AN ALGEBRAIC MODEL FOR CHAINS ON BG^p
DAVE BENSON
Abstract. We provide an interpretation of the homology of the loop space *
*on the p-
completion of the classifying space of a finite group in terms of represe*
*ntation theory,
and demonstrate how to compute it. We then give the following reformulati*
*on. If f is
an idempotent in kG such that f.kG is the projective cover of the trivial*
* module k, and
e = 1 - f, then we exhibit isomorphisms for n 2:
Hn( BG^p; k)~=Tore.kG.en-1(kG.e, e.kG)
Hn( BG^p; k)~=Extn-1e.kG.e(e.kG, e.kG).
Further algebraic structure is examined, such as products and coproducts,*
* restriction
and Steenrod operations.
1.Introduction
This paper grew out of an attempt to understand the relationship^between the *
*work of
Ran Levi [17, 18, 19, 20] (see also Cohen and Levi [6]) on H*( BGp; k), the hom*
*ology of
the loop space on the Bousfield-Kan p-completion of the classifying space of a *
*finite group
G with coefficients in a field k, and the papers [8, 9] of Dwyer, Greenlees and*
* Iyengar. The
result is a representation theoretic description of the homology of BG^p. For *
*reasonably
small groups, this is explicitly computable, and to illustrate the methods we g*
*ive the
results of some computations using the computer algebra system MAGMA [4].
In order to describe our results, we introduce the following operation in mod*
*ular repre-
sentation theory. Let k be a field of characteristic p, and G be a finite group*
*. Recall that
Op(G) is the smallest normal subgroup of G such that the quotient is a p-group.*
* If M is
a finitely generated kG-module, we define [Op(G), M] to be the k-linear span of*
* the set
{g(m) - m | g 2 Op(G), m 2 M}.
This is the smallest submodule of M such that Op(G) acts trivially on the quoti*
*ent, or
equivalently the smallest submodule such that the quotient has a filtration whe*
*re G acts
trivially on the filtered quotients.
Inductively define P0 = N0 to be the projective cover P (k) of the trivial kG*
*-module k,
and for i 1, Mi-1= [Op(G), Ni-1], Piis the projective cover of Mi-1, and Ni= *
* (Mi-1),
the kernel of Pi! Mi-1. This construction gives us a complex of projective kG-m*
*odules
. .!.P2 ! P1 ! P0 ! 0
with the property that Op(G) acts trivially on the homology, and the Pi with i *
* 1 do
not have the projective cover of k as a direct summand. We call this a left k-*
*squeezed
resolution for G, and we write H* (G, k) for the homology of this complex. A mo*
*re formal
definition is given in Section 3.
1
2 DAVE BENSON
The dual of this construction gives us a complex of injective kG-modules
0 ! I0 ! I-1 ! I-2 ! . . .
such that I0 is the injective hull of the trivial module (which is isomorphic t*
*o P (k)),
Op(G) acts trivially on the homology, and the I-i with i 1 do not have P (k) *
*as a direct
summand. We call this a right k-squeezed resolution for G, and we write H*(G, k*
*) for the
homology of this complex, with the indexing negated to make it cohomological.
Our first theorem, which we prove in Section 5, gives a homological interpret*
*ation for
the construction. Let f be an idempotent in kG such that f.kG is isomorphic to*
* the
projective cover of the trivial module, and set e = 1 - f.
Theorem 1.1. For n 2, we have isomorphisms
Tore.kG.en-1(kG.e, e.kG)~=Hn (G, k)
Extn-1e.kG.e(e.kG, e.kG)~=Hn(G, k).
There are exact sequences
0 ! H1 (G, k) ! kG.e e.kG.ee.kG ! kG ! H0 (G, k) ! 0
0 ! H0(G, k) ! kG ! Hom e.kG.e(e.kG, e.kG) ! H1(G, k) ! 0.
Notice that we can rewrite the left hand sides of the isomorphisms in Theorem*
* 1.1 as
Tore.kG.en-1(f.kG.e, e.kG.f) and Ext n-1e.kG.e(e.kG.f, e.kG.f).
This makes computations slightly easier because e.kG.f and f.kG.e are smaller t*
*han e.kG
and kG.e.
Our second theorem gives a topological interpretation of H* (G, k) and H*(G, *
*k).
Theorem 1.2. For i 0, Hi (G, k) ~=Hi( BG^p; k) and Hi(G, k) ~=Hi( BG^p; k).
It is this theorem that we use for the MAGMA computations; examples can be fo*
*und
in the next section. ^
We prove Theorem 1.2 in Section 6 by making a model of BGp that admits a free
action of G with acyclic^quotient. In Section 7, we provide another proof by in*
*terpreting
the chains on BGp as a best approximation to kG by chain complexes built from *
*an
injective resolution of k. This proof^uses more machinery, but makes it easy to*
* interpret
the loop multiplication in H*( BGp; k) algebraically.
Combining the statements of Theorems 1.1 and 1.2, we obtain isomorphisms for *
*n 2:
^
(1.3) Tore.kG.en-1(f.kG.e, e.kG.f)~=Hn( BGp; k)
^
(1.4) Extn-1e.kG.e(e.kG.f, e.kG.f)~=Hn( BGp; k).
Finally, in the last section we discuss some of the problems that motivated t*
*his^work.
These involve the boundary between polynomial and exponential growth for H*( BG*
*p; k).
We completely solve the problem in characteristic two for groups with elementar*
*y abelian
Sylow 2-subgroup E. In this situation, we show that polynomial growth occurs if*
* and only
if H*(G, k) is a complete intersection. This condition is equivalent to the sta*
*tement that
NG (E)=O20NG (E) is a direct product of copies of Z=2, A4 and (Z=2)3o F21, wher*
*e F21 is
a Frobenius group of order 21.
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 3
2.Examples
We illustrate the main theorems with some examples.
Example 2.1. Let G = A4, the alternating group of degree four, and let k be a f*
*ield of
characteristic two. We write k, ! and ~!for the three simple kG-modules. The pr*
*ojective
indecomposable kG-modules have the following structure:
k ! ~!
! ~! ~! k k !
k ! ~!
P (k) P (!) P (~!)
We have P0 = N0 = P (k),
M0 = ! k ~! P1 = P (!) P (~!) N1 = ~! ! k ~! !
M1 = ~!! !~! P2 = P (~!) P (!) N2 = k~! k!
M2 = ~! ! P3 = P (~!) P (!) N3 = ~! ! k k !~ !
Thereafter, the construction has period two, so that Mi~= Mi-2, Pi~= Pi-2and Ni*
*~= Ni-2.
So according to Theorem 1.2, Hn( BG^p; k) has dimension one for n = 0 and n = 1*
*, and
it has dimension two if n 2.
It is shown in Levi [19] that H*( BG^p; k) does not always have polynomial gr*
*owth.
Using the computer system MAGMA [4], some examples of exponential growth were i*
*n-
vestigated using Theorem 1.2, and the following table gives the dimensions of t*
*he first few
homology groups for these examples.
__________________________________________
|Degree |
________________|________________________________________|_
| Group |p |0 1 2 3 4 5 6 7 8 9 10 |
|_____________|__|________________________________________|
| (Z=2)3 o Z=72||1 0 1 3 3 6 12 18 33 57 96 |
| || |
| (Z=3)2 o Z=23||1 1 5 12 32 84 220 576 |
| || |
| (Z=3)2 o Z=43||1 1 3 6 12 22 42 80 150 |
| || |
| (Z=5)2 o Z=35||1 1 3 6 14 30 64 138 |
|______________||_________________________________________|
In each case the action in the semidirect product is faithful, and in the secon*
*d entry the
involution acts to invert every element of (Z=3)2.
Example 2.2. Take the group (Z=3)2o Z=2 given by the second entry in the table *
*above,
with k a field of characteristic three. Then A = e.kG.e is the five dimensional*
* commutative
algebra
A = k[u, v, w]=(uw - v2, u2, uv, vw, w2).
4 DAVE BENSON
This is a Koszul algebra, whose Koszul dual is
A!= Ext*A(k, k) = k=(fffl + flff + fi2).
The module M = e.kG.f is the four dimensional module given in terms of generato*
*rs and
relations as
M = (Ax Ay)=(vx - uy, wx - vy, ux, wy).
Here are pictures of A and M:
O @
u"""|@@w@
"" v| @ O NwNNNupOpp
A = O @ O O M = v | ppNNNp|v
@@ v|""" O|pp N |O
w@@ |""u
O
It is not hard to compute the syzygies of M as an A-module. Since A is a symme*
*tric
algebra with J(A)3 = 0, each nM is an indecomposable module with Rad2 nM = 0 a*
*nd
Soc( nM) = Rad( nM) ~= n-1M=Rad ( n-1M). Set bn = dimk nM=Rad ( nM) and
cn = dimkRad ( nM). Then we have b0 = c0 = 2, cn+1 = bn and bn+1 = 3bn - cn.
This gives us the recurrence relation
b0 = 2, b1 = 4, bn+1 = 3bn - bn-1 (n 2),
and hence the generating function
X 2(1 - t)
bntn = __________2= 2 + 4t + 10t2 + 26t3 + 68t4 + . . .
n 0 1 - 3t + t
Thus bn = 2F2n+1, where Fn is the nth Fibonacci number.
Lemma 2.3. For every homomorphism from nM to M, for n 1, the image lies in
Rad(M) = Soc(M).
Proof (outline).The first step is to prove that there are short exact sequences
0 ! nM ! n+1M ! nk nk ! 0
for all n 0. The second step is to prove by induction on n that for every n *
* 0, every
endomorphism of nM can be written as a multiple of the identity plus an endomo*
*rphism
whose image lies in Rad ( nM). The final step is to compose the given homomorph*
*ism
from nM to M with the composite of the inclusions given by the short exact seq*
*uences
above, M ,! M ,! . .,.! nM to obtain an endomorphism of nM, and examine the
image.
It follows from the lemma that for n 2 there are short exact sequences
0 ! Extn-1A(M, M=Rad (M)) ! ExtnA(M, Rad(M)) ! ExtnA(M, M) ! 0.
Thus for n 2 we have dimkExt nA(M, M) = 2bn - 2bn-1 = 4F2n. So
X X
dimkExtnA(M, M) = -2b1t2 + 2(1 - t) bntn.
n 2 n 2
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 5
Finally, adjusting to give the correct dimensions for n 2, Theorem 1.1 give*
*s us
X ^ X
dim kHn( BGp; k)tn = 1 + t + 5t2 + 2t(1 - t) bntn
n 0 n 2
(1 - t + t2)2
= ___________
1 - 3t + t2
= 1 + t + 5t2 + 12t3 + 32t4 + 84t5 + 220t6 + . . .
p_
This function has poles at (3 5)=2; since one of these is in the interior of*
* the unit
circle in the complex plane, it follows that dimk Hn( BG^p; k) is growing expon*
*entially
with n. Alternatively, note that for n 3 we have dimkHn( BG^p; k) = 4F2n-2, a*
*nd the
Fibonacci numbers grow exponentially.
3. Squeezed resolutions
Let G be a finite group, and k be either a field of characteristic p. We also*
* denote by k
the trivial kG-module, namely the ring k regarded as a kG-module with trivial G*
*-action.
Thus if IG is the augmentation ideal of kG, generated by the elements g - 1 for*
* g 2 G,
then k ~=kG=IG .
If M is a kG-module, we define
[Op(G), M] = {g(m) - m | g 2 Op(G), m 2 M}.
This is the smallest submodule of M such that the quotient has a filtration wit*
*h G acting
trivially on the filtered quotients. We have
Hom kG([Op(G), M], k) = 0.
Dually, we define MOp(G) to be the fixed points of Op(G) on M. This is the l*
*argest
submodule of M that has a filtration such that G acts trivially on the filtered*
* quotients.
We have p
Hom kG(k, M=MO (G)) = 0.
Definition 3.1. A left k-squeezed resolution is a complex
. .!.P2 d2-!P1 d1-!P0 ! 0
of kG-modules satisfying the following conditions.
(i)Each Pn is projective,(
k n = 0
(ii)Hn(P* kG k) ~=
0 n 6= 0,
(iii)[Op(G), Hn(P*)] = 0 for all n.
We can construct a complex satisfying these conditions inductively by the pro*
*cess de-
scribed in the introduction. The dual definition is as follows.
Definition 3.2. A right k-squeezed resolution is a complex
d-1 d-2
0 ! I0 d0-!I-1 --! I-2 --! . . .
of kG-modules satisfying the following conditions.
6 DAVE BENSON
(i)Each In is injective,(
k n = 0
(ii)Hn Hom kG(k, I*) ~=
0 n 6= 0,
(iii)[Op(G), Hn(I*)] = 0 for all n.
We can construct a complex satisfying these conditions inductively by the dua*
*l of the
process described in the introduction.
Theorem 3.3 (Comparison Theorem). (i) Let
. .!.P2 ! P1 ! P0 ! 0
. .!.P20! P10! P00! 0
be complexes together with augmentations P0 ! k and P00! k satisfying:
(a) each Pi a projective kG-module, and the augmentation P0 ! k induces an is*
*omor-
phism H*(P* kG k) ! k;
(b) [OpG, H*(P*0)] = 0.
Then there exists a map of complexes P* ! P*0lifting the identity map on k:
. ._.___//P2___//_P1___//_P0___//k___//_0
| | | ||
| | | ||
fflffl| fflffl| fflffl|||
. ._.___//P20__//_P10__//_P00__//k___//_0
Any two such comparison maps are chain homotopic.
(ii) Let
0 ! I0 ! I-1 ! I-2 ! . . .
0 ! I00! I0-1! I0-2! . . .
be complexes together with coaugmentations k ! I0 and k ! I00satisfying:
(a) [OpG, H*(I*)] = 0;
(b) each I0-iis an injective kG-module, and the augmentation k ! I00induces a*
*n iso-
morphism k ! H*Hom kG(k, I0*).
Then there exists a map of complexes I* ! I0*extending the identity map on k:
0_____//k___//_I0___//I-1___//_I-2__//_. . .
||
|| | | |
|| | | |
|| fflffl| fflffl| fflffl|
0_____//k___//_I00__//I0-1__//_I0-2_//_. . .
Any two such comparison maps are chain homotopic.
Proof.This is an exercise in diagram chasing.
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 7
Corollary 3.4. (i) Given two left k-squeezed resolutions P* and P*0, an isomorp*
*hism
Hom kG(P0, k) ~=Hom kG(P00, k),
lifts to a chain homotopy equivalence P* ! P*0, and hence to an isomorphism
H*(P*) ~=H*(P*0).
(ii) Given two right k-squeezed resolutions I* and I0*, an isomorphism
Hom kG(k, I0) ~=Hom kG(k, I00),
extends to a chain homotopy equivalence I* ! I0*, and hence to an isomorphism
H*(I*) ~=H*(I0*).
Proof.This follows from the theorem.
Definition 3.5. Given a finite group G, Corollary 3.4 implies that the homology*
* of a
left k-squeezed resolution P* does not depend on the choice of resolution. So *
*we define
Hn (G, k) = Hn(P*). Similarly, if I* is a right k-squeezed resolution we define*
* Hn(G, k) =
H-n(I*).
4.Products, coproducts, subgroups, Steenrod operations
4.1. External coproducts. Let G and G0 be finite group. If P* and P*0are left*
* k-
squeezed resolutions for G and G0then P* kP*0is a left k-squeezed resolution fo*
*r G x G0.
This gives a K"unneth theorem for homology. This gives
H* (G, k) k H* (G0, k) ~=H* (G x G0, k).
Dually, if I* and I0*are right k-squeezed resolutions for G and G0then I* k I0*
**is a right
k-squeezed resolution for G x G0. This gives
H*(G, k) k H*(G0, k) ~=H*(G x G0, k).
4.2. Restriction and corestriction maps. Let H be a subgroup of a finite group *
*G. If
P* is a left k-squeezed resolution for G and Q* is a left k-squeezed resolution*
* for H then
the Comparison Theorem 3.3 gives a map Q* ! resG,HP*, and hence a corestriction*
* map
coresH,G:H* (H, k) ! H* (G, k).
Dually, there is a restriction map
resG,H:H*(G, k) ! H*(H, k).
4.3. Internal coproducts. Internal coproducts can be obtained by taking the ext*
*ernal
product for G x G and then restricting to the diagonal copy of G. This gives r*
*ise to a
comparison map P* ! P* P*, and hence to cocommutative coproducts
(4.4) H* (G, k) ! H* (G, k) k H* (G, k).
Dually, there are graded commutative products
(4.5) H*(G, k) k H*(G, k) ! H*(G, k).
8 DAVE BENSON
4.6. Products. Using the Comparison Theorem 3.3, an element of Hn (G, k) can be*
* lifted
to a map of complexes
. ._.___//_P1____//P0_____//_0
| | |
| | |
fflffl| fflffl| fflffl|
. ._.__//_Pn+1___//Pn____//Pn-1___//_._._.//_P0___//_0
which is unique up to chain homotopy. This allows us to use composition of maps*
* to define
a product
(4.7) H* (G, k) k H* (G, k) ! H* (G, k).
This product is associative, but usually not graded commutative.
4.8. Steenrod operations. The comparison map (4.4)is E1 . In particular, if k i*
*s a field,
then there is an action of the Steenrod algebra on H* (G, k) lowering degree. S*
*imilarly,
there is an action of the Steenrod algebra on H*(G, k) raising degree. We may d*
*evelop
this theory algebraically by analogy with the Evens norm map [1, Chapter 4] as *
*follows.
Let k be a field of characteristic p. Suppose that G is a finite group, and c*
*onsider the
group GxZ=p. Then we have Op(GxZ=p) = Op(G). So if J* is a right k-squeezed res*
*olu-
tion for G, then the tensor induced complex J*"GxZ=p satisfies [Op(G), H*(J*"Gx*
*Z=p)] = 0.
There is a subtle point here coming from the signs. If p = 2 the signs are not *
*a problem,
but if p is odd then we must remark that the action of G x Z=p on the cosets of*
* G lies
in the alternating group, and so the action of G on the homology of the tensor *
*product is
trivial.
It follows using the Comparison Theorem 3.3 that if I* is a right k-squeezed *
*resolution
for G and Z* is an injective resolution of k for Z=p then there is a comparison*
* map
J*"GxZ=p ! I* k Z*. So if x 2 Hn(G, k) = H-n(J*) then x p defines an element
of H-np(J*"GxZ=p) and hence by composition an element of H-np(I* k Z*). It may
be checked as in Lemma 4.1.1 of [1] that the resulting element is independent o*
*f the
representative of the homology class, and so we obtain a well defined map
Norm G,GxZ=p:Hn(G, k) ! H*(G, k) k H*(Z=p, k).
As in Section 4.4 of [1], picking out coefficients of elements of H*(Z=p, k) gi*
*ves the Steenrod
operations on H*(G, k). For example if k = F2 then we have H*(Z=2, F2) = F2[t] *
*with
|t| = 1. So if x 2 Hn(G, F2) then
Xn
Norm G,GxZ=2(x) = Sqj(x) tn-j
j=0
with Sqj(x) 2 Hn+j(G, F2). The corresponding formulae for p odd are more compli*
*cated,
but are essentially the same as in [1, Proposition 4.4.3].
4.9. Yoneda interpretation. An n-fold k-squeezed extension for G is a complex *
*of
kG-modules
(4.10) M*: 0 ! k ! Mn-1 ! . .!.M0 ! k ! 0
such that
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 9
(i) [Op(G), H*(M*)] = 0,
(ii) the map k ! Mn-1 is injective, and
(iii) M0 ! k is surjective.
A map of n-fold k-squeezed extensions is a map of complexes which is the identi*
*ty on the
two copies of k. Two n-fold k-squeezed extensions are equivalent if there is a *
*chain of maps
connecting one with the other; in other words, this is the equivalence relation*
* generated
by the existence of a map of extensions.
Let P* be a left k-squeezed resolution for G. Given an n-fold k-squeezed exte*
*nsion as
above, the comparison theorem gives us a map of complexes
Pn-1 _____//._._._//P0____//k____//0
| | |
| | |
fflffl| fflffl|fflffl|
0_____//k___//_Mn-1____//_._._.//_M0___//_k___//0
and hence a map ^i:Ker(Pn-1 ! Pn-2) ! k. The map ^igives us a well defined elem*
*ent
i 2 Hn(G, k) that is an invariant of the equivalence class of the k-squeezed ex*
*tension.
Writing Li for the kernel of ^i, we obtain a map of k-squeezed extensions
0____//_k___//_Pn-1=Li___//_._._.//_P0___//_k___//_0
| | | |
| | | |
fflffl| fflffl| fflffl|fflffl|
0____//_k____//_Mn-1_____//_._._.//_M0___//_k___//_0
This way, we see that the equivalence classes of n-fold k-squeezed extensions a*
*re in one-one
correspondence with elements of Hn(G, k).
4.11. The antipode. If x 2 Hn(G, k) is represented by an n-fold k-squeezed exte*
*nsion
(4.10), then we can form a dual extension
0 ! k ! M*0! . .!.M*n-1! k ! 0.
Here, M* = Hom k(M, k) is the k-linear dual of the kG-module M. Let o(x) 2 Hn(G*
*, k)
be the class of this extension. Then o :H*(G, k) is an antiautomorphism of orde*
*r two for
both the product and the coproduct.
The coproduct (4.4), product (4.7)and antipode (4.11) make H*(G, k) into a co*
*mmu-
tative (but not cocommutative) Hopf algebra.
5.Homological reformulation
In this section we describe the homology of k-squeezed resolutions in terms o*
*f Tor and
Ext over a finite dimensional algebra closely related to the group algebra. Le*
*t f be an
idempotent in kG such that f.kG is the projective cover of the trivial module, *
*and let
e = 1 - f. Let A denote the algebra e.kG.e. The following is Theorem 1.1 from*
* the
introduction.
10 DAVE BENSON
Theorem 5.1. (i) We have isomorphisms Hn (G, k) ~=TorAn-1(kG.e, e.kG) for n 2*
* and
an exact sequence
0 ! H1 (G, k) ! kG.e A e.kG ! kG ! H0 (G, k) ! 0.
(ii) We have isomorphisms Hn(G, k) ~= Extn-1A(e.kG, e.kG) for n 2 and an ex*
*act
sequence
0 ! H0(G, k) ! kG ! Hom A(e.kG, e.kG) ! H1(G, k) ! 0.
Proof.(i) Given a projective resolution of e.kG as an A-module:
. .!.~P2! ~P1! ~P0! 0
we consider the complex of kG-modules
(5.2) . .!.kG.e A ~P2! kG.e A ~P1! kG.e A ~P0! kG ! 0
where the last non-zero map is the composite of the augmentation map and the mu*
*ltipli-
cation map
kG.e A ~P0! kG.e A e.kG ! kG.
It is not hard to show that (5.2)is a left k-squeezed resolution. Define P0 = *
*kG and
Pi= kG.e A ~Pi-1for i 1. Then P* is the mapping cone of the morphism of comp*
*lexes
kG.e A P~*! kG. The long exact sequence of this mapping cone gives the requir*
*ed
statements.
(ii) Dually, given a weakly injective resolution of e.kG by projective A-modu*
*les:
0 ! ~I0! ~I-1! ~I-2! . . .
we consider the complex of kG-modules
(5.3) 0 ! kG ! Hom A(e.kG, ~I0) ! Hom A(e.kG, ~I1) ! Hom A(e.kG, ~I2) ! . . .
where the first map is the composite
kG ! Hom A(e.kG, e.kG) ! Hom A(e.kG, ~I0).
It is not hard to show that (5.3)is a right k-squeezed resolution. Define I0 =*
* kG and
I-i = Hom A(e.kG, ~I-i+1) for i 1. Then I* is the mapping cone of the morphi*
*sm of
complexes kG ! Hom A(e.kG, ~I*). The long exact sequence of this mapping cone a*
*gain
gives the required statements.
6. Loops on BG^p
In this section, we give a homotopy theoretic interpretation of the squeezed *
*resolutions
described in Section 3. We begin by recalling that the Bousfield-Kan completion*
* functor^of
[5] with respect to the coefficient ring Fp takes BG to a space which we denote*
* BGp. There
is a map BG ! BG^pwhich is a mod p homology equivalence, and we have ss1(BG^p) *
*~=
G=Op(G). Applying the Hurewicz theorem to the universal cover of BG^p, we see *
*that
ss2(BG^p) ~=H2(Op(G), Zp).
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 11
Denote by Ap(G) the homotopy fibre of the completion map BG ! BG^p, and choose
a basepoint in Ap(G) mapping to the basepoint in BG. Then the long exact sequen*
*ce of
the fibration shows that there is an exact sequence
1 ! H2(Op(G), Zp) ! ss1(Ap(G)) ! Op(G) ! 1
and since Ap(G) is Fp-acyclic, it follows that this is the universal p-central *
*extension of
Op(G).
Now form the pullback Lp(G) as in the following diagram:
Lp(G) _____//EG
| ||
| |
^ fflffl| fflffl| ^
BGp _____//Ap(G)____//BG____//_BGp
Thus Lp(G) is homotopy equivalent to BG^p. The advantage of the space Lp(G), t*
*hough,
is that it comes with a free action of G with connected acyclic quotient Ap(G).*
* So the
chains C*(Lp(G); k) and the cochains C*(Lp(G); k) form complexes of free kG-mod*
*ules.
Theorem 6.1. C*(Lp(G); k) is a left k-squeezed resolution for G, and C*(Lp(G); *
*k) is a
right k-squeezed resolution for G.
Proof.The fact that G acts freely on Lp(G) gives condition (i) in Definitions 3*
*.1 and 3.2.
The fact that the quotient is connected and acyclic gives condition (ii). The *
*fact that
Op(G) acts trivially on H*(Lp(G); k) and on H*(Lp(G); k) gives condition (iii).
Corollary 6.2. There are natural isomorphisms
^
H* (G, k)~=H*( BGp; k)
^
H*(G, k) ~=H*( BGp; k).
Proof.This follows from Definition 3.5 and Theorem 6.1.
This completes the proof of Theorem 1.2 of the introduction.
7. Approximation in KInj(kG)
In this and the next section, we give another way of viewing the translation *
*between
the topology and the algebra.
We write KInj(kG) for the homotopy category of chain complexes of injective (*
*or equiva-
lently, projective) kG-modules. This category was studied in detail in Benson a*
*nd Krause
[2]. We recall some definitions from Dwyer and Greenlees [7], adapted to this *
*context.
Let ik denote an injective resolution of k, regarded as an object in KInj(kG), *
*where it
is well defined up to isomorphism. In KInj(kG), we say that an object N is ik-t*
*rivial if
the chain complex of homomorphisms Hom kG(ik, N) ' 0. A map is an ik-equivalenc*
*e if
its mapping cone is ik-trivial. We say that X is ik-torsion if we have Hom kG(X*
*, N) ' 0
for all ik-trivial N. We say that X is ik-complete if we have Hom kG(N, X) ' 0*
* for all
ik-trivial N. We write ik-Tors(kG) and ik-Comp (kG) for the subcategories of ik*
*-torsion
and ik-complete complexes.
12 DAVE BENSON
Remark 7.1. It is shown in Theorem 12.3 of [2] that localising subcategories of*
* KInj(kG)
are closed under filtered colimits. From this, it can be easily deduced that an*
* object X in
KInj(kG) is ik-torsion if and only if it is in the localising subcategory gener*
*ated by ik.
Given any object X in KInj(kG), there exists an ik-torsion object Cellik(X) a*
*nd an
ik-equivalence Cellik(X) ! X. Such an object and map are unique up to isomorphi*
*sm.
Similarly, given any object X in KInj(kG), there exists an ik-complete object*
* X^ikand
an ik-equivalence X ! X^ik. This is also unique up to isomorphism.
Lemma 7.2. (i) Let I* be an object in KInj(kG) which is bounded above and whose
homology satisfies [Op(G), H*(I*)] = 0. Then I* is ik-torsion.
(ii)Let P* be an object in KInj(kG) which is bounded below and whose homolog*
*y sat-
isfies [Op(G), H*(P*)] = 0. Then P* is ik-complete.
Proof.(i) Let n be the largest integer such that HnI* 6= 0. Since I* is bounde*
*d above,
it is equivalent to a complex I(0)* such that I(0)j = 0 for j > n. Let X(0)* b*
*e an
injective resolution of SocHnI(0)*, shifted in degree by n, so that X(0)* is a *
*direct sum of
copies of shifts of ik. Then the isomorphism HnX(0)* ! SocHnI(0)* extends to a *
*map of
complexes X(0)* ! I(0)*. Write I(1)* for the mapping cone, so that we have a tr*
*iangle
X(0)* ! I(0)* ! I(1)*. Now I(1)* has smaller homology than I(0), in the sense *
*that
either HjI(1)* = 0 for j n, or HjI(1)* = 0 for j > n and the radical length o*
*f HnI(1)*
is smaller than that of HnI(0)*.
So we repeat and get X(1)* ! I(1)* ! I(2)*, and so on. By the octahedral axio*
*m, we
have an object Y (1)* and triangles X(0)* ! Y (1)* ! X(1)* and Y (1)* ! I(0)* !*
* I(2)*.
Proceeding inductively, with Y (0)* = X(0)*, we obtain objects X(i)*, Y (i)* an*
*d I(i)*
fitting into commutative diagrams
Y (i - 1)*_____Y (i - 1)*
| |
| |
fflffl| fflffl|
Y (i)*_______//I(0)*_____//_I(i + 1)*
| | |
| | |
fflffl| fflffl| fflffl|
X(i)*________//I(i)*_____//I(i + 1)*
Taking the homotopy colimit of the sequence of triangles
Y (0)*____//Y (1)*__//_Y (2)*__//. . .
| | |
| | |
fflffl| fflffl| fflffl|
I(0)*______I(0)*_____I(0)*______. . .
| | |
| | |
fflffl| fflffl| fflffl|
I(1)*_____//I(2)*___//I(3)*____//. . .
we have lim-!I(i)* ' 0 and so I* ' I(0)* ' lim-!Y (i)* is in the localising sub*
*category of
i i
KInj(kG) generated by ik.
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 13
(ii) The proof of this is dual. There is just one more point to be made, whic*
*h is that
generally the limit of an inverse system of triangles is not a triangle. Howeve*
*r, the inverse
systems in question are eventually constant in any particular degree, so the in*
*verse limit
of the triangles is indeed a triangle.
Theorem 7.3. (i) If I* is a right k-squeezed resolution then I* ~=Cellik(kG) in*
* KInj(kG).
(ii) If P* is a left k-squeezed resolution then P* ~=kG^ikin KInj(kG).
Proof.(i) Lemma 7.2 (i) shows that I* is ik-torsion. There is a map I* ! kG whi*
*ch is an
ik-equivalence, because its mapping cone is equivalent to a complex consisting *
*of injective
modules which admit no homomorphisms from k.
(ii) Lemma 7.2 (ii) shows that P* is ik-complete. There is a map kG ! P* whic*
*h is an
ik-equivalence for the same reason as before.
Now recall that if X is simply connnected, or k is a field of characteristic *
*p and ss1(X)
is a finite p-group, then the Eilenberg-Moore construction gives equivalences
REnd C*(X;k)(k)~=C*( X; k)
L *
k C*(X;k)k~=C ( X; k)
(see x4.22 of [8]). We can't apply this directly to BG with G a finite group, *
*because
ss1(BG) = G is not necessarily a finite p-group. However, ss1(BG^p) = G=Op(G) i*
*s a finite
p-group, and we have C*(BG^p; k) ' C*(BG; k). In particular, we have
^
(7.4) C*( BGp; k) ~=REnd C*(BG;k)(k)
^ L
(7.5) C*( BGp; k) ~=k C*(BG;k)k.
Let EG be the differential graded algebra End kG(ik). Recall from Theorem 3.*
*3 and
Lemma 6.4 of Benson and Krause [2] that there is an adjunction
o-LEGiko_
KInj(kG) ________//Ddg(EG ) ' Ddg(C*(BG; k))
HomkG(ik,-)
inducing an equivalence between Ddg(C*(BG; k)) and ik-Tors(kG). The composite i*
*s the
"cellularisation functor" in KInj(kG)
L
Cellik(X) = Hom kG(ik, X) EG ik.
We have Hom kG(ik, kG) ~=k in Ddg(EG ), so that under this correspondence, th*
*e object
k in Ddg(C*(BG; k)) corresponds to Cellik(kG) in ik-Tors(kG).
Similarly, the functor Hom kG(ik, -): KInj(kG) ! Ddg(C*(BG; k)) has a right a*
*djoint,
and this induces an equivalence between Ddg(C*(BG; k)) and ik-Comp (kG). The co*
*mpos-
ite is the "completion functor" in KInj(kG) sending X to X^ik.
Theorem 7.6. We have
(i)C*( BG^p; k) ~=EndkG (Cellik(kG)) ~=EndkG (kGi^k),
(ii)H*( BG^p; k) ~=H*(kG^ik),
14 DAVE BENSON
(iii)H*( BG^p; k) ~=H-*(Cellik(kG)).
Proof.Part (i) follows from the previous discussion.
(ii) Using the map kG ! kG^ik, we have isomorphisms
^ ^ ^ ^
H*(kGik) ~=Hom kG(kG, kGik)* ~=Hom kG(kGik, kGik)*.
(iii) follows from (ii) by dualising both sides.
Combining Theorems 7.3 and 7.6, we obtain our second proof of Theorem 1.2. T*
*he
advantage of this proof is that it makes it clear that the product (4.7)in H* (*
*G, k) cor-
responds to the loop product in H*( BG^p; k). This does not seem so apparent fr*
*om the
proof in Section 6.
8. Final remarks
Using work of F'elix, Halperin and Thomas [12], Levi [19] proves that H*( BG^*
*p; k) has
either polynomial or semi-exponential growth. Here, semi-exponential growth mea*
*ns that
there exists ~ > 1 such that for an infinitepnumber_of positive integers n the *
*dimension of
the nth homology group is at least ~ n. In the case of polynomial growth, the *
*homology
is finitely generated, nilpotent, and a finite module over a central polynomial*
* subalgebra
(cf. Theorem A of [10], Theorem B of [11]). No examples are known where the gr*
*owth
is semi-exponential but not exponential. We close with some remarks on the bou*
*ndary
between polynomial and semi-exponential growth.
A theorem of Gulliksen [14, 15] states that if R is a commutative Noetherian *
*local
ring with residue field k then R is a complete intersection if and only if Ext**
*R(k, k) has
polynomial growth.
In the case of connected graded commutative algebras, we have to modify the d*
*efinition
slightly to take account of the fact that odd degree elements anticommute inste*
*ad of
commuting. We say that a graded commutative ring R is of polynomial type over R*
*0 = k
if it can be generated by a set of elements x1, . .,.xr of positive degree subj*
*ect only to
the relations coming from graded commutativity. Notice that if xi is an element*
* of odd
degree then graded commutativity forces 2x2i= 0. If k is a field of characteris*
*tic two then
this definition just gives polynomial rings. If k is a field of odd characteris*
*tic, on the other
hand, then it defines a polynomial ring on the even degree generators tensored *
*with an
exterior algebra on the odd degree generators.
A homogeneous element y of a graded commutative ring R is said to be regular *
*if the
annihilator of y is (y - (-1)|y|y); i.e., it is the zero ideal if |y| is even, *
*and (2y) if |y| is
odd. A sequence of elements y1, . .,.ys of positive degree is said to be a regu*
*lar sequence
if each yi is regular in R=(y1, . .,.yi-1).
A connected graded k-algebra is said to be a complete intersection if it can *
*be written
as a quotient of a k-algebra of polynomial type by a regular sequence. With thi*
*s definition,
Gulliksen's theorem continues to hold (Avramov and Iyengar, unpublished).
Theorem 8.1. If H*(G, k) is a complete intersection then H*( BG^p; k) has polyn*
*omial
growth.
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 15
Proof.The theorem follows from Gulliksen's theorem and the Eilenberg-Moore spec*
*tral
sequence
^
Ext**H*(G,k)(k, k) ) H*( BGp; k).
We can prove the converse of Theorem 8.1 in the special case of an elementary*
* abelian
Sylow 2-subgroup in characteristic two. In this case, there is no ambiguity as *
*to what is
meant by a complete intersection since there are no signs to worry about.
Theorem 8.2. Suppose that k is a field of characteristic two, and that G has an*
* elemen-
tary abelian Sylow 2-subgroups E. Then the following are equivalent:
(i) H*( BG^2; k) has polynomial growth.
(ii) H*(G, k) is a complete intersection.
(iii) NG (E)=CG (E) is an odd order group generated by elements of order thre*
*e whose
fixed point set on E has index four.
(iv) NG (E)=O20NG (E) is a direct product of copies of Z=2, A4 = (Z=2)2 o Z=3*
* and
(Z=2)3 o F21, where F21 is the nonabelian group of order 21 and the action on (*
*Z=2)3 is
faithful.
(v) G=O20(G) is a direct product of copies of Z=2, A4, L2(q) (q = 4 or q 3,*
* 5 (mod 8)),
the Ree groups 2G2(32m+1) (m 1), (Z=2)3 o F21, L2(8) o Z=3 and the Janko grou*
*p J1.
Proof.First we prove that (i) is equivalent to (ii). If G has an elementary abe*
*lian Sylow
2-subgroup then C*(BG; k) is equivalent as a differential graded algebra to its*
* cohomology.
To see this, firstly it's true for G = Z=2 since the reduced bar complex is the*
* minimal
resolution. For a more general elementary abelian 2-group, we use tensor produc*
*ts of copies
of the reduced bar complex for the direct factors of the group to see that C*(B*
*G; k) is
equivalent to H*(G, k). In the case of a group with elementary abelian Sylow 2-*
*subgroup
E, C*(BG; k) is equivalent to the invariants of NG (E)=CG (E) on H*(E, k).
It follows that the Eilenberg-Moore spectralPsequence stops at the E2 page, s*
*o that
the dimension of Hn( BG^p; k) is equal to i+j=ndimk Exti,jH*(G,k)(k, k). Now *
*we can use
Gulliksen's theorem to see that (i) is equivalent to (ii).
Since H*(G, k) = H*(E, k)NG(E)=CG(E), the question of when (ii) holds reduces*
* to the
question of when the polynomial invariants of an odd order group over F2 form a*
* complete
intersection.
To prove that (ii) implies (iii), we remark that the general question of when*
* polynomial
invariants of a finite group form a complete intersection was studied by Gordee*
*v [13], Kac
and Watanabe [16] and Nakajima [21, 22, 23], among others. In particular, it is*
* known
that if the invariants form a complete intersection then the group is generated*
* by elements
whose fixed space has codimension at most two. Translating this into group theo*
*ry, this
means that NG (E)=CG (E) should be generated by elements of order three whose f*
*ixed
points on E have index four. This completes the proof that (ii) implies (iii).
The statement that (iv) implies (ii) is just a question of checking that the *
*cohomology
is a complete intersection in the cases given, and then using the K"unneth form*
*ula for the
cohomology of a product. The equivalence of (iv) and (v) is easy to deduce fro*
*m John
Walter's classification [24] of finite groups with an abelian Sylow 2-subgroup,*
* together
with Bombieri's proof [3] that the groups of Ree type are Ree groups.
16 DAVE BENSON
It remains to prove that (iii) implies (iv). I am grateful to Geoff Robinson *
*for supplying
me with a proof of this. We use the phrase "3-reflection" for an element of ord*
*er three in
~G= NG (E)=CG (E) whose centraliser in E has index four.
First we show that we may decompose ~Gas a direct product of groups ~Gi, and *
*E as a
direct product of Ei, so that ~Giacts trivially on Ej for j 6= i, and ~Giis gen*
*erated by a
single conjugacy class of 3-reflections. To see this, we note that if x and y a*
*re 3-reflections
in ~Gwhich generate non-conjugate subgroups then is isomorphic to an odd*
* order
subgroup of GL (4, 2) which is not isomorphic to F21, and which must therefore *
*be an
elementary abelian 3-group. It follows that the conjugates of x and the conjuga*
*tes of y
generate mutually centralising normal subgroups of ~G. Suppose that U is an irr*
*educible
summand of E on which both x and y act non-trivially. Setting F = EndF2~G(U), w*
*e can
regard U as an FG~-module. We can write U = U1 F U2 so that x acts as scalars *
*on U1
and non-trivially on U2, and y acts as scalars on U2 and non-trivially on U1. S*
*ince x and y
are 3-reflections, this implies that either U1 or U2 is one dimensional; then w*
*e deduce that
the other is also one dimensional. So x and y both act as scalars and U is one *
*dimensional
over F. Thus y acts as x or x-1 on U and trivially on the remaining summands of*
* E. So
we can remove y from the generating set without loss. This completes the proof*
* of the
decomposition.
This means that we may assume that the action of ~Gon E is faithful and irred*
*ucible.
If G~ is trivial then E ~= Z=2. Assuming we are not in this case, G~ is genera*
*ted by a
single conjugacy class of 3-reflections. The next step is to show that E is pri*
*mitive. If it
is induced from an F2H-module V for a proper subgroup H of ~Gthen x does not li*
*e in
the intersection of the conjugates of H, so V most be one dimensional, since ot*
*herwise x
could not act as a 3-reflection. But then V is the trivial module, and this con*
*tradicts the
irreducibility of E. Thus E is primitive.
As before, let F = F2e= EndF2~G(E), so that we may regard E as an FG~-module.*
* Since
~Gcontains a 3-reflection x, we have e = 1 or 2. If e = 2 then the fixed point*
*s of x
on E have F-codimension one. So for any g 2 ~G, is isomorphic to an od*
*d order
subgroup of GL(2, 4) ~= Z=3 x A5 generated by elements of order three. Hence <*
*x, xg>
is abelian. Since G~ is generated by the conjugates of x, it follows that G~ i*
*s cyclic, so
NG (E)=O20NG (E) ~=A4 and E ~=(Z=2)2. So if we are not in this case, we may ass*
*ume
that E is absolutely irreducible.
If E were absolutely primitive as an F2G~-module then using Clifford theory w*
*ith respect
to the Fitting subgroup F (G~) of the soluble group ~G, we see that CG~(F (G~))*
* = ZF (G~)
would be central in ~G, and hence by Schur's lemma it would be trivial, a contr*
*adiction. So
for some finite extension F of F2, F F2E is induced from some proper maximal s*
*ubgroup
H of ~G. If x is a 3-reflection in ~Gthen x must act as a single 3-cycle and po*
*ssibly some
fixed points on the cosets of H, or else the eigenvalues would be wrong. So the*
* permutation
action of ~Gon the cosets of H is generated by 3-cycles and has odd order, so t*
*hese 3-cycles
commute. So G modulo the intersection of the conjugates of H is abelian, and he*
*nce H is
normal of index 3. Since ~Gcontains 3-reflections, it follows that it's induced*
* from a one
dimensional representation of H, and so E is 3-dimensional. Thus ~Gis isomorphi*
*c to a
subgroup of GL(3, 2), and the only possibility is ~G~=F21.
^
AN ALGEBRAIC MODEL FOR CHAINS ON BGp 17
Remark 8.3. The smallest simple group with elementary abelian Sylow 2-subgroups*
*, and
not satisfying the conditions of the theorem is L2(8). Since the Sylow 2-normal*
*izer in this
group is (Z=2)3 o Z=7, this corresponds to the first row of the table in Sectio*
*n 2.
Conjecture 8.4. If G has abelian Sylow p-subgroups, then H*(G, k) is a complete*
* inter-
section if and only if H*( BG^p; k) has polynomial growth.
The conjecture fails badly when the Sylow p-subgroups are not abelian. For ex*
*ample,
there are plenty of p-groups whose cohomology rings are not complete intersecti*
*ons. In this
situation, we should use a "derived" notion of complete intersection. Dwyer, G*
*reenlees
and Iyengar [9] formulate the following definition, which we shall call "quasi *
*complete
intersection".
Definition 8.5. A differential graded augmented algebra ! k is said to be a q*
*uasi
complete intersection if, given any object X in D( ) with the augmentation idea*
*l in
its support, the thick subcategory it generates contains a non-zero object in t*
*he thick
subcategory generated by .
The following conjecture is due to John Greenlees:
Conjecture 8.6. For any finite group G, C*(BG; k) is a quasi complete intersect*
*ion if
and only if H*( BG^p; k) has polynomial growth.
Finally, we reiterate and strengthen a conjecture made by Levi [17].
Conjecture 8.7. For any finite group G, H*( BG^p; k) is a finitely generated k-*
*algebra.
Furthermore, either it has polynomial growth or it contains a free algebra on t*
*wo genera-
tors.
Acknowledgment. The author thanks the following people for stimulating convers*
*a-
tions about this work: Bill Dwyer, John Greenlees, Henning Krause, Ran Levi, Ma*
*rkus
Linckelmann, Geoff Robinson and Rapha"el Rouquier.
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Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE