LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY
J.P.C.GREENLEES
Abstract. The article describes the role of local homology and cohomology*
* in under-
standing the equivariant cohomology and homology of universal spaces. Thi*
*s brings to light
an interesting duality property related to the Gorenstein condition. The *
*phenomena are
studied and illustrated in several rather different families of examples.*
* Both topology and
commutative algebra benefit from the connection, and many interesting que*
*stions remain
open.
Contents
1. Overview 2
Part I. Equivariant K-theory. 2
2. The complex representation ring *
* 2
3. Free G-spaces. *
* 4
4. The Atiyah-Segal completion theorem and its dual. *
* 5
5. Duality for the representation ring. *
* 7
6. A proof for p-groups. *
* 8
7. Variations and extensions. *
* 11
Part II. Ordinary cohomology and graded connected k-algebras. 13
8. Cohomology of some classes of groups. *
* 13
8.1. Finite groups. *
* 13
8.2. Compact Lie groups. *
* 15
8.3. Arithmetic groups. *
* 16
8.4. p-adic Lie groups. *
* 16
9. The local cohomology theorem for group cohomology. *
* 17
10. An algebraic proof of the local cohomology theorem for ordinary cohomolo*
*gy. 18
11. Structural implications of the local cohomology theorem. *
* 20
Part III. Examples related to bordism. 22
12. The theorem for MU-modules. *
*22
13. The proof. *
*23
13.1. G-spectra. *
* 23
13.2. Commutative algebra with G-spectra. *
* 24
13.3. The map A is an equivalence. *
* 25
13.4. The map B is an equivalence. *
* 25
14. Homotopically Gorenstein rings. *
* 26
15. The chromatic case. *
* 27
16. Connective K theory. *
* 27
1
2 J.P.C.GREENLEES
References *
*29
1. Overview
The usefulness of local cohomology in equivariant topology is not just a supe*
*rficial phe-
nomenon. It arises because similar structures occur in both contexts. The aim*
* of these
lectures is to explain one particular connection that I am especially familiar *
*with, showing
the common structures in the process. This connection is useful in both directi*
*ons. I hope
to expose some interesting algebraic structures and recommend them for further *
*study, and
to display some topological phenomena that may be amenable to methods of commut*
*ative
algebra.
The main ingredients on the topological side are a group G and a G-equivarian*
*t cohomol-
ogy theory E*G(.). On the commutative algebra side we begin with an augmented k*
*-algebra
S, and the augmentation ideal J = ker(S -! k); in the topological examples S = *
*E*Gis the
equivariant coefficient ring and k = E* is the non-equivariant coefficient ring*
*. Aside from
the common structures, the reason for studying these phenomena is that the ring*
*s S have
an interesting duality property generalizing the notion of Gorenstein rings.
The paper is organized around families of examples of rings S arising from eq*
*uivariant
cohomology theories. We start in Part I with a rather simple instance (K theor*
*y and the
complex representation ring), where quite complete results are available in ver*
*y concrete
terms. This should give some life to the ideas. We then summarize the other e*
*xamples
we want to discuss, before considering each in more detail: these subsequent ex*
*amples all
correspond to complete rings. First in Part II we consider the case of ordinary*
* cohomology,
where S is a complete graded ring over a field k. In this simpler situation it *
*is possible to
give general geometric results of some substance. In Part III we turn to chroma*
*tic examples,
where S is finite over k; the geometry is more complicated here, so we restrict*
* attention to
rather crude structural phenomena and to some very special cases.
During the discussion, there will always be a group G in the background. Most*
* results are
interesting even for the group of order 2, and the reader may want to concentra*
*te on this
case to begin with. A few sections treat groups which are not finite, and reade*
*rs will lose
little by ignoring them.
Part I. Equivariant K-theory.
In Part I, we concentrate on a particular example which is not only very clos*
*e to algebra,
but can also be made completely explicit. Most of the phenomena we are concerne*
*d with
occur in this case in very concrete forms.
2. The complex representation ring
For further details of this section see [51, 52].
Let G be a compact Lie group and X a G-space (i.e. a topological space with c*
*ontinuous
left G-action). A complex G-equivariant vector bundle over X is a continuously*
* varying
family of complex vector spaces, parametrized by X. More precisely, it is a G-m*
*ap ss : -!
X so that for all x 2 X, the fibre x := ss-1(X) is a complex vector space, and *
*for each
g 2 G, the translation g : x -! gx is linear. We also require that is locally *
*trivial: for
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 3
each point x 2 X there is an open neighbourhood U over which ss is projection i*
*n the sense
that there is a commutative diagram
~=
ss-1(U)___________//GU x Cn
GGssGG wwww
GG wwprojw
GG## --ww
U
We let VectG(X) denote the set of isomorphism classes of vector bundles on X.*
* Direct sum
of vector spaces extends to vector bundles, making VectG(X) into a monoid. The *
*equivariant
K-theory of X is obtained by adjoining inverses to form an abelian group
K0G(X) = AbGp(VectG(X))=([ j] = [] + [j]):
Tensor product of vector spaces extends vector bundles, and this makes K0G(X) i*
*nto a com-
mutative ring.
Note that if X is a point, a vector bundle over X is just a vector space with*
* linear G
action, which is simply a complex representation:
K0G(pt) = R(G)
where R(G) is the complex representation ring. Indeed, tensor product with a re*
*presentation
makes K0G(X) into an R(G)-module, so from the algebraic point of view we are di*
*scussing
the ring R(G) and modules over it.
Example 2.1. (i) If G is cyclic of order n we may choose a faithful one dimensi*
*onal rep-
resentation ff and R(G) = Z[ff]=(ffn - 1). This is one dimensional, and using *
*cyclotomic
polynomials, we see that the irreducible components of spec(R(G)) correspond to*
* the sub-
groups of G.
(ii) If G is dihedral of order 8, character theory shows
R(D8) = Z[a; b; oe]=(a2 = b2 = 1; aoe = boe = oe; oe2 = 1 + a + b + ab*
*):
For an arbitary compact Lie group G, R(G) is Noetherian. Segal has shown [51]*
* that its
dimension is 1+rank(G) where rank(G) is the dimension of the maximal torus in G*
*, and the
irreducible components of spec(R(G)) correspond to the conjugacy classes of top*
*ologically
cyclic subgroups of G.
Now in fact the functor K0Gextends to an equivariant cohomology theory K*G: G*
*-spaces-!
AbGp . We will not explain in detail what this means, but it is a contravarian*
*t functor with
good exactness properties (analogous to those of functors which take exact sequ*
*ences of
modules to long exact sequences of cohomology groups), and which takes sums to *
*products.
The extension can be given by stating that K*Gis 2-periodic in the sense that K*
*nG= Kn+2G
and the odd part is given by K-1G(X) = ker(K0G(S1 x X) -! K0G({1} x X)). The fa*
*ct that
this gives a suitably exact functor is Bott periodicity. For example
K*G(pt) = R(G)[u; u-1]
where u is a unit of degree 2; the value K*G(X) on a G-space X is a module over*
* K*G(pt), and
multiplication by u gives the periodicity. Henceforth we adopt the convenient a*
*bbreviation
K*G= K*G(pt) for the coefficient ring.
The other consequence of having a cohomology theory is that there is an assoc*
*iated ho-
mology theory KG*(X). This is more complicated to define, so we will be content*
* to say that
4 J.P.C.GREENLEES
it is related to cohomology in such a way that a form of Poincare duality holds*
* when X is
a manifold. The most trivial example of this is that KiG(pt) = KG-i(pt), so th*
*at, with the
usual convention for relating upper and lower indexing, we have
K*G= KG*= R(G)[u; u-1]:
Where necessary we refer to lower indices as degrees and upper indices as codeg*
*rees; for
example u 2 KG2= K-2Gis of degree 2 and codegree -2.
3. Free G-spaces.
For further details of this section see [52, 27].
The simplest sort of G-spaces are those with a free action (i.e. so that the*
* identity
element of the group is the only element fixing anything). In particular if X *
*is a free G-
space then equivariant vector bundles -! X over X correspond to non-equivarian*
*t bundles
j -! X=G over the quotient: given we take j = =G and given j we take to be the
pullback of j along the quotient map X -! X=G. This passes to K theory to say t*
*hat if X
is free
K*G(X) = K*(X=G):
For finite groups there is a similar statement
KG*(X) = K*(X=G)
for homology. However this is less elementary, and involves transfer arguments.*
* Accordingly,
if G is a general compact Lie group the statement must be modified by inserting*
* a suitable
kind of twist.
The other thing about free G-spaces is that there is a terminal free G space *
*EG in the
homotopy category. This means that for any free G-space X there is a G-map X : *
*X -!
EG, unique up to homotopy. In fact EG is characterized by two properties: it is*
* free and
non-equivariantly contractible. For example if G is cyclic we may view G as a s*
*ubgroup of
the unit complex numbers, and then EG = S(1C) is the unit sphere in the direct *
*sum 1C
of infinitely many copies of C. This may be more familiar to some through the *
*quotient
BG = EG=G, called the classifying space of G: for instance BC2 = RP 1.
Note in particular that for any free G-space X the universal map X induces a *
*diagram
K0G(X)Ooo____K0G(EG)OOO
~=|| ~=||
| |
K0(X=G) oo___K0(BG):
Thus a knowledge of K0G(EG) gives canonical characteristic classes in K0G(X) = *
*K0(X=G).
These can be useful invariants for distinguishing different G-spaces with the s*
*ame quotient
X=G. For example if G = C2 we may have X1 = G x RP 3whilst X2 = S(2C) (i.e. t*
*he
3-sphere with the antipodal action). In both cases the quotient is projective s*
*pace RP 3, but
the different characteristic classes distinguish them.
A similar motivation for studying KG*(EG) can be given, but a more convincing*
* one can
be given because quite powerful torsion invariants of free G-manifolds belong t*
*o this group
[55].
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 5
4.The Atiyah-Segal completion theorem and its dual.
For further details of this section see [3, 24].
In this section we describe the calculation of K*G(EG) and the associated hom*
*ology
KG*(EG). We present this in a quasi-historical way, as a series of theorems st*
*arting very
concrete and proceeding to greater generality at the cost of some additional ma*
*chinery.
Remembering the periodicity of K theory, we may work with the degree 0 part o*
*f the
coefficient ring S = R(G), and it suffices to describe the 0th and 1st K groups*
*. The ring S =
R(G) is augmented over k = R(1) by the map S = R(G) -! R(1) = Z taking the virt*
*ual
dimension, and we need to consider the augmentation ideal J = ker(R(G) -! R(1) *
*= Z) of
elements of virtual dimension 0.
Theorem 4.1. (Atiyah (1961) [1]) The equivariant K theory of EG is
K0G(EG) = K0(BG) = R(G)^J
and
K1G(EG) = K1(BG) = 0:
Example 4.2. (i) When G = C2, we have seen R(C2) = Z[ff]=(ff2 - 1), and since 1*
* and ff
are one dimensional representations, we see J = (1 - ff). Now a short calculati*
*on allows us
to identify the completion: (1 - ff)2 = 1 - 2ff + ff2 = 2(1 - ff), so that
(1 - ff)n+1 = 2n(1 - ff);
and
R(C2)^J= Z Z^2:
(ii) More generally, the Artin induction theorem easily shows that if G is a p-*
*group
R(G)^J= Z J^p:
(iii)It is not hard to see that for finite groups, J-completion is injective if*
* and only if G is a
p-group.
There is a more conceptual way of expressing the phenomena. We consider the p*
*rojection
map EG -! pt and the induced map
K*G(X) -! K*G(EG x X) = K*(EG xG X)
in K-theory.
Theorem 4.3. (Atiyah-Segal (1969) [3]) For any finite G-space X the natural map
K*G(X) -! K*G(EG x X) = K*(EG xG X)
is completion at the augmentation ideal J of R(G).
We pause for some remarks. Firstly, Theorem 4.1 is the special case X = pt, *
*since we
have seen that K*G(pt) = R(G)[u; u-1]. Next, we comment that the codomain K*G(E*
*G x X)
is a cohomology theory of X, so that for the statement to be plausible, it is n*
*ecessary
that K*G(X)^Jis a cohomology theory on finite complexes X. However, for each i*
*, the
R(G)-module KiG(X) is Noetherian provided X is a finite G-complex. This is cle*
*ar for
K0G(G=H) = K0H(pt) = R(H), and the general case follows by induction on the num*
*ber of
cells, using the exactness properties of a cohomology theory. Accordingly, the*
* Artin-Rees
lemma implies that J-completion is exact, and so K*G(X)^Jis a cohomology theory*
* on finite
6 J.P.C.GREENLEES
complexes X. Since J-completion is definitely not exact for arbitrary R(G)-mod*
*ules, the
statement must be modified if it is to cover arbitrary complexes X. Atiyah and *
*Segal used
the device of pro-groups, but for our purposes we prefer an alternative solutio*
*n.
Theorem 4.4. (Greenlees-May [31]) For any G-space X there is a natural short ex*
*act se-
quence
0 -! LJ1(K*G(X)) -! K*G(EG x X) -! LJ0(K*G(X)) -! 0;
where LJiis the ith left derived functor of J-completion.
The Artin-Rees lemma implies that for a Noetherian module M, we have LJ0M = M*
*^Jand
LJiM = 0 for i > 0, so that that if we take X to be a finite complex, we obtain*
* the previous
versions of the theorem.
It is a general phenomenon in topology that statements in homology have bette*
*r finiteness
properties than in cohomology, so it is natural to seek statements about K homo*
*logy. Finally,
local cohomology is about to make an appearance.
Theorem 4.5. (Greenlees (1993) [24]) For any finite group G and any G-complex X*
*, there
is a spectral sequence
H*J(KG*(X)) ) KG*(EG x X) = K*(EG xG X);
where J is the augmentation ideal of R(G).
We outline a proof for p-groups in Section 6, and a rather different proof fo*
*r all groups in
Part III (Section 13).
The next step is the major input from commutative algebra, and appears quite *
*magical
from the topological point of view. Grothendieck's vanishing theorem states tha*
*t local coho-
mology vanishes above the dimension; since R(G) is one dimensional, the spectra*
*l sequence
collapses.
Corollary 4.6. (Greenlees (1993) [24]) For any finite group G and any G-complex*
* X, there
is a short exact sequence
0 -! H1J(KGi+1(X)) -! KGi(EG x X) -! H0J(KGi(X)) -! 0:
In particular if X = pt
KG0(EG) = H0J(R(G)) = Z
and
KG1(EG) = H1J(R(G)):
Example 4.7. If G = C2 we may make the calculation completely explicit. We wor*
*k in
R(C2) = Z[ff]=ff2- 1, and J = (1 - ff). Since (1 - ff)(1 + ff) = 1 - ff2 = 0, i*
*nverting (1 - ff)
kills 1 + ff. Similarly, since (1 - ff)2 = 2(1 - ff), inverting 1 - ff inverts *
*2. Thus the stable
Koszul complex R(C2) -! R(C2)[1=(1 - ff)] becomes
Z Z -! Z[1=2];
showing H0J(R(C2)) = Z and H1J(R(C2)) = Z=21 .
The calculation of KG*(EG) is quite explicit in general. Indeed, H0J(R(G)) is*
* easily cal-
culated in terms of characters: J consists of representations whose characters *
*vanish at the
identity element e 2 G. Since characters separate conjugacy classes, H0J(R(G)) *
*consists of
representations whose characters vanish except at e. This consists of the integ*
*er multiples
of the regular representation.
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 7
It is not hard to calculate H1J(R(G)) either. Indeed, for any p-group, Artin*
*'s induction
theorem shows that the p-adic and J-adic topology on J coincide, and hence
H1J(R(G)) = R(G)=Z Z=p1 :
Remark 4.8. The theorem is also true for an arbitrary compact Lie group G. For *
*this one
must use the proof in Part III. The crucial ingredient is that the representing*
* spectrum of
K theory is a highly structured ring spectrum. Until recently this was only kno*
*wn for finite
groups, but M.Joachim [43] tells me that he has proved the result for arbitrary*
* compact Lie
groups.
Since the dimension of R(G) is 1 + rank(G), the spectral sequence will not no*
*rmally
collapse. However it still collapses when KG*(X) has depth rank(G).
Remark 4.9. An alternative way of motivating this whole connection is that by d*
*efinition J
acts as 0 on K*G(G) = K*(pt). By induction J acts nilpotently on K*G(X) and KG**
*(X) if X is
finite and free. Since EG is a direct limit of finite free complexes, we expect*
* K*G(EG) to be
the simplest possible "inverse limit" of J-power torsion complexes (hence it is*
* calculated by
the left derived functors of J-completion), and KG*(EG) to be the simplest poss*
*ible "direct
limit" of J-power torsion complexes (hence it is calculated by the right derive*
*d functors
of J-power torsion). Here `simplest possible' has a rather complicated meaning*
*, requiring
homotopy invariance, functoriality and exactness.
5. Duality for the representation ring.
In this section we introduce one of the central properties of equivariant coh*
*omology rings;
in this case we are considering the ring S = R(G) augmented over k = R(1) = Z. *
* For
the representation ring, this property appears rather commonplace, but at least*
* it is visible
to inspection. The idea is to combine the local cohomology theorems with the u*
*niversal
coefficient theorem to obtain a duality statement.
We are going to use the form appropriate to complete rings, and the completio*
*n theorem
in the non-equivariant form:
K0(BG) = R(G)^Jand K1(BG) = 0:
Similarly, we use the non-equivariant form of the local cohomology theorem; ass*
*uming G is
finite, this takes the form of the identifications
K0(BG) = H0J(R(G)) and K1(BG) = H1J(R(G)):
We have used the non-equivariant forms since there is a Universal Coefficient*
* Theorem [2]
for calculating K*(X) from K*(X). This is a short exact sequence
0 -! Ext1Z(Ki+1(X); Z) -! Ki(X) -! Hom Z(Ki(X); Z) -! 0:
We apply this to X = BG and substitute the algebraic expressions for its homolo*
*gy and
cohomology to obtain a short exact sequence
0 -! Ext1Z(H1J(R(G)); Z) -! R(G)^J-! Hom Z(H0J(R(G)); Z) -! 0:
This no longer mentions K theory: it is just a statement about the augmented *
*Z-algebra
R(G). It is reminiscent of local duality for Gorenstein rings, but of course R(*
*G) is not local or
even Cohen-Macaulay. It is also notable that local cohomology has separated the*
* Z-torsion
free H0Jfrom the torsion part H1J. The former gives the uncompleted part of R(G*
*)^J(namely
8 J.P.C.GREENLEES
multiples of the regular representation), and the latter gives the completed pa*
*rt (namely J^(p)
in the case of a p-group).
6. A proof for p-groups.
In this section I will outline a proof of the local cohomology theorem for p-*
*groups. The
reason for restricting to this case is that on the one hand it is a convincingl*
*y large family
of examples, whilst on the other, it is possible to give a rather concrete vers*
*ion of the proof
in which the correspondence between topological and commutative algebraic struc*
*tures is
highlighted.
In Part III we will see one method for going beyond p-groups using some quite*
* sophisticated
machinery. For the case of K-theory, and for finite groups, there is the more *
*elementary
approach used in [24]: the Burnside ring A(G) of finite G-sets is a good approx*
*imation to the
representation ring. Indeed the permutation representation homomorphism A(G) -!*
* R(G)
has the property that p ______________
J(A(G)) . R(G) = J(R(G))
so that we may replace the representation ring by the Burnside ring in the proo*
*f, considering
all R(G)-modules (such as R(G) itself) as modules over A(G). For topological r*
*easons,
manipulations with the Burnside ring are much more elementary. For infinite com*
*pact Lie
groups this method fails since A(G) is still 1-dimensional whilst the dimension*
* of R(G) is
equal to the rank of G. For the present we return to a method applying to p-gr*
*oups and
complex orientable theories.
We introduce notation by summarizing standard constructions in the commutativ*
*e algebra
of a ring S. The unstable Koszul complex is the cochain complex K(x) = (S - x!*
* S)
concentrated in codegrees 0 and 1. This is the desuspension of the mapping cone*
* of x:
K(x) = -1M(x) -! S -x! S:
To form the stable Koszul complex, we may now do the analogous construction for*
* powers
of x, assemble them into a direct system and pass to limits:
K(x) = -1M(x) -! S -x! S
# # #
2
K(x2) = -1M(x2) -! S -x! S
# # #
3
K(x3) = -1M(x3) -! S -x! S
# # #
.. . .
. .. ..
# # #
K(x1 ) = -1M(l) -! S -l! S[1=x]
Now given an ideal J = (x1; x2; : :;:xr) in S we may form the stable Koszul c*
*omplex
K(J1 ) = K(x11) K(x12) . . .K(x1r):
The notation is reasonable since the complex is independent of generators up to*
* quasi-
isomorphism (exercise). This allows us to define local cohomology by
H*J(M) := H*(K(J1 ) M);
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 9
and its dual local homology by
HJ*(M) := H*(Hom (P K(J1 ); M));
where P K(J1 ) is a complex of projectives quasi-isomorphic to K(J1 ). These d*
*efinitions
become interesting by Grothendieck's theorem that local cohomology calculates r*
*ight derived
functors of J-power torsion
H*J(M) = R*J(M)
where
J(M) = {x 2 M | JN x = 0 forN >> 0};
and the fact [32] that local homology calculates left derived functors of compl*
*etion
HJ*(M) = L*J(M)
where
J(M) = lim M=JkM:
k
Note also that Grothendieck vanishing and the evident universal coefficient the*
*orem
E*;*2= Ext*S(H*J(S); M) ) HJ*(M)
shows that the LiJ(M) = 0 for i > dim(S). This explains why only the zeroth and*
* first
derived functors entered into the Completion Theorem 4.4.
Now in the topological context, the analogue of the stable Koszul complex is *
*the universal
free space EG. Let us start with the case when G is a cyclic group, G C*. We n*
*ote that for
each k the group G acts freely on the unit sphere S(kC) in the k-fold direct su*
*mSkC. Since
S(kC) is a (2k - 1)-sphere (and hence (2k - 2)-connected), the union S(1C) = *
*kS(kC)
is contractible. Thus EG = S(1C) in this case.
Remark 6.1. For the best analogy with the above commutative algebra we need to *
*work
with G-spaces equipped with a G-fixed basepoint, so we need to describe the rou*
*tine trans-
lation between the based and unbased context.
For cohomology, it is in terms of the reduced cohomology: if Y is a G-space w*
*ith G-fixed
basepoint y0 we have maps {y0} -! Y - ! {y0} so that if we define the reduced c*
*ohomology
by eK*G(Y ) = ker(K*G(Y ) -! K*G(y0)), we have K*G(Y ) = eK*G(Y ) K*G(y0).
If X is an unbased G-space we form the based space X+, where the subscript + *
*denotes
the addition of a disjoint basepoint fixed by G. Thus
eK*G(X+) = K*G(X):
We need one more construction. In addition to the unit sphere S(V ) of a fini*
*te dimensional
inner product space V , we may form the one point compactification SV . A usefu*
*l feature of
the based context is that SV is the mapping cone of the map S(V )+ -! pt+ takin*
*g S(V )
to the non-base point. Consequently we may also say that S(V )+ is the desuspen*
*sion of the
mapping cone of the inclusion S0 -! SV .
10 J.P.C.GREENLEES
Now we have the ingredients to construct a diagram exactly analogous to the o*
*ne for the
stable Koszul complex
[x] C
EG(1)+ = S(C)+ -! S0 -! S
# # #
[x2] 2C
EG(3)+ = S(2C)+ -! S0 -! S
# # #
[x3] 3C
EG(5)+ = S(3C) -! S0 -! S
# # #
.. . .
. .. ..
# # #
EG+ = S(1C)+ -! S0 -l! S1C :
To complete the analogy, I should explain why it is reasonable to consider S0 a*
*nd SkC as
analogues of the same object S, and why we have written [xk] for the inclusion *
*map. For
this we may as well allow G to be arbitrary. The point is that equivariant Bott*
* periodicity
provides a specific isomorphism eK0G(SV ) ~=eK0G(S0) = R(G) for any complex rep*
*resentation
V . We then have a diagram
eKG0(S0)[x(V/)]/_eKG0(SV )
= || ~=Bott||
fflffl| fflffl|
R(G) K eKG0(S0)
KKK |
KKK |=
.O(V )K%%KKfflffl|
R(G)
This defines the element O(V ) 2 R(G) called the Euler class of V , and by tran*
*sitivity of
Bott periodicity isomorphisms, O(V W ) = O(V )O(W ). Thus the map marked [xk] *
*above
is O(kC) = O(C)k. For an arbitrary group G and representation V , the construct*
*ion of the
Bott map means that O(V ) is the alternating sum of exterior powers of V . It f*
*ollows that
when we apply eKG*(.) to
S(1V )+ -! S0 -! S1V
we obtain
Ke*G(S(1V )+) -! R(G) -! R(G)[1=O(V )]
and it is in this sense that KeG*(S(1V )+) is the analogue of the stable Koszul*
* complex
K(O(V )1 ).
Now, just as most ideals are not principal, so most groups are not cyclic. H*
*owever, we
can deal with this in the same way that we constructed a stable Koszul complex *
*for an ideal
with several generators. For example, if G = C1x C2x . .x.Cr is a product of cy*
*clic groups,
we have
EG = S(1C1) x S(1C2) x . .x.S(1Cr)
where Ci is the natural representation of the cyclic group Ci on C. It is not *
*hard to use
character theory to shown that if G is a p-group (or more generally supersolubl*
*e) then it has
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 11
complex representations V1; V2; : :;:Vr for some r so that G acts freely on S(V*
*1) x S(V2) x
. .x.S(Vr), and hence
EG = S(1V1) x S(1V2) x . .x.S(1Vr):
(Equally it is not hard to show that most groups do not have such representatio*
*ns; the
smallest example is A4, but see [49] for much more detailed information). Suppo*
*sing that we
do have such a model for EG, we may immediately outline a proof of the local co*
*homology
theorem.
Outline of proof of 4.5.
The theorem was stated with an unbased space X, but the proof works with base*
*d spaces,
like X+. The translation is routine as described in Remark 6.1. In particular E*
*G+ ^ X+ =
(EG x X)+ and eKG*(X+) = KG*(X).
We apply eKG*to the filtered space EG+ ^ X+. Because the filtration precisely*
* modelled
that of the stable Koszul complex, we immediately find a spectral sequence
E*;*1= K(E1 ) R(G)KG*(X) ) KG*(EG+);
where
E = (O(V1); O(V2); : :;:O(Vr))
is the ideal generated by the Euler classes. Again, the correspondence between *
*topology and
commutative algebra makes clear that d1 is the Koszul differential, so that
E*;*2= H*E(KG*(X)):
Finally we need to explain why E may be replaced by J. First, the inclusion S0 *
*-! SV is
obviously null-homotopic non-equivariantly if V 6= 0, so E J. In the cyclic ca*
*se we had
equality, and for a general p-group we have
p__
E = J:
This can be observed explicitly for the representation ring, since the primes a*
*re known by
[51], but in fact it follows from some finiteness assumptions [28].
Since local cohomology only depends on the radical it proves H*J(M) = H*E(M) *
*as re-
quired.
As far as any applications we have made are concerned, there is no particular*
* advantage
to using the augmentation ideal J rather than the Euler class ideal E. It is wh*
*en comparing
different groups that it comes into its own.
7. Variations and extensions.
There are two directions to develop the above ideas. The first replaces free *
*G-spaces (i.e.
spaces with all isotropy groups trivial) by spaces with isotropy in some other *
*family F of
subgroups. On the ring theoretic side, the ideal J is replaced by the ideal J(F*
*) of elements
restricting to zero in R(H) for all subgroups H in F. For equivariant K theory*
* this goes
smoothly. However there is no comparable expression in terms of the non-equivar*
*iant case,
and hence no duality statement. We will not pursue this variation any further h*
*ere.
12 J.P.C.GREENLEES
The second variation is to replace K theory by another cohomology theory; on *
*the ring
theoretic side this replaces the augmented ring R(G) by another. If K theory i*
*s replaced
by E-theory then R(G) is replaced by the coefficient ring S = E*G(or its degree*
* zero part
E0G), R(1) is replaced by k = E* (or its degree zero part E0) and J is replaced*
* by the kernel
J = ker(E*G-! E*) of the map forgetting equivariance.
Context 7.1. The correspondence between commutative algebra and topology is via
o S = E*G
o k = E*
o J = ker(E*G-! E*).
The remainder of this article is structured round a number of these. In this*
* section we
give an overview.
In the following table the first row summarizes the properties of K theory di*
*scussed above,
and each subsequent row is an analogous example. To avoid discussion of special*
* cases, the
information refers only to finite groups G. The final two columns refer to the*
* question of
whether the homology and cohomology of EG can be calculated using local cohomol*
*ogy and
completions as was the case for K theory.
______________________________________________________________________________*
*__||||
|_Cohomology_theory__|CoefficientsDimension_Augmented_over__|Homology___Cohom.*
*__|||||
||K-theory || R(G)0 1 ^ R(1) = Z | [24]| [3] *
* ||
||En-theory ||En(BG) n Zp[[u1; : :;:un-1]][|24,|20, 22]T *
* ||
||Stable cohomotopy |A(G)|+*nilp 1 A(1)*= Z ||FALSE! [15]*
* ||
||Ordinary ||H*(BG) rankp(G) H* = Fp || [26] T *
* ||
|_Connective_K-theory|ku_(BG)________2_________ku__=_Z[u]____|[24,_20,_22]T___*
*__|
Remark 7.2. (i) There are various ways of grouping these examples. First, K-the*
*ory and
En-theory are periodic, so we may work with ungraded rings. In the cohomotopy e*
*xample the
Burnside ring A(G) is in degree 0, and all other elements are nilpotent (Nishid*
*a's nilpotence
theorem); accordingly we may work over the degree zero part A(G) in this case t*
*oo. The
other examples are graded.
(ii) More significant is the fact that apart from K-theory and cohomotopy, all *
*the examples
are complete for the augmentation ideal. This means that the completion theore*
*m is a
tautology for finite complexes (hence the entries T for `true' and `tautology')*
* and concentrates
attention on local cohomology and the duality statement.
This is no disadvantage for present purposes, but in topology it is a major p*
*roblem to
identify natural rings of which these are completions. To see why it is valuab*
*le, one may
imagine we only knew the completion of R(G). We may then seek to refine it to R*
*(G) itself,
hoping thereby to invent representation theory.
(iii) In the first three examples, the ring is essentially finite over its none*
*quivariant counter-
part: in the other two we have finitely generated algebras over the non-equivar*
*iant coeffi-
cients.
(iv) All the examples for which the local cohomology theorem is known to be tru*
*e can be
proved for p-groups using the method of Part I. They can be proved for finite g*
*roups by the
method described in [24, Appendix], using [28] to establish finiteness properti*
*es. However,
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 13
the technology necessary for implementing the strategy was not introduced until*
* [20]; it was
shown to apply to these cases in [22].
(v) Stable cohomotopy is quite exceptional. The fact that local cohomology of s*
*s*Gdoes not
calculate ss*(BG) means that the type of proof for the completion theorem we wi*
*ll give in
the complex orientable case (by formally deducing it from the local cohomology *
*theorem)
cannot apply. Carlsson's proof of the completion theorem ("the Segal conjecture*
*") is not a
simple formality like the proofs we give here, and it seems that the substantia*
*l calculational
input is necessary. Other examples of this type occur in algebraic K theory, bu*
*t we shall not
be concerned with them further.
In Part II (Sections 8 to 11) we discuss ordinary cohomology: here the topolo*
*gical technol-
ogy is at a minimum, but the commutative algebra is correspondingly increased. *
*In Section
12 we discuss a family of examples arising from the topological theory of manif*
*olds: this
has the advantage of being a rather extensive family from the ring-theoretic po*
*int of view.
Finally, we look at two special cases of this. In Section 15 we discuss the En-*
*theory example
because an interesting interaction of Cousin filtrations occurs. In Section 16*
* we turn to
connective K theory; the interest here is that although it is very close to exa*
*mples where
rather complete results are available, some rather intricate commutative algebr*
*a appears.
Part II. Ordinary cohomology and graded connected k-algebras.
We spend Sections 8 to 11 considering ordinary cohomology. This has several *
*special
attractions, especially to algebraists. First, the cohomology of a point can be*
* described in
purely algebraic terms as an Ext algebra relevant to representation theory. Sec*
*ond, the ring
S is a connected commutative graded algebra over a field k, so it is especially*
* easy to apply
the methods of commutative algebra.
8. Cohomology of some classes of groups.
For the duration of Part II, S is an N-cograded algebra over a field k, and S*
*0 = k.
Properly speaking, S is graded commutative in the sense that ab = (-1)deg(a)deg*
*(b)ba, and
all the results we need can be proved in this context. Those not wishing to wo*
*rry about
graded commutativity can restrict attention to the case that char(k) = 2. For m*
*ost of the
discussion it suffices to replace S by its even-graded commutative subring.
In this section the augmentation ideal J = ker(S -! k) is the unique maximal *
*ideal of
the graded local ring S. To emphasize this we write m for it.
8.1. Finite groups. The first class of examples we discuss are the rings
S = H*(G; k) := Ext*kG(k; k)
for a finite group G, where kG denotes the group ring. General references for t*
*his subsection
are [5, 11]. One of the principal reasons for studying this is representation *
*theory. For
example the rate of growth of a minimal resolution of k is exactly captured by *
*the Hilbert
series X
h(G; k) := dimk(Hi(G; k))ti;
i0
for Noetherian rings this power series is easily seen to be the expansion of a *
*rational function
about t = 1. The cohomology ring H*(G; k) is known explicitly for a large numbe*
*r of groups,
and we tabulate some examples to bring the discussion down to earth. All the ex*
*amples are
14 J.P.C.GREENLEES
2-groups, and k is a field of characteristic 2. We use the convention that a s*
*ubscript on
a generator indicates cohomological degree. The Cohen-Macaulay defect for a lo*
*cal ring
S is defined by CM-defect(S) = dim(S) - depth(S). The groups are elementary ab*
*elian,
quaternion of order 8, dihedral of order 8, semidihedral of order 16, a certain*
* group of order
32, and the extra-special 2-groups.
_______________________________________________________________________________*
*_________||*|
|_Group_G__|____________H_(G;_k)_____________dim__depth__defect_______Hilbert_s*
*eries____||r|r|
|| (C2) || k[x(1)1; x(2)1;2. .;.x(r)1]23r3 r 0 1=(12- t*
*) |2 |
|| Q8 ||k[x1; y1; z4]=(x + xy + y ; x ; y1) 1 0 (1 + t + t )=(1*
* -2t)(1 + t|) |
|| D8 || k[x1; y1; u2]=(xy) 2 2 0 1=(1 -2t*
*) 2 | |
|| SD16 || See below 2 1 1 1=(1 - t) (*
*1 + t ) | |
|| 7a21+n|| 3 1 2 *
* ||
|___2______|k[x(1)1;_x(2)1;_:_:;:x(n)1]=I__k[i2n-r]rr______0___________________*
*_________|
Remark 8.1. (i) There is a copious supply of further examples on J.F.Carlson's *
*webpages
[14] (the cohomology of all but 5 of the 267 groups of order 64 are there, toge*
*ther with all
2-groups of smaller order).
(ii) Quillen [46] has shown that
dim (H*(G; k)) = rankp(G);
where the p-rank rankp(G) of G is the rank of the largest elementary abelian p-*
*subroup (Cp)r
in G.
(iii) For p-groups of small order, the Cohen-Macaulay defect tends to be quite *
*small. Indeed,
Duflot [16] has shown that
depth(H*(G; k)) rankp(Z(G))
where char(k) = p, and that in any case the depth is at least 1 [17].
(iv) The cohomology ring of the semidihedral group is
k[x1; y1; z3; t4]=(xy + y2; y3; yz; x3z + x3y + y2t + z2):
The cohomology ring of 7a2 has 8 generators and 18 relations, but the relevant *
*information
is well summarized in [6].
(v) The case of the extra special groups of order 21+2m is included because Qui*
*llen's calcu-
lation [47] is so elegant (see also [5, 5.5] for a brief account). The ideal I *
*is generated by a
regular sequence.
To be more precise, consider an extension 1 -! C2 -! G -! E - ! 1 where E is
elementary abelian of rank n. If C2 is the commutator subgroup and the centre o*
*f G then G
is called extraspecial, n = 2m is even and there are only two isomorphism types*
* of extraspecial
groups of this order, but the cohomology calculation applies to any extension o*
*f the given
form. As usual, r is the dimension of the cohomology ring, or equivalently the *
*rank of the
largest elementary abelian subgroup.
To describe the ideal, we use the fact that any such extension is classified *
*by the map
q : E -! C2 given by squaring the preimage in G of an element of E. This q is a*
* quadratic
form with associated bilineary form b. Now if e1; : :;:en is a basis of E as an*
* F2-vector space
and v = ieixithen I is generated by the sequence
q(v); q(v; F (v)); : :;:q(v; F n-r(v));
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 15
where F is the Frobenius map, and the sequence turns out to be regular. The cod*
*egrees are
2; 4; 8; : :;:2n-r+1, so it is easy to write down the Hilbert series.
8.2. Compact Lie groups. If G is a compact Lie group the ring
S = H*(BG; k)
is the cohomology ring of the classifying space of G that we met in Section 3. *
*One reason
for studying this is exactly parallel to the motivation we gave for K theory: i*
*t tells us about
characteristic classes. One often thinks of compact Lie groups such as the orth*
*ogonal and
unitary groups, but of course finite groups are perfectly good examples, and us*
*ing the bar
construction for BG one sees that it can be constructed with cells correspondin*
*g to copies
of kG in the bar resolution: thus H*(BG; k) = Ext*kG(k; k), and our notation is*
* consistent.
We may give a good supply of examples here too.
_______________________________________________________________________________*
*____||*|
|_Group_G__|_______H_(BG;_k)__________dim__depth__defect________Hilbert_series_*
*____||r|2r|
|| (U(1)) |k[x(1)2;|x(2)2; . .;.x(r)2]r r 0 1=(1 - t4) *
* ||
|| SU(2) || k[x4] 1 1 0 21=(1 - t4) *
* |2n |
|| U(n) || k[c1; c2; : :;:cn] n n 0 1=(1 - t4)(1 - t8) . .(*
*.1 - t4n) ||
|| O(2n) || k[p1; p2; : :;:pn] n n 0 1=(1 - t )(1 - t ) . .(*
*.1 -|t ) |
|__Spin(n)_k|[w2;_w3:_:;:wn]=I__k[i2n-r]r____r______0__________________________*
*____|
Remark 8.2. (i) By force of precedent, the Chern class ci is of cohomological d*
*egree 2i
and the Pontrjagin class pi is of cohomological degree 4i. Otherwise we have re*
*tained the
convention that subscripts refer to codegrees.
(ii) The field k can be of any characteristic except that char(k) 6= 2 for O(2n*
*) and char(k) = 2
for Spin(n).
(iii) The case O(2n) is included because its behaviour is slightly more complic*
*ated than its
polynomial cohomology ring suggests (see Example 9.5(iii)).
(iv) The cohomology of Spin(n) is also calculated in [47], and was Quillen's mo*
*tivation for
considering the extraspecial 2-groups. The ideal I is generated by the regular *
*sequence
n-r 2n-r-1 2 1
w2; Sq1w2; : :;:Sq2 Sq . .S.q Sq w2;
where Sqi is the ith Steenrod square. Quillen deduces this calculation by compa*
*rison with
the calculation for the extraspecial subgroup of Spin(n).
One other feature will be important. The group G acts on itself smoothly by c*
*onjugation.
Since it preserves the identity element e 2 G, a group element gives a self-map*
* of the tangent
space TeG, and we obtain the adjoint representation
ad : G -! GL(TeG);
and hence the k-orientation representation
sign *
G -ad!GL(TeG) det-!R* -! {+1; -1} -! k :
If the image is trivial we say that ad is orientable over k.
Evidently if char(k) = 2, the adjoint representation is orientable for every *
*group G. By
continuity, each connected component maps to the same point, so that if G is co*
*nnected,
or if it has an odd number of components then ad is orientable over any field. *
*However if
16 J.P.C.GREENLEES
char(k) 6= 2 then ad(O(2n)) is not orientable over k (this is easy to check for*
* n = 1, and
follows in general).
8.3. Arithmetic groups. For an arbitrary discrete group G we may consider the c*
*ohomol-
ogy ring
S = H*(G; k) = Ext*kG(k; k):
This can be arbitrarily unpleasant unless we place a restriction on G. We are i*
*nterested in
arithmetic groups, such as SLn(Z) and On(Z). However the appropriate level of g*
*enerality
is a little wider: virtual duality groups. A general reference for this subsect*
*ion is [11].
Before reaching virtual duality groups we should discuss duality groups. The *
*reader may
like to construct an analogy with Gorenstein local rings. A Gorenstein ring can*
* be character-
ized as one having finite injective dimension over itself, but perhaps the reas*
*on Gorenstein
rings are so important is their duality. Similarly, if we restrict attention to*
* groups G with the
finiteness condition that k admits a resolution by finitely generated projectiv*
*e kG modules,
we may characterize duality groups as torsion free groups G so that
Hi(G; kG) := ExtikG(k; kG) = 0 unlessi = n
for some n. The number n is called the dimension of G and the module
I = Hn(G; kG)
is called the dualizing module. Such groups then have a duality isomorphism
Hi(G; M) ~=Hn-i(G; M I):
We say that G is a Dn-group; if I is one dimensional over k we say G is a Poinc*
*are duality
group (P Dn-group), and if in addition G acts trivially on I we say it is orien*
*table.
A group G is said to be a virtual duality group if it has a subgroup G0of fin*
*ite index which
is a duality group. The virtual dimension of G is the dimension of G0(and is w*
*ell defined)
and the dualizing module for G is
I = Hn(G0; kG0) ~=Hn(G; kG)
where the isomorphism is Shapiro's lemma and this shows I is a G-module.
Example 8.3. (i) Borel and Serre [13] show that any torsion free arithmetic gro*
*up is a
duality group, and a general arithmetic group is a virtual duality group.
(ii) Fundamental groups of a knot complement are D3-groups.
(iii) Mapping class groups are virtual duality groups [41, 40].
(iv) Automorphism groups of free groups are virtual duality groups [10].
Thus the class of virtual duality groups is very extensive. However there ar*
*e very few
non-trivial examples where the cohomology ring is known explicitly.
8.4. p-adic Lie groups. It is possible to transpose the entire previous section*
* into the
category of profinite groups. If G is a profinite group we may consider module*
*s for its
(discrete) finite quotients. However it is important to be able to discuss inv*
*erse limits of
these (compact G-modules) and direct limits (discrete G-modules). It is possibl*
*e to put both
classes of modules into a reasonably well behaved category [54], and make most *
*of the usual
constructions of homological algebra in this category. In particular the comple*
*te group ring
k[[G]] behaves like a free module and the cohomology ring may be defined by
S := H*(G; k) = Ext*k[[G]](k; k):
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 17
Just as in the discrete case we may define duality groups and virtual duality*
* groups.
Virtual duality groups have a virtual dimension and a (compact) dualizing modul*
*e I =
Hn(G; k[[G]]).
Example 8.4. Lazard shows that any p-adic Lie group is a profinite virtual dual*
*ity group
[44]. As in the discrete case, the interesting examples that come up are automo*
*rphism groups
of suitable structures. Familiar examples are the groups SLn(Z^p), and another *
*class of exam-
ples of considerable interest to topologists are the Morava stabilizer groups (*
*automorphism
groups of certain formal groups).
For all these more general classes of groups, the ring S is known to share wi*
*th the case
of finite groups the properties that it is a finitely generated k-algebra and o*
*f dimension
rankp(G) [46].
9. The local cohomology theorem for group cohomology.
We return to the local cohomology theorem, treated in Part I for the represen*
*tation ring.
Now we consider what it says for ordinary cohomology, which is to say for the a*
*ugmented
rings k-algebras S discussed in Section 8. This case is much more interesting b*
*ecause S can
be of higher dimension, and exhibits a wide variety of behaviours as the group *
*varies.
We are writing m for the augmentation idea: it is the maximal ideal of positi*
*ve codegree
elements. We write
DM = Hom S(M; I(k)) = Hom k(M; k)
for the Matlis dual, where I(k) is the injective envelope of k.
Theorem 9.1. [26] If S = H*(G; k) for a finite group G there is a spectral sequ*
*ence
E*;*2= H*m(S) ) DS;
with differentials dr : Es;tr-! Es+r;t-r+1r. This is a spectral sequence of S-*
*modules in the
sense that dr : Es;*r-! Es+r;*ris a map of S-modules for all r and s.
We will outline the proof of the result in Section 10 below, but first we dis*
*cuss some special
cases.
Example 9.2. (i) If G is an elementary abelian 2-group of rank r and k is of ch*
*aracteristic
2 then S = k[x1; : :;:xr]. It is easy to calculate the local cohomology from th*
*e stable Koszul
complex, and we deduce
H*m(S) = Hrm(S) = -rDS:
More generally, whenever S is Cohen-Macaulay of dimension r, we find
H*m(S) = Hrm(S)
and hence the spectral sequence collapses to give
Hrm(S) = -rDS;
and hence S is Gorenstein. This recovers a theorem of Benson and Carlson [7].
(ii) The simplest non Cohen-Macaulay example is the semidihedral group of order*
* 16, of
dimension 2 and depth 1. The spectral sequence again collapses since the E2-ter*
*m is con-
centrated in two adjacent columns.
(iii) The simplest example with Cohen-Macaulay defect 2 is the group 7a2. The c*
*ohomology
is calculated by Rusin [50], and the relevant features are highlighted by Benso*
*n and Carlson
18 J.P.C.GREENLEES
[7]. One may show that the spectral sequence does not collapse, and one may ide*
*ntify the
differential.
Turning next to compact Lie groups, recall the conjugation of G induces the a*
*djoint
representation G -! k*, and let k(ad) denote k with this action.
Theorem 9.3. [9] If S = H*(BG; k) for a compact Lie group G of dimension d, the*
*n there
is a spectral sequence
H*m(S) ) -dH*(BG; k(ad));
of S-modules where k(ad) is the coefficient system in which each element g of t*
*he group acts
on k by +1 or -1 according to whether conjugation by g preserves or reverses th*
*e orientation
in a neighbourhood of the identity.
Remark 9.4. When G is orientable the spectral sequence is just like the finite *
*case with a
shift
H*m(S) ) -dH*(BG; k);
Example 9.5. (i) All the finite group examples discussed before are also exampl*
*es here.
(ii) If G is the circle group (of dimension d = 1) then S = k[x2] and if G = SU*
*(2) (of
dimension 3) then S = k[x4]. Both groups are connected, so that k(ad) = k. Thes*
*e examples
show how the dimension shift -d comes into play.
(iii) We have already commented that the adjoint representation of O(2n) is non*
*-trivial
if char(k) 6= 2. This accounts for the fact that the local cohomology of H*(BO*
*(2n)) =
k[p1; : :;:pn] is a copy of its dual shifted by -2n(n + 1), whereas the dimensi*
*on shift for an
orientable group of the same dimension (namely n(2n - 1)) and rank (namely n) w*
*ould be
-n - n(2n - 1). The difference of 2n is explained by the non-trivial adjoint ac*
*tion.
Finally, we turn to virtual duality groups.
Theorem 9.6. [8] If S = H*(BG; k) for a (discrete or profinite) virtual duality*
* group G of
dimension n there is a spectral sequence
H*m(S) ) nH*(G; I);
of S-modules where I is the dualizing module.
Remark 9.7. When G is an orientable Poincare duality group the spectral sequenc*
*e is just
like the finite case with a shift
H*(S) ) nH*(BG; k);
We will discuss the implications of these theorems for the structure of the r*
*ing S in the
Section 11, and in Section 10 we sketch a proof.
10. An algebraic proof of the local cohomology theorem for ordinary
cohomology.
In this section we outline the proof of the local cohomology theorem in the c*
*ase of finite
groups. This is the proof given in [26]; it is possible to give topological pro*
*ofs as in Part I
or Part III and another proof is given in [19].
In fact we will prove a more general result for cohomology of a kG-module M. *
*This states
that there is a spectral sequence
H*m(H*(G; M)) ) H*(G; M):
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 19
The case M = k recovers the theorem. Similarly there are versions of the result*
*s for other
classes of groups with modules of coefficients.
The simplest case is when G has periodic cohomology. This is known to be a ve*
*ry restrictive
assumption, but once this case is clear we should be able to describe the proof*
* in general.
The assumption means that there is a codegree n and an element x 2 S of codegre*
*e n so that
multiplication by x gives an isomorphism Sm ~=Sm+n whenever m 0. Thus S k[x].*
* Now
we may view x as an extension class x 2 ExtnkG(k; k) and represent it by an exa*
*ct sequence
___________________________________
0 -! k -! ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__||-! k -! 0:
If we begin a minimal resolution the C0; C1; : :;:Cn-2 are projective by constr*
*uction, and
Cn-1 is projective since Hn(G) ~=H0(G) = k. We thus consider the complex
___________________________________
C(x) = ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__||
and form a projective resolution P of k by concatenating copies of C(x), as we *
*may do
because Hn-1(C(x)) = k = H0(C(x)). We thus let
h ___________________________________
P = : :-:! ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__||-!
___________________________________ ___________________________________i
| Cn-1 -! Cn-2 -! . .-.! C1 -! C0 |-! | Cn-1 -! Cn-2 -! . .-.! C1 -! C0 |:
|__________________________________| |__________________________________|
Now we may write the stable Koszul complex
S -! lim 1=xkS = S[1=x]:
! k
It may help to visualize the term 1=xkS as a copy of S, displayed vertically, s*
*o that the
bottom is in degree -kn, and the maps in the direct system just include each co*
*lumn in the
next. We model this at the level of resolutions, at least after reversing arrow*
*s, to obtain
P = L0 - L1 = lim L[-kn; 1) = lim -knP:
k k
Again, one may view L[-kn; 1) as the chain complex L, displayed vertically, wit*
*h the
bottom is in degree -kn, and the maps in the inverse system just project each c*
*olumn onto
the previous one. Thus Lo = (L0 - L1) is a double complex, and the proof proce*
*eds by
considering the double complex
X = Hom kG(Lo; M)
There are two spectral sequences for calculating the cohomology of the double c*
*omplex.
The one in which we take Koszul cohomology first collapses to show
H*(X) = H*(H*Koszul(Hom kG(Lo; M))) = H*(Hom kG(HKoszul*(Lo); M))
since
HKoszul*(Lo) = H1(Lo) = P !;
where P != L1=P is the part of L1 in negative codegrees. Now
Hom k(P !; M) = P k M
so that
H*(X) = H*(G; M):
Taking the spectral sequence arising from the other filtration we obtain
H*Koszul(H*(Hom kG(Lo; M))) ) H*(X) = H*(G; M):
20 J.P.C.GREENLEES
This is the required spectral sequence. Indeed, by definition we have H*(Hom kG*
*(L0; M)) =
H*(G; M), and for L1 we calculate
H*(Hom kG(L1; M)) = H*(Hom kG(lim L[-kn; 1); M))
n
= lim H*(Hom kG(L[-kn; 1); M))
! n
= lim 1=xnH*(G; M))
! n
= H*(G; M))[1=x];
where the second equality used the fact that the limit is achieved in each degr*
*ee. Thus we
obtain the stable Koszul complex,
(H*(Hom kG(Lo; M))) = (H*(G; M) -! H*(G; M)[1=x]):
The cohomology of this complex is the local cohomology of H*(G; M).
When G does not have periodic cohomology, Noether normalization shows that S *
*is finite
over a polynomial subring k[x1; : :;:xr]. We replace the single complex C(x) by*
* C = C(x1)
. . .C(xr), which we can view as an r-dimensional box. This time the top module*
*s in each
complex C(xi) are not projective, but by the theory of support varieties [5, Ch*
*apter 5],
because S is finite over the polynomial subring, their tensor product is projec*
*tive. Now the
proof is directly analogous. We mimic the construction of the multigraded Koszu*
*l complex
by stacking boxes. For example with r = 2 we have
(L0 - L1 - L2) =
(P ([0; 1)x[0; 1)) - P ((-1; 1)x[0; 1))P ([0; 1)x(-1; 1)) - P ((-1; 1)x(-1; 1*
*)));
where P ([0; 1) x [0; 1)), for instance, is the result of stacking boxes in the*
* first quadrant.
11.Structural implications of the local cohomology theorem.
This section summarizes the contents of [30], outlining the implications for *
*a finitely
generated k-algebra S of the existence of a spectral sequence
H*m(S) ) aDS:
Duality is exact in the context so it is equivalent to say there is a spectral *
*sequence
DH*m(S) ) -aS:
We say that S has a local cohomology theorem with shift a, or that it is an LCT*
* aring.
First we repeat the immediate observation that if S is Cohen-Macaulay of dime*
*nsion r
then the spectral sequence collapses to give an isomorphism
DHrm(S) = r-aS;
so that S is Gorenstein.
Next, consider the case that S has Cohen-Macaulay defect 1 (in the sense that*
* its depth
is one less than its dimension). We write = DHrm(S) for the canonical module,*
* and
= DHr-1m(S) for the subcanonical module. Thus the spectral sequence collapses*
* to the
short exact sequence
0 -! a-r+1 -! S -! a-r -! 0:
Proposition 11.1. If S is an LCT aring with Cohen-Macaulay defect 1 then
o has depth r and Hrm() = r-aD, and
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 21
o has depth r - 1 and Hr-1m() = r-1-aD.
It is natural to say that a ring with Cohen-Macaulay defect 1 is almost Cohen*
*-Macaulay
and to say that it is almost Gorenstein if it satisfies the conclusion of the p*
*roposition.
Proof: Note first that the depth statements are equivalent to the vanishing of *
*local coho-
mology up to the top degree. To give the idea of proof, we will just explain th*
*e vanishing
of lower local cohomology groups. The isomorphisms for the top local cohomology*
* groups
follow by extending the analysis one more step.
Grothendieck's spectral sequence
Ep;q2= Hpm(DHr-qm(S)) ) DS
(which applies to an arbitrary commutative ring) gives Him() = Hi+2m() for i r*
* - 3.
Applying the local cohomology to the short exact sequence from the local cohomo*
*logy the-
orem gives Hi-1m() ~=Him() for i r - 2. Combining these gives the statements a*
*bout
depth.
Remark 11.2. It is worth recording some consequences of these results for an ar*
*bitrary
LCT aring. First, we recall that Grothendieck defined the dual localization fun*
*ctor L"M =
D(D(M)"), which is useful for Artinian modules like local cohomology modules. I*
*t is exact,
and an easy consequence of local duality is the fact that if " is a prime with *
*S=" of dimension
d then L"Him(M) = Hi-dm(M"). Accordingly, we may apply L" to see that if S is a*
*n LCT a
ring then S" is an LCT a-dring. Hence we conclude that for any minimal prime ",*
* the ring
S" is Gorenstein (one says S is Gorenstein in codimension 0), and that if " is *
*of height 1,
the ring S" is almost Gorenstein (one says S is almost Gorenstein in codimensio*
*n 1).
On a more concrete level, if S is a finitely generated k-algebra in codegrees*
* 0, we may
obtain conditions on its Hilbert series [S](t) = idim k(Si)ti.
Corollary 11.3. (i) (Stanley [53]) If S is a Cohen-Macaulay LCT aring then
[S](1=t) = t-a(-t)r[S](t):
(ii) If S is an almost Cohen-Macaulay LCT aring then
[S](1=t) - t-a(-t)r[S](t) = (-1)r-1(1 + t)[](t)
and
[](1=t) = ta(-t)1-r[](t):
Remark 11.4. In [30] the functional equation for is misstated: two signs are n*
*egated,
one in the statement and one in the proof.
Proof : To give the idea, we present the proof of Part (i) in a way that sugges*
*ts that of
Part (ii), referring the reader to [30] for further details. We first observe t*
*hat the result is
elementary for a polynomial ring. Now by Noether normalization we may find a po*
*lynomial
subring "S S over which S is a finitely generated module. We may work entirely*
* with
S"-modules in the rest of the proof. By the Auslander-Buchsbaum formula, since *
*S is Cohen-
Macaulay, S = F0 k "Sas "S-modules, where F0 is a finite dimensional graded vec*
*tor space.
Thus for Hilbert series, [S] = [S"][F0]. Now calculate
[S](1=t) = [F0_][S"](1=t) = (-1)r[F0_][DHrm(S")] = (-1)r[DHrm(S)] = (-1)rt*
*r-a[S];
22 J.P.C.GREENLEES
where the last equality is the local cohomology theorem. The proof in the almos*
*t Cohen-
Macaulay case uses exactly the same ingredients, now working with a short exact*
* sequence
0 -! F1 k "S-! F0 k "S-! S -! 0.
Example 11.5. We consider the semidihedral group SD16. It is of dimension 2 an*
*d has
Hilbert series f(t) = 1=(1 - t)2(1 + t2). We calculate
t4 t2 t2
f(1=t) - (-t)2f(t) = ______________- ______________= -(1 + t)_____________:
(1 - t)2(1 + t2)(1 - t)2(1 + t2) (1 - t)(1 + t2)
It is then easy to check that with ffi(t) = t2=(1 - t)(1 + t2) we have
ffi(1=t) = (-t)-1ffi(t)
as required.
It is interesting to note that f(t) also satisfies the single functional equa*
*tion
f(1=t) = t2(-t)2f(t)
just as if it was the the Hilbert series of a Cohen-Macaulay LCT ring with shif*
*t a = -2.
Part III.Examples related to bordism.
There is a rather general class of cohomology theories for which one can give*
* a uniform
treatment very close to ideas from commutative algebra. Curiously, these examp*
*les arise
from the study of a certain class of manifolds, but these geometric antecedents*
* are not
relevant here. Even in topology, Quillen's theorem [48] that there is an intim*
*ate relation
between bordism and the algebraic theory of formal groups means that much discu*
*ssion of
bordism is conducted in purely algebraic terms.
12. The theorem for MU-modules.
There is an equivariant version of bordism constructed from certain manifolds*
* with group
action. The cohomology theory MU*G(.) is called equivariant (homotopical) compl*
*ex cobor-
dism. Its value MU*G(X) on a G-space X is a module over the coefficient ring S *
*= MU*G.
Theorem 12.1. (Greenlees-May [37]) If M*G(.) is module valued over MU*Gthen for*
* any
finite group there are spectral sequences
H*J(MG*(X)) ) MG*(EG x X) = M*(EG xG X)
and
HJ*(M*G(X)) ) M*G(EG x X) = M*(EG xG X)
where J = ker(MU*G-! MU*) is the augmentation ideal.
Remark 12.2. (i) In fact MU*Gis not Noetherian, and J is not known to be of fin*
*ite
arithmetic rank. Accordingly there is work involved in showing the initial ter*
*ms of the
spectral sequences make sense: one shows that H*J0(.) is independent of J0 for *
*all sufficiently
large finitely generated ideals J0 J.
(ii) We shall only be applying this in the case that M*Gis a Noetherian ring in*
* its own right,
and one may replace J with JM = ker(M*G- ! M*). We view this as an example in*
* the
form of Context 7.1 by replacing S with SM = M*G, and k with kM = M*.
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 23
(iii) The sense in which M*G(.) is required to be module valued will be explain*
*ed in Section
13.
Example 12.3. The easiest class of examples to describe are those which are J-c*
*omplete
in a suitable sense. Fortunately these are the ones we intend to discuss later.
For this class we first define the non-equivariant theory. Indeed the coeffi*
*cient ring of
complex cobordism is a polynomial ring
MU* = Z[x1; x2; x3; : :]:
on infinitely many variables, where xihas degree 2i. For any flat module M* ove*
*r MU* the
definition
M*(X) = MU*(X) MU* M*
gives an exact functor of X, and is therefore a homology theory, and hence repr*
*esented by a
spectrum M. For topological reasons the flatness condition can be weakened to s*
*omething
much more easily verified (Landweber exact functor theorem). There are also man*
*y module
valued theories for which M* is not flat, but other means must be used to const*
*ruct them.
From any such non-equivariant theory one may form the complete theory
M*G(X) = M*(EG xG X):
One family of well known examples are the complete Johnson-Wilson theories M *
*= En
with
E*n= Z^p[[u1; : :;:un-1]][u; u-1];
where the ui are of degree 0 and u is of degree 2. This may be constructed usi*
*ng the
Landweber exact functor theorem. One may show that for any finite group G, E*n(*
*BG) is
finite over E*n.
Another example has ku* = Z[x1], and here ku*(BG) is only finite over a compl*
*etion of
ku*. The coefficient ring does not satisfy the hypotheses of the Landweber exa*
*ct functor
theorem, so other methods must be used to construct it.
We will return to an algebraic investigation of these two last examples in Se*
*ctions 15 and
16, but first we spend Section 13 explaining the idea of the proof of the the l*
*ocal cohomology
theorem for Noetherian MU-algebras. This is so closely analogous to commutative*
* algebra
in the derived category that it inspires a number of definitions of interest in*
* algebra as well
as topology, and we briefly introduce them in Section 14.
13.The proof.
The main thing is that there is a good category to work in [20]. Working the*
*re, the
proof is essentially formal and just like working in an algebraic derived categ*
*ory. This is the
category of G-spectra, briefly introduced in Subsection 13.1. In Subsection 13.*
*2 we transpose
some commutative algebra into the category of G-spectra and in the final two su*
*bsections
we complete the proof.
13.1. G-spectra. A G-spectrum may be thought of as a generalized based G-space.*
* The
purpose of the generalization is to ensure that any equivariant cohomology theo*
*ry E*G(.) is
represented in the category of G-spectra. This means that there is a G-spectrum*
* E so that
for any unbased G-space X,
E*G(X) = [X+; E]*G
24 J.P.C.GREENLEES
where the expression on the right denotes G-homotopy classes of G-maps of G-spe*
*ctra.
Sufficiently well behaved cohomology theories (such as MU) are represented by r*
*ing objects
R in the category of G-spectra, and there is a category of module spectra over *
*R. These
rings R are analogous to differential graded algebras, and the homotopy categor*
*y of modules
over R is directly analogous to the derived category of differential graded mod*
*ules over the
differential graded ring. This homotopy category is where we work. Thus for a m*
*odule M
over R, and an unbased G-space X,
M*G(X) = [X+; M]*G= [R ^ X+; M]*R;G
and
MG*(X) = [S0; X+ ^ M]G*= [R; X+ ^ M]R;G*;
where the decoration R; G refers to equivariant homotopy classes of R-module ma*
*ps.
Warning 13.1. Even if R*G(.) is represented by a ring R, it is not automatic th*
*at the
representing G-spectrum M of a module valued cohomology theory M*G(.) is an R-m*
*odule.
However this is true in many cases, and specifically for the complete theories *
*formed from
En and ku discussed above.
The proof of the the local cohomology theorem described in this section does *
*not in fact
apply to R = MU. It requires us to work with a cohomology theory R*G(.) represe*
*nted by a
ring G-spectrum R, with two properties. Firstly, the coefficient ring R*Gmust b*
*e Noetherian
(and for all subgroups H the modules R*Hmust be finitely generated), and second*
*ly, it must
be complex oriented (this is equivalent to saying that R is an MU-algebra up to*
* homotopy,
but it has a more concrete meaning that will be described at the appropriate po*
*int). The
coefficient ring of MU is not Noetherian, so the main obstacle is the construct*
*ion of enough
elements of the ideal J: this is interesting but not relevant to our applicatio*
*ns. In fact the
complete theories of both En and ku are both represented by ring spectra R to w*
*hich the
argument of this section does apply.
13.2. Commutative algebra with G-spectra. In the category of modules over R one
can mimic most constructions in the derived category. For example we can const*
*ruct the
homotopy I-power torsion functor. If x 2 RG*= [R; R]G;R*we may form
(x)R := fibre(R -! R[1=x])
and then if I = (x1; : :;:xn) we take
IR := (x1)R ^R (x2)R ^R : :^:R(xn)R:
Up to equivalence, IR does not depend on the generators used, and it only depen*
*ds on the
radical of the ideal I: this is an easy exercise exactly as in the algebraic de*
*rived category of
modules over a commutative ring (see Section 6). The case of a principal ideal *
*is constructed
as a fibre and so comes with a filtration of length 1, and hence IR has a filtr*
*ation of length
n. Because the construction is modelled on the stable Koszul comples, it is cle*
*ar that the
homotopy spectral sequence of the filtration takes the form
E*;*2= H*I(M*G) ) [R; IM]R;G*:
To prove the local cohomology theorem it therefore suffices to establish the *
*two equiva-
lences
R ^ EG+ -A JR ^ EG+ -B! JR
of equivariant R-modules.
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 25
We may deduce the local cohomology theorem by applying (M ^ X+) ^R (.) to obt*
*ain the
equivalence
M ^ EG+ ^ X+ ' JR ^R M ^ X+:
The homotopy of the left hand side is MG*(EG x X) and that of the right hand si*
*de is
calculated by a spectral sequence with E2 term H*J(MG*(X)).
We may deduce the completion theorem by applying Hom R (. ^ X+; M) to obtain *
*the
equivalence
Hom R(R ^ EG+ ^ X+; M) ' Hom R(JR ^ X+; M):
The homotopy of the left hand side is M*G(EG x X), and the homotopy of the righ*
*t hand
side may be calculated by a spectral sequence with E2-term HJ*(M*G(X)).
It remains to prove that the maps A and B are equivalences of equivariant R-m*
*odules.
13.3. The map A is an equivalence. Since EG+ is free, it is built from free G-c*
*ells
Sn ^ G+, so it suffices to show that R ^ G+ - JR ^ G+ is an equivalence, which*
* is to say
that R - JR is a non-equivariant equivalence. But resG1J = 0 by definition, so
resG1JR = resG1JR = 0R = R
as required.
13.4. The map B is an equivalence. We show the mapping cone of B is contractibl*
*e. In
fact we may assume by induction on the group order that it is H-contractible fo*
*r all proper
subgroups H. This uses the fact that resGHJ has the same radical as J(H) = ker(*
*R*H-! R*)
as follows in the Noetherian setting from [28]. From this we deduce
resGHJR = resGHJR = J(H)R:
Now the mapping cone of B is JR ^ "EG where "EG is the mapping cone of EG+ -!*
* S0.
Because (E"G)G = S0, there is unique map E"G -! S1V which is the identity in G*
*-fixed
points, where V is the reduced regular representation. It suffices to show that*
* JR ^ S1V
is contractible. Indeed, if we define Q by the cofibre sequence
E"G -! S1V -! Q;
we see that Q is built from cells G=H+ for proper subgroups H. It follows from *
*the inductive
hypothesis JR ^ Q ' *.
Since JR^S1V is obviously H-contractible for all proper subgroups H it suffi*
*ces to show
its G-homotopy is zero. Now for any R-module M we may define Euler classes O(V *
*) exactly
as was done for K-theory in Section 6. Indeed, it is immediate from the constru*
*ction of MU
that MU ^SV ' MU ^S|V,|(where |V | denotes V with the trivial G-action) so that*
* we may
take O(V ) to be the pullback of 1 2 MU|VG|= [S|V;|MU]*Gunder the inclusion S0 *
*-! SV .
From the definition we see that
ssG*(M ^ S1V ) = lim ssG*(M ^ SnV) = ssG*(M)[1=O(V )]:
! n
On the other hand, since the inclusion S0 -! SV is non-equivariantly nullhomoto*
*pic, O(V ) 2
J so that inverting it kills J-local cohomology and therefore ssG*(JR ^ S1V ) =*
* 0.
This completes the proof.
26 J.P.C.GREENLEES
14.Homotopically Gorenstein rings.
For further details see [29, 19].
To smooth the transition between topology and commutative algebra we now writ*
*e k*(X) =
E*(X) for the non-equivariant cohomology theory and S*(X) = E*G(X) for the equi*
*variant
one. In view of the completion theorem we may also write ^S*(X) = E*(EG xG X) f*
*or the
complete theory. We restrict attention to finite groups G, and therefore to th*
*e case with
shift a = 0 (in the notation of Section 11).
We have obtained duality statements by combining the local cohomology theorem*
* with
the universal coefficient theorem. In the first instance these were spectral se*
*quences
H*J(k*(BG)) ) k*(BG)
and
Extk*(k*(BG); k*) ) k*(BG);
but we have already reformulated the first as
JF (EG+; k) ' k ^ EG+;
and the second can be written
Hom k(k ^ BG+; k) ' F (BG+; k);
which we would like to think of as the fixed points of an equivariant equivalen*
*ce
Hom k(k ^ EG+; k) ' F (EG+; k):
The analogy is with the algebraic derived category, so Hom corresponds to the *
*total right
derived functor of ordinary homomorphisms. Substituting in the local cohomology*
* theorem
expression for k ^ EG+ ' F (EG+; k) ^ EG+, and writing ^S= F (EG+; k) we find
Hom k(JS^; k) ' ^S:
More generally we can consider this condition on any ring ^Sup to homotopy equi*
*pped with
ring maps k -! ^S- ! k making it into a supplemented k-algebra. We may say tha*
*t a
supplemented ring ^Sis homotopically Gorenstein if there is such an equivalence*
*. If k is a
field we may take homotopy and deduce that ^S*is an LCT ring. If k is not a fie*
*ld the left
hand side is the composite of two functors. Each of these functors could be cal*
*culated with
a spectral sequence, but it is harder to extract information about their compos*
*ite. However
when k is of small injective dimension we can still get striking duality proper*
*ties for ^S*.
For complete rings it is quite often equivalent to consider the statement
JS^' Hom k(S^; k);
and this is better behaved for non-complete rings, so we may say that a supplem*
*ented k-
algebra S is homotopically Gorenstein if
JS ' Hom k(S; k):
It turns out [19] that under quite weak hypotheses this is equivalent to requir*
*ing
Hom S(k; S) ' k
as modules over Hom S(k; k), which may be a more familiar form of the Gorenstei*
*n condition.
This is related to the notion of Gorenstein differential graded commutative alg*
*ebras consid-
ered by Avramov and Foxby [4], and to the ideas of Felix, Halperin and Thomas [*
*23] but
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 27
they only require the isomorphism as S-modules. This circle of ideas is investi*
*gated further
by Dwyer, Greenlees and Iyengar [18, 19].
15.The chromatic case.
For further details see [39].
We have explained that it is of interest to calculate the local cohomology mo*
*dules H*J(S)
where S = R*G= E*n(BG), because they give the E2-term of a spectral sequence fo*
*r calcu-
lating the more subtle invariant (En)*(BG). Playing this spectral sequence off *
*against the
universal coefficient theorem implies that the ring R*Ghas very special duality*
* properties.
Now here we have k = R* = Z^p[[u1; : :;:un-1]][u; u-1], and we may consider t*
*he Landweber
sequence of prime ideals
0 (p) (p; u1) (p; u1; u2) . . .(p; u1; : :;:un-1):
It turns out that a natural ring of endomorphisms acts on R* and that the Landw*
*eber
sequence is the unique maximal sequence of invariant prime ideals. For any modu*
*le M we
may form the Cousin complex
C(M) = (M[1=p] -! M=(p1 )[1=u1] -! M=(p1 ; u11)[1=u2] -! . .-.! M=(p1 ; u11; : *
*:;:u1n-1):
If (p; u1; : :;:un-1) is M-regular we say M is good; in this case M ' C(M). It *
*is then natural
to use this filtration to approach the calculation of local cohomology.
Theorem 15.1. [39] If E*n(BG) is good, then local cohomology is trivial on pure*
* chromatic
strata in the sense that
H*J(E*n(BG)) = H*(JC(E*n(BG)):
Remark 15.2. (i) The module E*n(BG) is known to be good in many cases, includin*
*g all
abelian groups G, and all symmetric groups [42].
(ii) The complex JC(E*n(BG)) is highly non-trivial. Indeed, JC(E*n(BG)) is usua*
*lly non-
zero up to degree n: for example if G is abelian then in degree i it is |G|iCi(*
*E*n). On the
other hand if G is abelian of rank r, then J has arithmetic rank r, and hence *
*the local
cohomology is trivial above degree r. If n > r the exactness of the complex abo*
*ve degree r
must involve interesting differentials.
(iii) The case of periodic K-theory gives something we have already seen. It st*
*ates
H*J(R(G)^p) = H*(JR(G)^p[1=p] -! JR(G)=p1 ) = H*(Z^p[1=p] -! R(G)=p1 ):
This is very effective, and recovers the local cohomology calculations of Secti*
*on 4.
16.Connective K theory.
For further information see [12, Chapter 4].
In this section we present a very concrete example where the commutative alge*
*bra is very
intricate and very striking. The local cohomology theorem states there is a spe*
*ctral sequence
H*J(ku*(BG)) ) ku*(BG);
for any finite group G, but we want to discuss a case where the entire behaviou*
*r is understood.
We take G = V an elementary abelian 2-group of rank r. In fact there is a sho*
*rt exact
sequence
0 -! T -! ku*(BV ) -! Q -! 0:
28 J.P.C.GREENLEES
Here Q is the extended Rees ring for the J-completion ^R(V ) of the representat*
*ion ring
R(V ) = Z[ff1; : :;:ffr]=(ff21= . .=.ff2r= 1):
This completion has the effect of 2-adic completion on J, and leaving alone the*
* Z summand
generated by the regular representation. Thus
Q = ^R(V )[v; y1; y2; : :;:yr] ^R(V )[u; u-1]
where yi= (1-ffi)=u, and we may take J to be generated by elements mapping to y*
*1; : :;:yr.
One may calculate H*J(Q) without too much difficulty: H0J(Q) = Z[u] and H1J(Q) *
*is zero
below degree -2r and the order of its Z-torsion increases with degree.
On the other hand T is a bit more complicated. It is a P -submodule of F2[x1*
*; : :;:xr]
whereQP = F2[y1; : :;:yr] and x2i= yi. It turns out that it is generated by el*
*ements the
qS = i2Sxi(j;k2Sxjx3kas S ranges over subsets of {1; 2; : :;:r} with at least*
* 2 elements.
Now we can decompose T = T2 T3 . . .Tr, where Ti is generated by the qS with
|S| = i. These P -modules Tishow some remarkable behaviour.
Proposition 16.1. (i) Ti is of dimension r and depth i.
(ii) Hjm(Ti) = 0 unless j = i or r.
(iii) Him(Ti)_ is 1-dimensional if i < r.
(iv) Hrm(Ti)_ = Tr-i+2(-r + 4) if i < r.
(v) Hrm(Tr)_ = P (-r + 4).
Thus in codimension r - 2, the module T is CM and Hr(T )_ = T (-r + 4): it is*
* very
nearly Gorenstein.
However it turns out the structure of ku*(BV ) is bound even more tightly.
Lemma 16.2. The modules Him(Ti)_ are the subquotients in the 2-adic filtration *
*of H1J(Q):
Him(Ti)_(i - 2) = (2i-2H1J(Q)=2i-1H1J(Q))_
for i = 2; 3; : :r:- 1.
It turns out this is just what is required for ku*(BV ) to be well-behaved. T*
*here is a short
exact sequence
0 -! -4T _-! fku*(BV ) -! -1(2r-1H1J(Q)) -! 0
and furthermore this corresponds under the universal coefficient theorem to the*
* exact se-
quence above for ku*(BV ) in the sense that
Ext2ku*(-2T _; ku*) = T
and
Ext 1ku*(2r-1H1J(Q); ku*) = Q=(Z[u]):
This shows that even when we are not working over a field, the implications of *
*the homotopy
Gorenstein condition are very striking.
LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 29
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Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk