COMMUTATIVE ALGEBRA FOR COHOMOLOGY RINGS OF
CLASSIFYING SPACES OF COMPACT LIE GROUPS.
D. J. BENSON AND J. P. C. GREENLEES
Abstract. We apply the techniques of highly structured ring and module sp*
*ectra to
prove a duality theorem for the cohomology ring of the classifying space *
*of a compact Lie
group. This generalizes results of Benson-Carlson [2, 3] and Greenlees [1*
*0] in the case of
finite groups. In particular, we prove a functional equation for the Poin*
*care series in the
oriented Cohen-Macaulay case.
1. Introduction
Some time ago,1 based on joint work with Carlson [3] on finite group cohomolo*
*gy, the first
author made the following conjecture. Let G be a compact Lie group, BG its clas*
*sifying
space, and k any field of coefficients. Then, provided that H*(BG; k) is Cohen-*
*Macaulay,
the Poincare series
X
pG(t) = tidim kHi(BG; k);
i0
regarded as a rational function of t, satisfies the functional equation
pG(1=t) = td(-t)rpG(t):
Here, d = dim(G) denotes the dimension of G as a manifold, and r = rp(G) denote*
*s the
maximal rank of a finite elementary abelian p-subgroup of G if char(k) = p is a*
* prime,
and the Lie rank r0(G) if char(k) = 0; Quillen has shown this is the Krull dime*
*nsion of
H*(BG; k). In particular, this conjecture implies that if H*(BG; k) is Cohen-M*
*acaulay
then it is Gorenstein.
Even if H*(BG; k) is not Cohen-Macaulay, the conjecture goes on to say that f*
*or any
choice of a homogeneous set of parameters i1; : :;:ir 2 H*(BG; k) with ii in co*
*degree2 ni,
there is a spectral sequence of the form described in [3], convergingPto the co*
*homology of
a finite Poincare duality complex of formal dimension dim(G) + ri=1(ni- 1). T*
*he results
of [3] verify the conjectures when G is finite, but the methods do not appear t*
*o extend.
In the meanwhile, also for finite groups G, the second author [10] applied th*
*e methods
of [3] to construct another spectral sequence using Grothendieck's local cohomo*
*logy of
H*(BG; k) with respect to the ideal J of elements of positive codegree, and con*
*verging to
H*(BG; k). This gives the same information in the Cohen-Macaulay case, and is c*
*losely
related to what happens in the spectral sequence of [3] in the limit as the gen*
*erators are
replaced by higher and higher powers.
___________
1The Summer of 1991, to be precise.
2Because we wish to view cohomology as homology with the degrees negated (and*
* vice-versa), we use
the word degree to denote homological degree, and codegree to denote cohomologi*
*cal degree.
1
2 D. J. BENSON AND J. P. C. GREENLEES
In fact, it turns out that the conjecture is false, but for subtle reasons to*
* do with ori-
entation. The simplest counterexample is the orthogonal group O(2) over a fiel*
*d k with
char(k) 6= 2. The problem comes from the fact that the adjoint representation A*
*d (G) of
G is not orientable. In this paper k will denote an arbitrary commutative Noeth*
*erian ring
unless otherwise stated, and we describe a spectral sequence which gives a sort*
* of global
duality for the ring H*(BG; k). In case Ad (G) is orientable, the statement is *
*as follows.
Theorem 1.1. If G is a compact Lie group of dimension d with the property tha*
*t the
adjoint representation Ad (G) is orientable over the ring k, there is a spectra*
*l sequence of
the form
H*;*J(H*(BG; k)) =) -dH*(BG; k):
Here, -d denotes a shift of d in degree, and H*;*Jdenotes local cohomology wi*
*th respect
to J (we recall the definition in Section 2, and the grading conventions are ma*
*de explicit in
Corollary 5.2). More generally, without the orientability assumption, the spect*
*ral sequence
converges to a twisted form of the homology of BG (see Theorem 5.1). Namely, th*
*e adjoint
representation may be regarded as a group homomorphism G ! O(d) to the orthogon*
*al
group of the tangent space at the identity. Compose this homomorphism with the *
*determ-
inant homomorphism O(n) ! {1}, to get a homomorphism : G ! {1} kx whose
kernel is a subgroup H of index one or two in G. The subgroup H contains the co*
*nnected
component of the identity in G, so induces a homomorphism from ss1(BG) ~=ss0(G*
*) to
{1} kx , and hence a local system " on BG. The spectral sequence then takes th*
*e form
H*;*J(H*(BG; k)) =) -dH*(BG; "):
Notice that if k is a field and H 6= G then k does not have characteristic two,*
* and in this
case, H*(BG; k) and H*(BG; ") are the +1 and the -1 eigenspaces of the action o*
*f G=H
on H*(BH; k) respectively.
We remark that there is still no known analogue for compact Lie groups of the*
* resolutions
constructed in [3] for finite groups. However, the above theorem gives enough i*
*nformation
to deduce what we want about Poincare series. Indeed since local cohomology de*
*tects
depth, if H*(BG; k) is Cohen-Macaulay and Ad (G) is orientable over k the theor*
*em states
that Hr;*(H*(BG; k)) is the (d + r)th desuspension of H*(BG; k). If k is a fiel*
*d this is the
canonical module and so H*(BG; k) is also Gorenstein. It also has the following*
* implication
about Poincare series.
Theorem 1.2. Suppose that Ad (G) is orientable overPa field k, and that H*(BG*
*; k) is
Cohen-Macaulay. Then the Poincare series pG(t) = i0 dimk Hi(BG; k), regarded *
*as a
rational function of t, satisfies the functional equation
pG(1=t) = tdim(G)(-t)rp(G)pG(t):
We remark that the assumption of orientability of Ad(G) is satisfied whenever*
* G is finite,
or the component group of G is of odd order, or k has characteristic two. It is*
* not satisfied
for the orthogonal group O(2) unless char(k) = 2.
We use the method outlined in [9], which can be implemented in the category o*
*f highly
structured module spectra over a highly structured ring spectrum introduced by *
*[6]. It
COHOMOLOGY RINGS OF CLASSIFYING SPACES 3
is proved in the companion paper by Elmendorf and May [8] that Borel cohomology*
* is
represented by a highly structured ring spectrum; using this, it is rather rout*
*ine to complete
the proof using Venkov's theorem [16, 17] that the cohomology of the classifyin*
*g space is a
Noetherian ring.
The rest of the paper is arranged as follows. We begin in Section 2 by recall*
*ing the algebra
necessary to make sense of the statement of the main theorem. In Section 3 we i*
*llustrate
the use of the theorem by giving a number of calculations. We then begin to int*
*roduce the
method of proof by giving a quick summary of relevant facts about the Elmendorf*
*-Kriz-
Mandell-May category of highly structured modules. This prepares us for the pro*
*of itself;
we recall the strategy from [9], and verify the the relevant algebraic hypothes*
*es in Section
5.
2. Local cohomology
In this section we summarize the basic definitions and properties of Grothend*
*ieck's local
cohomology. The basic reference is [12], but an expository summary in a form su*
*itable for
our use is given in [11].
Suppose given a ring R, which is either ungraded and commutative, or graded a*
*nd graded
commutative, and which need not be Noetherian, and suppose given a finitely gen*
*erated
ideal J = (fi1; fi2; : :;:fin). If R is graded the fii are required to be homog*
*eneous.
For any element we may consider the stable Koszul cochain complex
Ko(fi) = (R ! R[1=fi])
concentrated in codegrees 0 and 1. Notice that we have a fibre sequence
Ko(fi)- ! R -! R[1=fi]
of cochain complexes. We may now form the tensor product
Ko(fi1; : :;:fin) = Ko(fi1) : : :Ko(fin):
It is clear that this complex is unchanged if we replace some fi by a power, an*
*d it is not hard
to check that if we invert any element of the ideal J the complex becomes exact*
*. Therefore,
up to quasi-isomorphism Ko(fi1; : :;:fin) depends only on the radical of the id*
*eal J, and we
henceforth write Ko(J) for it. Notice that there is an augmentation Ko(J) -! R *
*obtained
by tensoring the augmentations of the factors.
Following Grothendieck we define the local cohomology groups of an R-module M*
* by
H*J(M) := H*(Ko(J) M):
It is easy to see that H0J(M) is the submodule J(M) := {m 2 M|JN m = 0 for some*
* N}
of J-power torsion elements of M. If R is Noetherian it is not hard to prove di*
*rectly that
H*J(R; .) is effaceable and hence that local cohomology calculates right derive*
*d functors of
J(.). It is clear that the local cohomology groups vanish above codegree n, bu*
*t in the
Noetherian case Grothendieck's vanishing theorem shows the powerful fact that t*
*hey are
zero above the Krull dimension of R. The other fact we shall use is that if fi*
* 2 J then
H*J(R; M)[1=fi] = 0; this is a restatement of the exactness of Ko(J)[1=fi].
4 D. J. BENSON AND J. P. C. GREENLEES
When R and M are graded the local cohomology group HsJ(M) is itself graded, a*
*nd we
write Hs;tJ(M) for the codegree t part in the standard way.
3. Sample calculations
For the examples we restrict attention to the case when k is a field. The fir*
*st case to look
at is where G is connected and the cohomology H*(G; k) of G as aPmanifold is an*
* exterior
algebra ("i1; : :;:"ir) with deg("ii) = ni-1, so that dim(G) = ri=1(ni-1). In*
* this case, by
Theoreme 19.1 of Borel [5], the cohomology H*(BG; k) is a polynomial ring on ge*
*nerators
ii of codegrees ni. In particular, it is Cohen-Macaulay, and it is easy to che*
*ck that the
functional equation (Theorem 1.2) holds for
Yr
1
pG(t) = ______ni:
i=11 - t
For a more non-trivial Cohen-Macaulay example, we can look at the spinor grou*
*ps
G = Spin(n) in characteristic two. Quillen [15] has calculated the cohomology i*
*n this case,
and the answer is
H*(BSpin(n); F2) = F2[w2; : :;:wn]=(j2; j3; j5; : :;:j2n-r-1+1) F2[i2n*
*-r]:
Here, F2[w2; : :;:wn] is a polynomial ring in the n - 1 Stiefel-Whitney classes*
* for BSO(n),
r = r2(G) is roughly half of n but varies in a way that depends on the residue *
*class of n
modulo eight, j2j-1+1(1 j n - r) are elements of codegrees 2j-1+ 1 which form*
* a
regular sequence in F2[w2; : :;:wn], and i2n-ris an independent generator in co*
*degree 2n-r.
Thus the Poincare series is
n-r-1+1
(1 - t2)(1 - t3)(1 - t5) . .(.1 - t2 )
pG(t) = ____________________________________n-r
(1 - t2) . .(.1 - tn)(1 - t2 )
and
dim (G) = n(n - 1)=2 = 1 + 2 + . .+.(n - 1) + (2n-r - 1) - 1 - 2 - 4 - . .-.2*
*n-r-1;
so that an easy check shows that the functional equation (Theorem 1.2) is again*
* satisfied
in this case.
In the Cohen-Macaulay case, Hs;*J(H*(BG; k)) is zero except when s = r (= rp(*
*G)), and
then
Hr;*J(H*(BG; k)) ~=-(d+r)H*(G; k):
Recalling our convention that homology is just negatively graded cohomology (an*
*d the
suspension is cohomological), this means that the E2 page of the spectral seque*
*nce sits in
the fourth quadrant, and consists of H*(BG; k) in the rth column, starting in d*
*egree -d-r
and working downwards.
Theorem 1.2 is now readily verified using the fact that in the Cohen-Macaulay*
* case
H*(BG; k) is finitely generated and free over the polynomial subring on generat*
*ors i1; : :;:ir
which generate an idealQwith radical J. Thus the Poincare series has the form *
*pG(t) =
q(t)r(t) where q(t) = ri=11=(1 - tni) and r(t) is a polynomial. The Poincare*
* series of
Hr;*(H*(BG; k)) is readily checked to be t-nr(t)q(1=t) where n = n1 + n2 + . .+*
*.nr, and
COHOMOLOGY RINGS OF CLASSIFYING SPACES 5
the Poincare series of -(d+r)H*(BG; k) is t-(d+r)r(1=t)q(1=t). Hence r(1=t) = t*
*d+r-nr(t),
and as remarked above q(1=t) = (-1)rtnq(t).
For a family of examples in which the orientation problem interferes with the*
* functional
equation, look at the orthogonal groups G = O(2n), with k a field which does no*
*t have
characteristic two. Let H = SO(2n), the connected normal subgroup of index two *
*in G. In
this case, H*(BH; k) is a polynomial algebra on n generators k[p1; p2; : :;:pn-*
*1; e], with pi
in codegree 4i and e in codegree 2n (pi are the Pontrjagin classes and e is the*
* Euler class,
which satisfies e2 = pn). The group G=H ~=Z=2 acts on this ring by fixing the P*
*ontrjagin
classes and negating the Euler class. Thus
H*(BG; k) ~=H*(BH; k)G=H = k[p1; : :;:pn]:
Although this is a Cohen-Macaulay ring, and even a Gorenstein ring, the dualizi*
*ng class
is in the wrong degree. The functional equation satisfied is
2+2n-1 n
pG(1=t) = t4n (-t) pG(t);
whereas dim(G) = 4n2 - 1. The reason for this is that elements of G which are n*
*ot in H
act on the adjoint representation Ad(H) = Ad(G) with a reverse in orientation. *
*So instead
of computing H*(BG; k), with G=H = ss1(BG) acting trivially on k, we should mak*
*e it act
as the sign representation ". Then
H*(BG; ") = k[p1; : :;:pn] . e;
the free module of rank one over H*(BG; k) generated by the Euler class. The s*
*hift in
degree of 2n effected by this takes care of the dualizing degree.
In general, the stable Koszul complex may be regarded as an E1 page for the s*
*pectral
sequence. It consists of H*(BG; k) in the zeroth column (in non-negative degre*
*es), the
direct sum of the rings obtained by inverting each iiin turn in the first colum*
*n, and so on,
until the rth and last column consists of H*(BG; k) with all the ii inverted.
For an example which is not Cohen-Macaulay, we examine the (simply connected)*
* com-
pact Lie group E6 of dimension 78, in characteristic two. The cohomology was ca*
*lculated
by Kono and Mimura [13], and the answer is
H*(BE6; F2) = F2[y4; y6; y7; y10; y18; y32; y34; y48]=R;
where deg(yi) = i and R is the ideal generated by y7y10, y7y18, y7y34and y234+y*
*210y48+y218y32+
possibly y34y18y10y6.3 The Poincare series of this ring is
ae oe
1 1 + t34 t7
pG(t) = _____________________________ _______________+ ______ :
(1 - t4)(1 - t6)(1 - t32)(1 - t48)(1 - t10)(1 - t18)1 - t7
A homogeneous set of parameters is given by the elements y4, y6, y32, y48, y107*
*+ y710and
y18. The first five of these form a regular sequence, while the last is a zero *
*divisor. So the
depth of H*(BE6; F2) is five.
___________
3This is an ambiguity in the answer given by Kono and Mimura, which does not *
*affect our Poincare
series calculations.
6 D. J. BENSON AND J. P. C. GREENLEES
Again the E2 page is equal to the E1 page in the spectral sequence, and consi*
*sts of zero
except in columns five and six. The Poincare series for column five is
X t-90
t-idim F2H5;-iJ(H*(BE6; F2)) = _________________________________________*
*-4-6-7-32-48;
i0 (1 - t )(1 - t )(1 - t )(1 - t )(1 -*
* t )
while the Poincare series for column six is
X
t-idim F2H6;-iJ(H*(BE6; F2))
i0
t-84(1 + t-34)
= __________________________________________________:
(1 - t-4)(1 - t-6)(1 - t-10)(1 - t-18)(1 - t-32)(1 -*
* t-48)
We conjecture that in general, in the oriented case where the depth and Krull*
* dimension
of H*(BG; k) differ by one, the appropriate functional equation is
pG(1=t) - td(-t)rpG(t) = -(1 + t)p0G(t);
where
X
p0G(t) = tidim kHr-1;-iJ(H*(BG; k)):
i0
The latter would then satisfy the subsidiary functional equation
p0G(t) = td(-t)r-1p0G(1=t):
These are the analogues of the functional equations given in Benson and Carlson*
* [4] in the
finite case.
4. Highly structured ring and module spectra
In this section we say the minimum amount possible to make sense of the struc*
*ture of
our proof, referring the reader to [6] and [11] for further details.
Our proof proceeds by considering the cohomology ring H*(BG; k) as the coeffi*
*cients of
an equivariant cohomology theory. In fact, for unbased G-spaces X, we may consi*
*der Borel
cohomology
X 7- ! H*(EG xG X; k):
The coefficient ring is the value (namely H*(BG; k)) this takes when X is a poi*
*nt, and the
projection X -! * makes H*(EG xG X; k) into a module over this ring.
It will be convenient to work from now on with the reduced theory on based G-*
*spaces
X, for which we use the notation
b*G(X) := H*(EG xG X; EG xG *; k) ~=H"*(EG+ ^G X; k):
In the based formulation the coefficient ring is the value of the theory on S0:
b*G~=H"*(BG+; k) ~=H*(BG; k):
For formal reasons, Borel cohomology is represented by a G-spectrum b in the se*
*nse that
b*G(X) = [X; b]*G, where the right hand side denotes G-homotopy classes of maps*
* of G-
spectra in the sense of [14]. Indeed, if H represents ordinary cohomology with *
*coefficients
COHOMOLOGY RINGS OF CLASSIFYING SPACES 7
in k, we may build in non-trivial representations to form the G-spectrum i*H an*
*d calculate
using [14, II.4.5]
[X; F (EG+ ; i*H)]*G= [EG+ ^ X; i*H]*G= [EG+ ^G X; H]*;
so that the G-spectrum b = F (EG+ ; i*H) represents b*G(.). It is not hard to *
*deduce, from
the fact that H*(EGxG X; k) is a graded commutative ring, that the representing*
* spectrum
b is a commutative ring object in the homotopy category of G-spectra (a `commut*
*ative ring
G-spectrum').
The idea is that it would be useful if we could construct some form of derive*
*d category
of modules over b. One could then work in this category to provide analogues o*
*f the
constructions of Section 2, and hence exploit the formal properties of the alge*
*bra. The
spectral sequence would then arise by taking the analogue of the homology of a *
*filtered
chain complex.
The first problem with this is that for an arbitrary commutative ring spectru*
*m R there is
no way to put an R-module structure on the mapping cone of an R-map between R-m*
*odules.
The solution is to restrict the class of ring spectra R and endow them with ext*
*ra structure.
The problem arises from choices involved in the homotopies used to prove commut*
*ativity
and associativity. The reason for these choices is that we have only worked in *
*the homotopy
category; the traditional solution is to continue as far as possible in the hom*
*otopy category
and assume that these and all higher homotopies are unique up to homotopy. One*
* thus
reaches the definition of an E1 ring as a ring spectrum with extra coherence co*
*nditions on
the commuting and associating homotopies. The more satisfying solution is to a*
*ttribute
the problem to premature passage to homotopy, and to ask that the spectrum R is*
* actually
a ring spectrum in a category of spectra before passage to homotopy; however th*
*is only
makes sense if there is a smash product which is commutative, associative and u*
*nital before
passage to homotopy. Such a category of spectra and such a smash product have r*
*ecently
(and unexpectedly) been constructed by Elmendorf-Kriz-Mandell-May [6], and they*
* show
that a spectrum which is an algebra over the sphere spectrum at the point set l*
*evel is
essentially the same as an E1 ring spectrum in the traditional sense. We shall *
*be content
to treat the Elmendorf-Kriz-Mandell-May category as a black box delivering cons*
*tructions
with certain properties we need. We shall refer to an algebra R over the sphere*
* G-spectrum
as a highly structured ring G-spectrum. Elmendorf and May [8] write SG for the *
*0-sphere
G-spectrum, and would thus refer to R as an SG-algebra. When emphasis is necess*
*ary we
refer to a module over R as a highly structured R-module.
Now suppose R is a highly structured ring G-spectrum and M is a highly struct*
*ured
module spectrum over it. Following Section 2, we shall explain how to define a*
* highly
structured module spectrum which is the analogue of the `right derived J-power *
*torsion
functor' in the derived category: this suggests notations RJM or HJ(M), but we *
*shall
use the simpler notation JM since in our context it is not ambiguous. Beginning*
* with the
principal case, for fi 2 ssG*R we define (fi)(R) by the fibre sequence
(fi)(R) -! R -! R[1=fi]:
fi fi
Here R[1=fi] = holim(R -! R -! . .).is a module spectrum and the inclusion of*
* R is a
!
module map; thus (fi)(R) is an highly structured module. Analogous to the filtr*
*ation at
8 D. J. BENSON AND J. P. C. GREENLEES
the chain level we have an R-module filtration of (fi)(R) by viewing it as -1(R*
*[1=fi][CR),
where CR denotes the cone on R.
Next we define the J-power torsion spectrum [9, 11] for the sequence fi1; : :*
*;:fin by
(fi1;:::;fin)(R) = (fi1)(R) ^R : :^:R(fin)(R):
Using the same proof as in the algebraic case we conclude that (fi1;:::;fin)(*
*R) depends
only on the radical of J = (fi1; : :;:fin); we therefore write J(R) for it. It *
*is then natural
to define the J-power torsion spectrum of M by
J(M) := J(R) ^R M:
To calculate the homotopy groups of J(R; M) we use the product of the filtratio*
*ns of
(fii)(R) given above. Since the filtration models the algebra precisely, the ho*
*motopy spec-
tral sequence of the filtered spectrum JR gives us a useful means of calculatio*
*n.
Lemma 4.1. There is a spectral sequence
Es;t2= Hs;tJ(RG*; MG*) ) ssG-s-t(J(M))
*
* __
with differentials dr : Es;tr! Es+r;t-r+1r: *
* |__|
The one other property we need is good behaviour under restriction. Indeed, *
*if H
is a subgroup of G we may view the highly structured ring G-spectrum R as a hig*
*hly
structured ring H-spectrum by neglect of structure, and the change of groups is*
*omorphism
[G=H+; R]*G= [S0; R]*H= R*Hallows us to construct the restriction homomorphism *
*resGH:
R*G- ! R*Has the map induced by projection G=H -! G=G. Given any ideal J of R*G
we may consider the ideal resGHJ in R*Hgenerated by the image of J, noting that*
* this is
also generated by the restrictions of any set of generators of J. With this no*
*tation, the
behaviour under restriction is immediate by construction.
Lemma 4.2. In the above situation there is an equivalence of highly structure*
*d module
H-spectra
__
resGH(JM) ' resGHJresGHM : |__|
5. Strategy
In this section we prove the main theorem modulo the fact that the representi*
*ng spectrum
b (which is only determined up to homotopy type) may be chosen to be a highly s*
*tructured
ring G-spectrum. This is proved by Elmendorf-May in the companion paper; in fac*
*t they
prove the stronger result that the G-spectrum i*H is weakly G-equivalent to a h*
*ighly
structured ring [8, 2.4].
For any highly structured ring G-spectrum R we may take J to be the augmentat*
*ion
ideal J = ker(resG1: RG*! R*), and attempt to implement the strategy below; the*
*re are
only three places where further assumptions are necessary, but for definiteness*
* we shall take
R = b throughout, referring the reader to [11] for further discussion of the ge*
*neral case. By
definition the coefficient ring is the ring of interest to us
b*G= "H*(BG+; k) ~=H*(BG; k);
COHOMOLOGY RINGS OF CLASSIFYING SPACES 9
and the augmentation ideal J is the ideal of positive degree elements. Since b**
*Gis Noetherian
by Venkov's finite generation theorem [16, 17], the ideal J is finitely generat*
*ed, and we
may construct the spectrum J(b). For the rest of the section we work entirely w*
*ith highly
structured modules over b.
Since resG1(J) = (0) and, since for any R the augmentation gives an equivalen*
*ce (0)R '
R, it follows from 4.2 that the natural augmentation J(b) -! b is nonequivarian*
*t equi-
valence, so that EG+ ^ J(b) ' EG+ ^ b; the map collapsing EG to a point thus gi*
*ves a
map
: EG+ ^ b -! Jb
whose mapping cone is E"G ^ J(b), where E"G is the mapping cone of the projecti*
*on
EG+ -! S0.
The main theorem is proved by showing that is a G-equivalence. The point is *
*that the
homotopy groups of the codomain are calculated by the spectral sequence of 4.1,*
* whereas
those of of the domain are closely related to the homology of BG. In fact, sinc*
*e EG+ ^ b =
EG+ ^ F (EG+ ; i*H) ' EG+ ^ i*H, it is immediate from the Adams isomorphism *
*[14,
II.7.2] that
ssG*(EG+ ^ b) = "H*(EG+ ^G SAd(G); k);
where Ad (G) is the adjoint representation, and SAd(G) is its one point compact*
*ification
with the new point as its G-fixed basepoint.
Theorem 5.1. For any compact Lie group G and commutative Noetherian ring k th*
*ere is
a spectral sequence
Es;t2= Hs;tJ(H*(BG; k)) ) "H-s-t(EG+ ^G SAd(G); k)
of modules over H*(BG; k) with differentials dr : Es;tr-! Es+r;t-r+1r:
We say that the adjoint representation is orientable over k if it can be repl*
*aced by the
G-fixed representation of the same dimension so that we have an isomorphism "H**
*(EG+ ^G
SAd(G); k) ~=H"*(dBG+; k) of modules over H*(BG; k). A Serre spectral sequence *
*argu-
ment shows this is the case if G acts trivially on Hd(SAd(G); k), which is cert*
*ain if G is
finite, the component group is of odd order or k is of characteristic two. How*
*ever, the
adjoint representation may not be orientable; for example that of O(2) is not o*
*rientable
over k unless char(k) = 2.
Corollary 5.2. For any compact Lie group G of dimension d and commutative Noeth*
*erian
ring k over which the adjoint representation is orientable, there is a spectral*
* sequence
Es;t2= Hs;tJ(H*(BG; k)) ) H-s-t-d(BG; k)
*
* __
of modules over H*(BG; k) with differentials dr : Es;tr-! Es+r;t-r+1r. *
* |__|
It remains to prove that is a G-equivalence, or equivalently that its mappin*
*g cone is
G-contractible. Since all descending chains of subgroups in a compact Lie group*
* are finite,
we may suppose by induction that the analogous statement has been proved for al*
*l proper
subgroups H of G. To make use of this assumption we need to know augmentation i*
*deals
are compatible in a sense we now make precise.
10 D. J. BENSON AND J. P. C. GREENLEES
Lemma 5.3. For any subgroup H of G the augmentation ideals J(G) of H*(BG; k) *
*and
J(H) of H*(BH; k) are related by
q ___________
resGH(J(G))= J(H):
Proof:The proof of Venkov's finite generation theorem [16, 17] shows that the r*
*estriction
map H*(BG; k) -! H*(BH; k) is finite, and the assertion follows. Indeed if p is*
* a prime
of H*(BH; k) containing resGHJ(G) then (resGH)-1(p) J(G) = (resGH)-1(J(H)); by*
*_the
Going Up Theorem p J(H). |_*
*_|
It follows from 4.2 that we have equivalences of H-spectra
resGHJ(G)b ' resGHJ(G)b ' J(H)b;
so that we may safely write J without qualification. In particular, by untwisti*
*ng [14, II.4.8]
and the inductive hypothesis we have
G=H+ ^ "EG^ Jb ' G nH (E"H ^ Jb) ' *
for any proper subgroup H of G, and hence
T ^ "EG^ Jb ' *
whenever T is built out of cells G=H+ ^ Sn with H a proper subgroup.
The extreme example of such a space T is the space EP+ . Here EP is the univ*
*ersal
space for the family P of proper subgroups characterized by the property that E*
*P is H-
contractible for any proper subgroup H, but (EP)G = ;. The cofibre sequence
EP+ -! S0 -! "EP
and the inductive hypothesis show that it is enough to prove that "EP^Jb is G-c*
*ontractible.
Since E"P is H-contractible for any proper subgroup H, it follows from the Whit*
*ehead
theorem that it is enough to show ssG*(E"P ^ Jb) = 0.
At this point we must recall that for any complex representation V there is a*
* Thom class
t(V ) 2 "H|V(|EG+ ^G SV ; k), giving rise to Thom isomorphisms
"H*(EG+ ^G X; k) -~=!"H*(EG+ ^G V X; k)
by external multiplication. In particular, taking X = S0, the image of the uni*
*t is the
Euler class O(V ) 2 b|VG|= "H|V(|BG+; k); equivalently the inclusion e(V ) : S0*
* ! SV gives a
diagram
e(V )*n 0
bnG(SV ^ X) -! bG(S ^ X)
~="Thom k
.O(V ) n
bn-|VG|(X) -! bG(X):
The represented manifestation of the Thom isomorphism is a G-equivalence SV ^b *
*' S|V^|b.
Using this, there is a useful reduction.
Lemma 5.4. (Carlsson's reduction) It is sufficient to show ssG*(S1V ^ J(b)) *
*= 0 for a
single chosen complex representation V provided V G = 0.
COHOMOLOGY RINGS OF CLASSIFYING SPACES 11
Proof: Since V G = 0, we have an equivalence S1V ^ "EP ' "EP, so that it is en*
*ough to
show ssG*(S1V ^ "EP ^ J(b)) = 0. However E"P can be constructed as a direct l*
*imit of
spheres SW where W runs over complex representations without trivial summand. *
*Since
SW ^ J(b) ' S|W| ^ J(b) from the Thom isomorphism, the hypothesis ensures that*
* __
ssG*(S1V ^ SW ^ J(b)) = 0, and hence the direct limit of these groups is also*
* zero. |__|
It is now easy to complete the proof of Theorem 5.1; indeed we may calculate
ssG*(S1V ^ J(b))= lim ssG*(SkV ^ Jb)
! k
= lim ssG*Jb; O(V )
!
= (ssG*Jb)[1=O(V )]:
But O(V ) 2 J since e(V ) is nonequivariantly null-homotopic and so
H*J(bG*) [1=O(V )]= 0;
from the spectral sequence 4.1 we see that
__
ssG*(S1V ^ J(b)) = 0: |__|
References
[1]J. F. Adams. "Prerequisites (on equivariant stable homotopy) for Carlsson'*
*s lecture." Lecture Notes
in Mathematics 1051, Springer-Verlag (1984).
[2]D. J. Benson and J. F. Carlson. "Complexity and multiple complexes." Math.*
* Z. 195 (1987), 221-238.
[3]D. J. Benson and J. F. Carlson. "Projective resolutions and Poincare duali*
*ty complexes." Trans.
American Math. Soc. 342 (1994), 447-488.
[4]D. J. Benson and J. F. Carlson. "Functional equations for Poincare series *
*in group cohomology." To
appear in Bull. London Math. Soc.
[5]A. Borel. "Sur la cohomologie des espaces fibres principaux et des espaces*
* homogenes de groupes de
Lie compacts." Ann. Math. 57 (1953), 115-207.
[6]A. Elmendorf, I. Kriz, M. Mandell and J. P. May. "Rings, modules and algeb*
*ras in stable homotopy
theory." 260 pp, Hopf preprint archive (Hopf@math.purdue.edu).
[7]A. Elmendorf, I. Kriz, M. Mandell and J. P. May. "Modern foundations for s*
*table homotopy theory."
Handbook of Topology (ed. I. M. James) North Holland (1995).
[8]A. Elmendorf and J. P. May. "Algebras over equivariant sphere spectra." Pr*
*eprint (1995)
[9]J. P. C. Greenlees "The K-homology of universal spaces and local cohomolog*
*y of the representation
ring." Topology 32 (1993), 295-308.
[10]J. P. C. Greenlees. "Commutative algebra in group cohomology." J. Pure and*
* Applied Algebra 98
(1995) 151-162
[11]J. P. C. Greenlees and J. P. May. "Completions in algebra and topology." H*
*andbook of Topology
(ed. I. M. James), North Holland (1995).
[12]A. Grothendieck (notes by R. Hartshorne). "Local cohomology." Lecture Note*
*s in Mathematics 41,
Springer-Verlag (1967).
[13]A. Kono and M. Mimura. "Cohomology mod 2 of the classifying space of the c*
*ompact connected Lie
group of type E6." J. Pure and Applied Algebra 6 (1975), 61-81.
[14]L. G. Lewis, J. P. May and M. Steinberger (with contributions by J. E. McC*
*lure). "Equivariant stable
homotopy theory." Lecture Notes in Mathematics 1213, Springer-Verlag (1986*
*).
[15]D. G. Quillen. "The mod 2 cohomology rings of extra-special 2-groups and t*
*he spinor groups." Math.
Ann. 194 (1971), 197-212.
12 D. J. BENSON AND J. P. C. GREENLEES
[16]B. B. Venkov. "Cohomology algebras for some classifying spaces." (Russian)*
* Dokl. Akad. Nauk. SSSR
127 (1959), 943-944.
[17]B. B. Venkov. "Characteristic classes for finite groups." (Russian) Dokl. *
*Akad. Nauk. SSSR 137
(1961), 1274-1277.
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
E-mail address: djb@byrd.math.uga.edu
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk