi(xi) is identified with -br*(xi) in*
* E*;03=
^*(x1; : :;:xn) hence fi induces the same map as -br* on E*;03. Thus we see tha*
*t the
final expression in equation 8 becomes -br*O br*O OE*|V *. As OE* is an isomorp*
*hism,
we can conclude that d3|fi(V *)= 0 and hence d3 = 0 if and only if br* O br* = *
*0 on
W *= ^1(W *) or in other words if and only if (W *; br*) is a co-Lie algebra. H*
*ence
we conclude:
Lemma 2.11. d3 = 0 if and only if Log(G) = (W; br) is a Lie algebra.
When this happens E3 = E4 but then for grading reasons, dr|fi(V *)= dr|W* =
0 for all r 4. Hence as the differentials are derivations which vanish on the
generators, all further differentials in the spectral sequence must be 0. We h*
*ave
proven
Proposition 2.12.E3 = E1 if and only if Log(G) is a Lie algebra. In this case,
E*;*3= E*;*1= ^*(W *) Fp[fi(V *)]:
8 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
Due to the free nature of this graded ring, one can then show in a routine ma*
*nner
that
H*(G; Fp) = ^*(W *) Fp[S*]
where we have abused notation a bit and identified W *with its image under the
map ss* : H*(W ; Fp) ! H*(G; Fp). S* is a subspace which restricts isomorphical*
*ly
to the subspace fi(V *) of H*(V ; Fp). Hence we have proved theorem 2.10.
Let us define LGrp to be the full subcategory of BGrp whose objects are the
bracket groups G where the associated Log(G) is a Lie algebra. Let Lie be the
full subcategory of Brak whose objects are the Lie algebras. Then restricting
the natural functors we had before we see LGrp and Lie are naturally equivalent
categories.
2.4. A formula for the Bockstein on H*(G; Fp). From now on we consider
groups G 2 Obj(LGrp ). By the previous theorem
H*(G; Fp) = ^*(W *) Fp[S*]
where W = G=1(G):
We wish to study the Bockstein fi : H*(G; Fp) ! H*+1(G; Fp). The reason for
this is that one can determine the groups H*(G; Z) from knowledge of fi and the
"higher Bocksteins" on H*(G; Fp).
Recall that fi is a derivation, i.e. , for homogeneous elements u; v 2 H*(G; *
*Fp)
we have
fi(uv) = fi(u)v + (-1)deg(u)ufi(v):
Also recall that fi O fi = 0. Since fi is a derivation we need only describe it*
* on the
generating subspaces W *and S* of H*(G; Fp).
Now recall the central short exact sequence
1 ! 1(G) = V !i G ss!W ! 1
Then we have seen that the subalgebra ^*(W *) of H*(G; Fp) is the image of the
map ss* : H*(W ; Fp) ! H*(G; Fp). In the L.H.S.-spectral sequence of the last
section, we can identify this image with E*;01= E*;03. We saw there that fi agr*
*ees
with -br* on this subalgebra.
Thus fi|W* = -br*. It is easy to check that the differential complex (^*(W *)*
*; fi)
is the standard Koszul resolution used to calculate the Lie algebra cohomology
H*(L; Fp), where L = (W; br). Also, we see that one can obtain the Lie algebra
structure of L from knowledge of fi on the exterior algebra part of H*(G; Fp).
Now we are left with finding fi|S* which is harder. Let us develop some more
notation. Purely from considering degree restrictions one has:
fi|S* : S* ! (W * S*) ^3(W *):
Thus we can write
(9) fi|S* = fi1 + fi2
where fi1 : S* ! W * S* and fi2 : S* ! ^3(W *). If we let j = fi-1 O i* : S* ! *
*V *
be the natural isomorphism, we can define ^fi1= (1j)Ofi1Oj-1 : V *! (W *V *)
and ^fi2= fi2 O j-1 : V *! ^3(W *).
Before we go on, we should note an inherent ambiguity. Note ^fi1and ^fi2deter*
*mine
fi, however their definition requires a choice for the subspace S*. We would li*
*ke that
all our expressions involving fi be determined from knowledge of the underlying*
* Lie
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 9
algebra L = Log(G). Note W = G=1(G) can be identified with the underlying
vector space of L and hence so can V = 1(G) via the p-power isomorphism OE.
Thus the subalgebra ^*(W *) of H*(G; Fp) is unambiguously determined by L.
However the subspace S* which generates the polynomial part of H*(G; Fp) is only
determined by the fact it restricts isomorphically to the natural subspace fi(V*
* *) of
H*(V ; Fp). We can see that given a homomorphism : S* ! ^2(W *), then the
image of S* under j = Id + works just as well, and that all possible other cho*
*ices
for this generating subspace arise in this way. Let us see how changing ones ch*
*oices
effects the decomposition of fi in equation 9.
It is easiest to see this by using a basis {s1; : :s:n} for the space S*. If *
*we put
this basis in a column vector: 0 1
s1
s = B@...CA
sn
then we have that
fi(s) = s + fi2(s)
where is an n x n matrix with entries in ^1(W *).
If we change our choice of subspace S* using the homomorphism mentioned
above, we get a new subspace S* = j(S*) with basis s0 = s + (s). We then
calculate:
fi(s0) = fi(s + (s)) = s + fi2(s) + fi O (s) = s0+ (fi2(s) + (-br* - )((s))):
(Here we have used again that fi = -br* on ^*(W *).) So if we decompose fi =
fi01+ fi02using this new subspace S* and note that (1 ^ ) O fi1(s) = (s), we see
that:
(10) (1 j-1) O fi01O j = fi1 andfi02O j = fi2 + (-br* O - (1 ^ ) O fi1)
or
(11) f^i01= ^fi1and^fi02= ^fi2+ (-br* O - (1 ^ ) O ^fi1)
(Here we have made the natural identification of as a map : V *! ^2(W *).)
Thus ^fi1is not ambiguous while ^fi2depends on the particular subspace S* chose*
*n,
as indicated in equation 11. We will return to this later.
Let us now concentrate on ^fi1which is well defined given the Lie algebra L. *
*Since
fi O fi = 0 we have:
(12) 0 = fi O fi|S* = fi O fi1 + fi O fi2:
However since fi is a derivation and fi = -br* on ^*(W *) , it follows easily f*
*rom
equation 12 that
(13) 0 = (-br* 1) O fi1 - (1 fi1) O fi1 - (1 fi2) O fi1 - br* O fi2:
Taking components in the usual way, we get the equations:
(14) (-br* 1) O fi1 = (1 fi1) O fi1
and
(15) (1 fi2) O fi1 = -br* O fi2:
10 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
Equation 14 gives us the following commutative diagram:
^fi1
V * -----! W * V *
? ?
-^fi1?y ?y1-^fi1
*1
W * V * br----!^2(W *) V *
Thus, -f^i1equips V *with the structure of a comodule over the co-Lie algebra
(W *; br*). Thus, taking duals, we see that -f^i*1equips V with the structure o*
*f a
module over the Lie algebra (W; br). However, we can naturally identify V with
W using the p-power map, thus for every Lie algebra L = (W; br) we obtain a map
: L ! gl(L) from the module structure above. Furthermore it follows, that if we
give gl(L) the canonical Lie algebra structure, then this map will be a map of
Lie algebras.
These maps obtained above are natural in the following sense:
Claim 2.13. Let L; L02 Obj(Lie) and 2 Mor(L; L0) then
[0( (x))]( (y)) = ([(x)](y)):
(The notation [(x)](y) means of course (x) 2 gl(L) applied to the element y.)
Proof.Suppose L = (W; br) and L0= (W ; br). Then : W ! W induces a unique
morphism Exp( ) : Exp(L) ! Exp(L0) in the category LGrp . This is a class
of group homomorphisms where any two representatives of this class induce the
same mapping on the 1 level as mentioned earlier. It is easy to check that any
representative of this class will induce the dual map * from
W * = H1(Exp(L0); Fp) ! W *= H1(Exp(L); Fp):
Since the Bockstein is a natural operation, it will commute with any such map on
cohomology and from this, one obtains the following commutative diagram:
^
V * --fi1---!W *V *
? ?
*?y ?y * *
^fi1
V * -----! W * V *
The claim follows easily now by taking the dual of the diagram above and recall*
*ing_
that the map was defined via -f^i*1. |__|
Notice that knowledge of the map is equivalent to knowledge of the component
^fi1of the Bockstein. We have isolated enough properties of the maps and will
now set out to determine them explicitly.
2.5. Self-representations. In this section, let k be a finite field with char(k*
*) 6= 2.
Let Lie be the category of finite dimensional Lie algebras over k with morphisms
the lie algebra maps. (We will of course be interested in the case k = Fp).
Definition 2.14.A natural self-representation on Lieis a collection of linear m*
*aps
L : L ! gl(L), which satisfy the naturality condition
[L0( (x))]( (y)) = ([L(x)](y))
for all L; L0 2 Obj(Lie), 2 Mor (L; L0) and x; y 2 L. One says that a self-
representation is strong if the maps L : L ! gl(L) are maps of Lie algebras.
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 11
Examples:
(a) If we set = 0 we see easily that we get a strong self-representation which*
* we
call the zero representation.
(b) By claim 2.13, we see that the maps fit together to give a strong natural
self-representation which we will call .
(c) If we set = ad where ad : L ! gl(L) is given by ad(x)(y) = [x; y] for
all x; y 2 L, then it is easy to see that each map ad is a map of Lie algebras.
Furthermore if L0 is another Lie algebra and : L ! L0 is a morphism of Lie
algebras then we have for x; y 2 L :
[ad0( (x))]( (y)) = [ (x); (y)] = ([x; y]) = ([ad(x)](y)):
So we see this assignment defines a strong natural self-representation which we*
* call
the adjoint representation.
Now we will study self-representations so that we can show is the adjoint
representation. Note that if 0; 1 are two self-representations, then if they as*
*sign
the same map L ! gl(L) for some lie algebra L, (we will say they agree on L) th*
*en
they assign the same map for all lie algebras isomorphic to L. So we will impli*
*citly
use this from now on without mention.
Lemma 2.15. If 0; 1 are two self-representations, then if they agree on a lie
algebra L, they agree on all Lie subalgebras of L.
Proof.Follows from the naturality of self-representations. |*
*___|
Recall gln is the Lie algebra of n x n matrices and sln is the lie algebra of
n x n matrices with trace zero. Define N to be the 3-dimensional Lie algebra wi*
*th
basis {x; y; z} and bracket given uniquely by [x; y] = z; [x; z] = [y; z] = 0. *
* Note
that x; z generate a 2-dimensional abelian subalgebra of N. Define S to be the *
*2-
dimensional Lie algebra with basis {x; y} and bracket given uniquely by [x; y] *
*= x.
Note as char(k) 6= 2 this Lie algebra embeds into sl2by the Lie algebra map
where
(x) = 01 00
(y) = 1=20 -10=2 :
Lemma 2.16. Let 0; 1 be two self-representations, then if 0 and 1 agree on
sl2and N then 0 = 1 as self-representations.
Proof.Assume 1; 0 are two self-representations satisfying the assumptions above.
By the Ado-Iwasawa theorem, all Lie algebras embed as subalgebras of glnfor some
n. So by lemma 2.15, to show 1 = 0 one need only show they agree on glnfor all
n. A basis for glnis {ffii;j: i; j = 1; : :n:} where ffii;jis the matrix with e*
*ntry one at
the (i,j)-position and zero elsewhere. One calculates easily that:
8
>>>0 ifj 6= l; m 6= i
i;m
>>:-ffil;j ifj 6= l; m = i
ffii;i- ffij;jifj = l; m = i:
Let Lnij;lmbe the Lie subalgebra of glngenerated by ffii;jand ffil;mfor all i; *
*j; l; m 2
{1; : :;:n}. Suppose we have shown 1 and 0 agree on Lnij;lmfor all i; j; l; m 2
12 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
{1; : :;:n}. Then by an easy linearity argument it follows that 1 and 0 agree on
glnfor all n and hence are equal. So we see that if we can show 1 and 0 agree on
Lnij;lmfor all n 2 N; i; j; l; m 2 {1; : :;:n} we are done. (Since gln gln+1we *
*see
we can assume n 5 say.) Now let us identify the Lie algebras Lnij;lm. We have
the following cases:
1. Lij;lmwhere j 6= l; m 6= i. Here we have [ffii;j; ffil;m] = 0 so this Lie *
*algebra is
an abelian Lie algebra of dimension 2. Hence it is contained inside N.
2. Lij;jmwhere m 6= i; m 6= j; j 6= i. This Lie algebra has basis {ffii;j; ff*
*ij;m; ffii;m}
and has bracket relations, [ffii;j; ffij;m] = ffii;m, [ffii;j; ffii;m] = [*
*ffij;m; ffii;m] = 0.
Thus it is easy to see this Lie algebra is isomorphic to N.
3. Lij;jjwhere j 6= i. Here we have [ffii;j; ffij;j] = ffii;j. So we see this*
* Lie algebra
is isomorphic to S and so is inside sl2.
4. Ljj;jmwhere m 6= j. We see easily we get the same case as in case 3.
5. Lij;liwhere j 6= l. Here we note that Lij;li= Lli;ijso it occurs as one of*
* the
cases 2 - 4.
6. Lij;jiwhere i 6= j. Here we have that Lij;jihas basis {ffii;j; ffij;i; ffi*
*i;i-ffij;j= }.
The bracket is given by [ffii;j; ffij;i] = , [; ffii;j] = ffii;j+ ffii;j= *
*2ffii;jand
[; ffij;i] = -ffij;i- ffij;i= -2ffij;iand so we see easily that this Lie a*
*lgebra is
isomorphic to sl2.
7. Lii;ii. Obviously this Lie algebra is 1-dimensional and hence lies in eith*
*er sl2
or N.
All the Lie algebras Lnij;lmfit into one of these cases and hence embed in eith*
*er
sl2or N. As 1 and 0 agree on these two Lie algebras by assumption, they must *
*__
agree on all the Lnij;lmand hence must be equal as argued before. |*
*__|
Now we will study how a self-representation must look in certain particular c*
*ases.
Lemma 2.17. Let be a self-representation. If x; y 2 L have [x; y] = 0 then
[(x)](y) = 0. So in particular = 0 on abelian Lie algebras.
Proof.By naturality, it is enough to show = 0 on abelian Lie algebras of dimen-
sion 2. Fix L an abelian Lie algebra of dimension n 2. Let fl 2 k - {0; 1}.
Then multiplication by fl induces an automorphism of the Lie algebra L. Given
x; y 2 L then if [(x)](y) = z it follows from naturality that
[((x))]((y)) = (z)
fl2z = flz
However, fl2 6= fl so z = [(x)](y) = 0. Since x; y were arbitrary, the_lemma_
follows. |__|
Corollary 2.18.Let be a self-representation. Then for a Lie algebra L and
x; y 2 L one has [(x)](x) = 0 and [(x)](y) = -[(y)](x).
Proof.As [x; x] = 0, [(x)](x) = 0 follows from lemma 2.17. The second part __
follows easily from [(x + y)](x + y) = 0. |__|
Recall S is the Lie algebra with basis {x; y} and [x; y] = x. Let ss : S ! S *
*be a
linear map given by ss(x) = 0 and ss(y) = y. It is easy to check that ss is a m*
*ap of
Lie algebras.
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 13
Lemma 2.19. Let be a self-representation. Let S be the Lie algebra with basis
{x; y} and bracket given by [x; y] = x. Then there is a number a() 2 k such that
one has [(u)](v) = a()[u; v] for all u; v 2 S.
Proof.By corollary 2.18, one sees it is enough to show [(x)](y) = a()x for some
a() 2 k. One knows a priori that
(16) [(x)](y) = a()x + b()y;
so we want to show b() = 0. Applying the map of Lie algebras ss, mentioned in
the paragraph preceding the lemma, to equation 16 and using naturality, we_get_
0 = b()y and so it follows that b() = 0. |__|
Recall sl2has basis {ffi1;2; ffi2;1; ffi1;1- ffi2;2}. Let us relabel as follows*
*: H = ffi1;1- ffi2;2,
X+ = ffi1;2and X- = ffi2;1then it is easy to verify that the bracket on sl2is g*
*iven
by
(17) [X+ ; X- ] = H; [H; X+ ] = 2X+ and [H; X- ] = -2X- :
Define linear maps mix : sl2! sl2, neg: sl2! sl2as follows:
(18) neg(H) = H; neg(X+ ) = -X+ ; neg(X- ) = -X- ;
and
(19) mix(H) = H + 2X+ ; mix(X+ ) = -X+ ; mix(X- ) = H + X+ - X- :
It is easy to see that negis a map of Lie algebras, and a routine calculation s*
*hows
mix is one as well. Now we are ready to prove:
Lemma 2.20. Let be a self-representation. Then there is a() 2 k which is the
same as in lemma 2.19, such that = a()ad on sl2, i.e., [(x)](y) = a()[x; y] for
all x; y 2 sl2. Furthermore, if is strong, then a() = 0 or 1.
Proof.One has [(H)](H) = [(X+ )](X+ ) = [(X- )](X- ) = 0 as usual. Now
H; X+ span a Lie subalgebra of sl2which is isomorphic to S via H ! -2y and
X+ ! x. Thus by naturality and lemma 2.19 we have [(H)](X+ ) = a()[H; X+ ].
A similar argument works for X- in place for X+ . So we conclude easily that
(20) [(H)](.) = a()[H; .]:
Thus it follows that [(X )](H) = a()[X ; H]. Now [(X+ )](X- ) = ffH +fiX+ +
flX- . Applying the map of Lie algebras negto this equation, and using naturali*
*ty
one gets:
[(neg(X+ ))](neg(X- )) =ffH - fiX+ - flX-
ffH + fiX+ + flX-f=fH - fiX+ - flX- :
So fi = fl = 0. Thus [(X+ )](X- ) = ffH = ff[X+ ; X- ]. So if we can show ff = *
*a()
we can conclude that
[(X )](.) = a()[X ; .]
and together with equation 20 this implies = a()ad on sl2. So it remains
only to show ff = a(). Applying the Lie algebra map mix to the equation
14 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
[(X+ )](X- ) = ffH one gets:
[(mix(X+ ))](mix(X- )) =ffmix(H)
-[(X+ )](H) + [(X+ )](X- )f=f(H + 2X+ )
-a()[X+ ; H] + ffH =ffH + 2ffX+
2a()X+ = 2ffX+ :
From which one concludes that a() = ff as desired.
Now if = a()ad is a map of Lie algebras, then we have
([X+ ; X- ])= (X+ ) O (X- ) - (X- ) O (X+ )
a()ad([X+ ; X- ])= a()2ad(X+ ) O ad(X- ) - a()2ad(X- ) O ad(X+ )
a()ad([X+ ; X- ])= a()2ad([X+ ; X- ]):
Since ad([X+ ; X- ]) is nonzero, it follows that a() = a()2 or in other_words, *
*that
a() = 0 or 1. |__|
Recall N has basis {x; y; z} and bracket given by [x; z] = [y; z] = 0; [x; y]*
* = z.
Define linear maps x; y : N ! N by
x(y) = x(z) = 0; x(x) = x
and
y(x) = y(z) = 0; y(y) = y:
It is easy to verify that these maps are maps of Lie algebras.
Lemma 2.21. Let be a self-representation. Then there is () 2 k such that
= ()ad on N.
Proof.As z is central one sees by lemma 2.17, that (z) = 0 = ad(z). Now
[(x)](y) = ffx + fiy + z. Applying the Lie algebra map x to this equation and
using naturality one gets 0 = ffx. So one concludes ff = 0. Similarly using y
instead one concludes fi = 0. So [(x)](y) = z = [x; y]. So now it is easy to see
[(x)](.) = [x; .] and [(y)](.) = [y; .] via lemma 2.17 and corollary 2.18._Thus
setting () = we are done. |__|
Recall son gln, the subset of skew-symmetric n x n matrices, is a Lie subalgebr*
*a.
For char(k) 6= 2, so3is 3-dimensional with basis {X = ffi1;2-ffi2;1; Y = ffi1;3*
*-ffi3;1; Z =
ffi2;3- ffi3;2} and bracket given by the relations [X; Y ] = -Z,[Y; Z] = -X,[Z;*
* X] =
-Y . As we are over a finite field of odd characteristic, so3is isomorphic to s*
*l2via
the map which sends X ! X+-X-_2; Y ! aX++X-_2+ bH_2; Z ! bX++X-_2- aH_2. Here
a; b 2 k are such that a2 + b2 = -1.
Lemma 2.22. Let be a self-representation. Then = ()ad = a()ad on so3
where () is the same as that of lemma 2.21 and a() is the same as lemma 2.20.
In particular, a() = ().
Proof.The fact that = a()ad follows from lemma 2.20 and the fact that so3is
isomorphic to sl2. So it remains to show that = ()ad also. Consider the Lie
algebra gl3. In the proof of lemma 2.16 we saw that
{ffii;j; ffij;k; ffii;k}
formed a basis of a Lie subalgebra isomorphic to N for i; j; k distinct. Thus
[(ffii;j)](ffij;k) = ()[ffii;j; ffij;k] = ()ffii;k
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 15
for all i; j; k distinct. Also [(ffii;j)](ffil;m) = 0 if j 6= l; i 6= m as unde*
*r these conditions
[ffii;j; ffil;m] = 0. So
[(ffi1;2- ffi2;1)](ffi1;3-[ffi3;1)(=ffi1;2)](ffi1;3) - [(ffi1;2)](ffi3*
*;1)
- [(ffi2;1)](ffi1;3) + [(ffi2;1)](ffi3;1)
= 0 - (-()ffi3;2) - (()ffi2;3) + 0
= ()(ffi3;2- ffi2;3)
= ()[ffi1;2- ffi2;1; ffi1;3- ffi3;1];
and similarly
[(ffi1;2- ffi2;1)](ffi2;3- ffi3;2) = ()[ffi1;2- ffi2;1; ffi2;3- ffi3*
*;2]
and
[(ffi1;3- ffi3;1)](ffi2;3- ffi3;2) = ()[ffi1;3- ffi3;1; ffi2;3- ffi3*
*;2]:
So with these 3 equations and corollary 2.18 one concludes that = ()ad_on
so3. |__|
We now state:
Theorem 2.23. Let be a strong self-representation. If 6= 0 on sl2then = ad
as self-representations. If = 0 on sl2then = 0 as self-representations.
Proof.If 6= 0 on sl2then we have a() = () = 1 from lemmas 2.20 and 2.22.
Thus = ad on sl2and on N. Thus by lemma 2.16 we have that = ad as *
*__
self-representations. the case = 0 on sl2proceeds similarly. |*
*__|
For use later on, we need to state theorem 2.23 in slightly more generality. *
*To
do this we need to introduce the categories Liek(p) for k 1 and p a prime. The
objects of this category will be Lie algebras over Z=pkZ, that is free Z=pkZ-mo*
*dules
R of finite rank equipped with a bilinear, alternating form [.; .] : R x R ! R *
*which
satisfies the Jacobi identity. The morphisms will be the Z=pkZ-module maps which
preserve the brackets. Thus Lie1(p) is just the same category Lieconsidered bef*
*ore
for k = Fp.
For t k, we have reduction functors Liek(p) ! Liet(p). These reduction
functors are obtained in the obvious way from the reduction map of Z=pkZ to
Z=ptZ. For L 2 Obj(Liek(p)) we denote L to be its reduction in Obj(Lie1(p)). We
will also use the bar notation for the reduction of a morphism. Note that there
is the canonical reduction homomorphism which is a map of abelian groups which
preserves the bracket from L to L, we will refer to maps of abelian groups which
preserves brackets as Lie algebra maps for the rest of this section.
Definition 2.24.A natural self-representation on Liek(p) is a collection of maps
L : L ! gl(L ) of abelian groups, which satisfy the naturality condition
[L0( (x))]( (y)) = ([L(x)](y))
for all L; L02 Obj(Liek(p)), 2 Mor(L; L0), x 2 L and y 2 L. We say the natural
self-representation is strong if the maps L : L ! gl(L ) are maps of Lie algebr*
*as.
Note this definition extends the previous one.
It is easy to check that if we assign L : L ! gl(L ) to be the composition of
the reduction homomorphism with ad : L ! gl(L ) then we obtain a strong self-
representation which we will call ad.
16 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
The Lie algebras gln, sl2,N; S and so3will denote the obvious Lie algebras in
Obj(Liek(p)). For example, glnis the Lie algebra of n x n matrices with Z=pkZ
entries. Theorem 2.23 is true for these more general self-representations. We s*
*tate
this in the next theorem.
Theorem 2.25. Let p be an odd prime and let be a strong self-representation
over Liek(p). If 6= 0 on so3and sl2, then = ad as self-representations.
Proof.The proof proceeds as before with only some modifications so we leave it *
*to
the reader (see [Pak]). The only thing to note is there is a generalized Ado-Iw*
*asawa
theorem (see [W4 ]) which says that any L 2 Obj(Liek(p)) embeds as a subalgebra
of gln(Z=pkZ). __
|__|
2.6. Bockstein spectral sequence implies = ad. Recall we had a strong
self-representation over Fp which was defined using the Bockstein. In this sect*
*ion
we show = ad as self-representations.
By Theorem 2.23, it is enough to show is not identically zero on sl2. To do
this we will need the Bockstein spectral-sequence.
Let X be a topological space (with finitely generated integral homology). Then
the coefficient sequence
0 ! Z p!Z ! Z=pZ ! 0
gives a long exact sequence
. .-.! H*(X; Z) -p!H*(X; Z) mod-!H*(X; Fp) -ffi!H*+1(X; Z) -! . . .
which gives a spectral sequence in a standard way if we think of it as an exact
couple. This spectral sequence has B*1= H*(X; Fp) with differential given by the
Bockstein fi (as defined previously) and converges to Fp(H*(X; Z)=torsion). If *
*we
apply this to the case X=BG where G is a finite group, then the Bockstein spect*
*ral
sequence must converge to zero in positive gradings, as the integral cohomology*
* of
a finite group is torsion in positive gradings.
Now we will apply the Bockstein spectral sequence to show that is not zero on
sl2.
Let {H; X+ ; X- } be the usual basis for sl2and let G = Exp(sl2). Let {h; x+ *
*; x- }
be the obvious dual basis for sl2. Then as we have seen before,
B*1= H*(G; Fp) = ^*(h; x+ ; x- ) Fp[sh; s+ ; s- ]
where we have chosen the s-basis to correspond in the obvious way. Then we see
using the formulas found before that on the exterior algebra part the Bockstein*
* is
given by
fi(h)= -[.; .]H = -x+ x-
fi(x+=)-[.; .]X+ = -2hx+
fi(x-=)-[.; .]X- = 2hx- :
So as B11= ^1(h; x+ ; x- ) one sees that B12= 0. Now note B21= ^2(h; x+ ; x- )
span(sh; s+ ; s- ) and by the above formulas, the exterior part contributes not*
*hing to
B22. If is zero on sl2then this means fi maps span(sh; s+ ; s- ) into ^3(h; x+*
* ; x- )
which is 1-dimensional generated by hx+ x- . So we see fi must have at least a
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 17
2-dimensional kernel on the span of {sh; s+ ; s- }. Thus dim(B22) 2. Now B31is
equal to
^3(h; x+ ; x- ) span({shh; shx+ ; shx- ; s+ h; s+ x+ ; s+ x- ; s- h; s- x+ ;*
* s- x- }):
We will show fi is injective on the second part so that dim(B32) 1. Note that
fi(s*) = a*hx+ x- , for some a* 2 Fp, where * stands for an arbitrary subscript*
*. So
fi(s*)x* = 0 and one has:
fi(s*x*)f=i(s*)x* + s*fi(x*)
= s*fi(x*)
So because {fi(xh); fi(x+ ); fi(x- )} are linearly independent, and {sh; s+ ; s*
*- } are
algebraically independent, it is easy to see fi is injective on the second part*
* of B31
as claimed. Thus we have B12= 0, dim(B22) 2 and dim(B32) 1. However the
Bockstein spectral sequence converges to zero in positive gradings implying we *
*need
dim(B22) dim(B32) which we don't have. Thus we have a contradiction and our
assumption that = 0 must be false.
Thus we finally conclude = ad for all Lie algebras. Now let us chase through
the definitions to get formulas for the fi1 part of fi. So fix a Lie algebra L*
* over
Fp. Fix a basis E = {e1; : :;:en} of L and let G = Exp(L). Then H*(G; Fp) =
^*(x1; : :;:xn)Fp[s1; : :;:sn] where x1; : :;:xn is the dual basis of E and s1;*
* : :;:sn
is the corresponding basis for the polynomial part of the cohomology. Let ckij=
[ei; ej]k be the structure constants of L with respect to the basis E. We have *
*shown
that n
X
[(ei)](ej) = ckijek:
k=1
Recalling that is the dual of -fi1 : S* ! W * S*, we see easily that
Xn
-fi1(sk) = ckijxisj:
i;j=1
Using that ckij= -ckji, one can easily deduce the following equation for fi:
Xn
(21) fi(sk) = ckijsixj+ fi2(sk):
i;j=1
Thus we can also recover the Lie algebra structure of L from knowledge of the
fi1 component of fi|S*. It remains to study fi2 which is the remaining term of *
*the
Bockstein. Before doing that, we will digress to give an application of equatio*
*n 21.
2.7. Comodule algebra structure. Let G 2 Obj(LGrp ) then the multiplication
in G induces a homomorphism : 1(G) x G ! G and
= * : H*(G; Fp) ! H*(1(G); Fp) H*(G; Fp)
gives H*(G; Fp) the structure of a comodule algebra over the Hopf algebra
H*(1(G); Fp): (See [W2 ].)
In order to describe , it is enough to describe it on algebra generators of
H*(G; Fp). To do this let us write
H*(G; Fp) = ^*(x1; : :;:xn) Fp[s1; : :;:sn]
and H*(1(G); Fp) = ^*(t1; : :;:tn) Fp[s1; : :;:sn] with si= fi(ti) for all i. *
*Fur-
thermore we have made our choices so that j*(si) = siwhere j is the inclusion m*
*ap
18 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
of 1(G) into G and also so that equation 21 holds. Recall that the Hopf algebra
structure of 1(G) is one where all the generators siand ti are primitive. Now we
can state:
Corollary 2.26.Let G 2 Obj(LGrp ) and ckijbe the structure constants of Log(G)
in a suitable basis. Using the notation of the paragraph above, one has the fol*
*lowing
formulas which determine :
(xk) = 1 xk
X
(sk) = sk 1 + 1 sk + ckijti xj
i;j
Proof.Precomposing with the natural inclusions of 1(G) and G into 1(G)xG,
it is not hard to show that = * must take the following form on the generators:
(xk) = 1 xk
X
(sk) = sk 1 + 1 sk + akijti xj:
i;j
Thus it is enough to show akij= ckijfor all i; j; k. As the Bockstein fi is a n*
*atural
operation, and since is induced from a homomorphism of groups, they must
commute, that is O fi = (fi 1 + 1 fi) O . Applying fi 1 + 1 fi to the form*
*ula
for (sk) and using equation 21, we get
Xn
(22) (fi 1 + 1 fi)((sk)) = 1 fi(sk) + akij(si xj- ti fi(xj))
i;j=1
but
nX
(fi(sk))= cklm(slxm ) + (fi2(si))
l;m=1
nX X
= cklm(sl xm + 1 slxm + aluvtu xvxm ) + 1 fi2(si)
l;m=1 u;v
Now we know the expressions in equation 22 and the last one are equal so
equating their H2(1(G); Fp) H1(G; Fp) components we get:
nX nX
akijsi xj = cklmsl xm
i;j=1 l;m=1
which immediately gives akij= ckijfor all i; j; k which is what we desired._
|__|
Thus we see that the comodule algebra structure of H*(G; Fp) also determines the
Lie algebra corresponding to G.
2.8. The class [j] and lifting uniform towers. In this section we will show
that the fi2 component for the Bockstein, defines a cohomology class in a suita*
*ble
cohomology group.
Fix L = (W; br) 2 Obj(Lie). Recall the existence of a Koszul complex:
0 ! ^0(L; ad) d!: :!:d^n(L; ad) ! 0
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 19
whose cohomology is H*(L; ad). Here ^i(L; ad) denotes the L-valued alternating
i-forms on L and d is given by
X
(d!)(u0; : :;:ul) =(-1)i+j!([ui; uj]; u0; : :;:^ui; : :;:^uj; : :;:ul)
i 1, xi is induced from Gi-1) and deg(si) = 2 (and
for i > 1, si "comes" from 1(Gi)), and H*(GN ; Fp) is given by the same formula
if and only if the tower extends to a uniform tower {G1; : :;:GN ; GN+1 }.
Proof.By the comments in the preceding paragraph and lemma 2.31 one obtains
the first part of this theorem by induction on i where 1 i < N. More explicitly
note that one has the result for G1. So fix i > 1 and assume we have shown the
first part for all j < i. Then we have the central short exact sequence:
1 ! 1(Gi) ! Gi!ssiGi-1! 1:
So the E*;*2term of the L.H.S.-spectral sequence of this extension is given by
^*(t1; : :;:tn) Fp[s1; : :;:sn] ^*(x1; : :;:xn) Fp[y1; : :;:yn]
where the first two terms are the cohomology of 1(Gi) and the last two terms are
the cohomology of Gi-1by induction. One has as usual d2(si) = d2(xi) = d2(yi) =
0 and d2(ti) are the elements representing the extension. By the proof of lemma*
* 2.31
and the fact Giis an 1 extension of Gi-1one sees (after changing basis for y's)*
* that
these elements are of the form d2(ti) = yi+ i for all i where i2 ^*(x1; : :;:xn*
*).
Thus as in the proof of theorem 2.10, one gets E*;*3= Fp[s1; : :;:sn]^*(x1; : :*
*;:xn)
with d3(xi) = 0. Now E0;*1is the image of i* : H*(Gi; Fp) ! H*(1(Gi); Fp) and
by lemma 2.31 this contains n algebraically independent elements in degree 2. S*
*ince
E0;22has dim n we see by necessity that d3 = 0 on E0;22, which implies d3 = 0. *
*Then
as in theorem 2.10 one sees that H*(Gi; Fp) is as stated. So we are done with t*
*he
first part. If the tower extends to a uniform tower of length one more, then on*
*e can
perform the same argument on GN to conclude its cohomology is as stated. On the
other hand if its cohomology is as stated, image i* : H*(GN ; Fp) ! H*(1(GN ); *
*Fp)
has n algebraically independent elements in degree 2 and so GN has the EP. Using
these elements as in the proof of lemma 2.31, one constructs an extension_to_a
uniform tower of length N + 1. |__|
Examples:
Definition 2.33.Fix p an odd prime number, then n;k(p) is defined as the kernel
of the reduction homomorphism
GLn(Z=pk+1Z) ! GLn(Z=pZ)
for all n; k 1. Similarly, ^n;k(p) is the group obtained by replacing GLn with
SLn in the definition above.
One shows by induction on k that the n;k(p) are p-groups and it is then easy
to show that for fixed n they fit together to give an infinite uniform tower:
! n;k(p) ! n;k-1(p) ! . .!.n;1(p) ! 1:
A similar statement holds for the ^n;k(p).
Theorem 2.32 has the following immediate corollary:
22 WILLIAM BROWDER AND JONATHAN PAKIANATHAN
Corollary 2.34.For n; k 1 and p an odd prime we have:
H*(n;k(p); Fp) ~=^*(x1; : :;:xn2) Fp[s1; : :;:sn2]
and
H*(^n;k(p); Fp) ~=^*(x1; : :;:xn2-1) Fp[s1; : :;:sn2-1]
where deg(xi) = 1 and deg(si) = 2 for all i.
Recall that if G 2 Obj(BGrp) then G = G2 of a uniform tower {G1; G2}. This
tower extends to a uniform tower of length 3 if and only if Log(G) is a Lie alg*
*ebra.
This follows form theorem 2.32 and theorem 2.10. Next, we will show that this
uniform tower {G1; G2} extends to a tower of length 4 if and only if [j] = 0 wh*
*ere
[j] 2 H3(L; ad) is the cohomology class defined earlier.
Now suppose we have a uniform tower {G1; G2; G03} so G2 2 Obj(LGrp) and has
Fp-cohomology given by theorem 2.10. We will now show that there is a uniform
tower {G1; G2; G3; G4} if and only if [j] = 0 where we say G3 instead of G03sin*
*ce
we might have to choose a different 1 extension of G2 to do this. Note the tower
{G1; G2} extends to one of length 4 as desired if and only if G2 has a uniform 1
extension G3 which itself has the EP. By theorem 2.32, this happens if and only*
* if
G3 has Fp-cohomology ^*(x1; : :;:xn) Fp[s1; : :;:sn] where deg(xi) = 1; deg(si*
*) =
2 and n = dim(G1).
So let us study the cohomology of an 1 extension G3 of G2. So H*(G2; Fp) =
^*(x1; : :;:xn) Fp[s1; : :;:sn] where deg(si) = 2; deg(xi) = 1 as usual. Then *
*G3 is
represented in H2(G2; 1(G3)) = [H2(G2; Fp)]n by (s1+1; : :;:sn+n) where i2
^2(x1; : :;:xn) under a suitable choice of basis of 1(G3). However now note that
due to the ambiguous nature of siwe can shift them by elements of ^2(x1; : :;:x*
*n)
so without loss of generality the extension element of G3 is (s1; : :;:sn). Not*
*e this
implies a particular choice of s-basis. Let j be the Bockstein term for this ch*
*oice of
s-basis. Then as we saw in the first sections, when we look at the L.H.S.-spect*
*ral
sequence for
1 ! 1(G3) ! G3 ss3!G2 ! 1
Then
E*;*2= ^*(t1; : :;:tn) Fp[fi(t1); : :;:fi(tn)] ^*(x1; : :;:xn) Fp[s1; : :*
*;:sn]
where the first two terms are the cohomology of 1(G3) and the last two are that
of G2. Again d2(fi(ti)) = d2(xi) = d2(si) = 0 and d2(ti) = si. Thus,
E*;*3= Fp[fi(t1); : :;:fi(tn)] ^*(x1; : :;:xn)
with d3(xi) = 0 and d3(fi(ti)) = {fi(d2(ti))} = {fi(si)} in E3. So by what we s*
*aid
before G3 has the EP if and only if {fi(si)} = 0 for all i. So let us calculate*
* this
explicitly. There are elements i;l2 ^1(x1; : :;:xn) such that:
Xn Xn
fi(si) = i;lsl+ fi2(si) = i;l(0) + fi2(si)
l=1 l=1
where in the last step we used that si= 0 in E3 due to the fact d2(ti) = si. Th*
*us
we get d3(fi(ti)) = fi2(si) for all i. So that G3 defined by the extension ele*
*ment
(s1; : :;:sn) has the EP if and only if fi2 = 0. As we have argued that the 1
extensions of G2 are exactly those defined by such extension elements, we see t*
*hat
there is an 1 extension G3 of G2 such that G3 has the EP if and only if there is
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 23
a choice of s1; : :;:sn via changing up to adding elements in ^2(x1; : :;:xn) s*
*o that
the corresponding fi2 = 0, i.e., if and only if [j] = 0. Thus we have shown:
Theorem 2.35. Let {G1; G2} be a uniform tower. Then it extends to a uniform
tower of length 3 if and only if L = Log(G2) is a Lie algebra. In this case it*
* is
possible to extend {G1; G2} to a uniform tower of length 4 if and only if [j] 2
H3(L; ad) is zero.
2.9. Formulas for B*2. In this section we will derive explicit formulas for B*2*
*of
the Bockstein Spectral Sequence of a group in LGrp . These formulas will express
B*2in terms of various Lie algebra cohomologies of the Lie algebra corresponding
to the group.
Let G 2 Obj(LGrp ) and Log(G) = L. Let n = dim(L) and assume [j] = 0.
Then we have that
H*(G; Fp) = ^*(x1; : :;:xn) Fp[s1; : :;:sn]:
The Bockstein is given by
X
fi(xi) = - cilmxlxm
l k:
Furthermore the jt define a cohomology class [j] 2 H3(L ; ad) which vanishes if*
* and
only if the Lie algebra L has the EP. When this is the case, one can drop the j
terms in the formula above.
Here the definition of EP has been extended to finite Zp-Lie algebras in gene*
*ral
by saying that L has the EP if there is a Zp-Lie algebra L such that L=1(L ) = *
*L.
3.2. Calculating B*2in various cases. Recall that there is a natural equivalence
between the categories Lieand LGrp . Under this equivalence, Abelian Lie algebr*
*as
correspond to abelian groups so are not so interesting. Let us work out B*2for *
*the
nonabelian Lie algebra S which has basis {x; y} and bracket given by the relati*
*on
[y; x] = y. Let the corresponding group in LGrp be denoted G(S). This is a gro*
*up
of order p4 and exponent p2. Then we have the structure constants cyy;x= 1 and
cxy;x= 0. Thus using theorem 3.14, we see that
H*(G(S); Fp) ~=^*(x; y) Fp[X; Y ]
with
fi(x) = 0f,i(y) = xy
fi(X) = 0 ,fi(Y ) = Y x - Xy
where we used that the class [j] vanishes as S lifts to a Lie algebra over the *
*p-adic
integers. Let A be the subalgebra of H*(G(S); Fp) generated by x; yY p-1; X; Y *
*p.
This is isomorphic as graded algebras, to the graded algebra
^*(x; z) Fp[X; Z]
where deg(x) = 1; deg(z) = 2p - 1; deg(X) = 2; deg(Z) = 2p. It is easy to check
that fi vanishes on A. Thus there is a well defined map of graded-algebras from
A to B*2. A direct calculation shows that this is an isomorphism. By comparisons
(restricting to suitable subgroups of G(S)), it is easy to show then that we can
choose z; Z so that fi2 is given by fi2(x) = X; fi2(z) = Z. Thus B*3= 0 for * >*
* 0.
Thus we can conclude that
exp(H* (G(S); Z)) = p2:
Now let us consider the Lie algebra sl2with basis {h; x+ ; x- } with bracket gi*
*ven
by [h; x+ ] = 2x+ ,[h; x- ] = -2x- and [x+ ; x- ] = h. Let G(sl2) = ^2;2(p) be *
*the
corresponding group. Again we note this Lie algebra lifts to the p-adics so by
theorem 3.14 we conclude:
H*(G(sl2); Fp) ~=^*(h; x+ ; x- ) Fp[H; X+ ; X- ]
COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS 29
with Bockstein given by
fi(h) = -x+ x-f,i(H) = X+ x- - X- x+
fi(x+ ) = -2hx+f,i(X+ ) = 2(Hx+ - X+ h)
fi(x- ) = 2hx-f,i(X- ) = -2(Hx- - X- h):
Using this explicit formula and viewing B*2as the sum of Lie algebra cohomologi*
*es
as was shown before, one computes H*(sl2; Fp) is zero except in dimensions 0
and 3 where it is one dimensional. Also H*(sl2; S1) = 0 and H0(sl2; S2) is one
dimensional generated by the Killing form of sl2. This implies the following fa*
*cts:
dim(B12) = dim(B22) = 0, dim(B32) = 1 generated by hx+ x- and dim(B42) = 1
generated by the Killing form 8(H2 + X+ X- ). By comparisons, fi2(hx+ x- ) = 0 *
*so
that B32= B33and thus by necessity (since the Bockstein spectral sequence must
eventually converge to zero), B42= B43. Thus in particular, B*36= 0 in positive
dimensions. Thus we conclude: exp(H* (G(sl2); Z)) > p2. On the other hand,
G(sl2) contains G(S)xZ=pZ as a subgroup of index p. Thus by transfer arguments,
exp(H* (G(sl2); Z)) p exp(H* (G(S); Z)) = p3:
Thus
exp(H* (G(sl2); Z)) = p3:
Note this is neither the order nor the exponent of G(sl2) as G(sl2) has order p*
*6 and
exponent p2.
The authors would like to thank T. Weigel for many useful discussions.
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