MULTIPLICATIVE EQUIVARIANT FORMAL GROUP LAWS.
J.P.C.GREENLEES
Abstract. The universal ring for multiplicative equivariant formal group *
*laws is shown to
be closely related to the Rees ring of the representation ring at the aug*
*mentation ideal, but
only equal to it if the group is topologically cyclic.
1. Introduction
The notion of an A-equivariant formal group law [2] for a compact abelian Lie*
* group A
was introduced to study complex oriented A-equivariant formal group laws, but h*
*as some
intrinsic algebraic interest. The theorem [4] that the coefficient ring of equi*
*variant complex
bordism is the universal ring for equivariant formal group laws establishes tha*
*t the definition
is the correct one. We shall be concerned here with a very special class of eq*
*uivariant
formal group laws: the multiplicative ones, which appear to play a privileged r*
*ole amongst
all equivariant formal group laws. However our principal motivation for consid*
*ering this
case is its importance in understanding equivariant K-theories, and its close r*
*elationship to
representation theory. Much of the algebra presented here is closely mirrored i*
*n [1], and the
author is grateful to R.R. Bruner for useful discussions and comments.
We recall the definition of an A-equivariant formal group law to establish no*
*tation.
Definition 1.1. [2] If A is a finite abelian group, an A-equivariant formal gro*
*up law over a
commutative ring k is a*commutative topological k-algebra R with a coproduct :*
* R -!
R ^R, a map : R -! kA and an orientation y(ffl) 2 R with the following proper*
*ties.
Firstly, makes R into a bicommutative Hopf algebra, and is a homomorphism of *
*Hopf
algebras. Secondly,
(i) y(ffl) is regular
(ii)R=(y(ffl)) ~=k (induced by (ffl)) and
(iii)The topology on R is complete and defined by the ideal = ker().
Remark 1.2. (i) If A is a general abelian compact Lie group the definition is*
* the same
except that in Condition (iii), R is required to be complete with respect to th*
*e system of
principal ideals given*by finite intersections of kernels of the components (ff*
*) : R -! k of
. The Hopf algebra kA is topologized as the product of discrete rings k.
(ii) The element y(ffl) is called the orientation of the formal group law. If t*
*he orientation is
not specified, the resulting structure is called an equivariant formal group.
(iii) One view is that an equivariant formal group law encodes the formal prope*
*rties of the
Euler classes e(ff) = (y(ffl))(ff-1).
The additive structure of every equivariant formal group law is topologically*
* free, and
we may therefore express the structure maps of R in terms of the basis. To des*
*cribe this
basis, we note the element y(ffl) determines elements y(ff) for ff 2 A* by the *
*formula y(ff) =
1
2 J.P.C.GREENLEES
((ff) 1)(y(ffl)). The completeness is thus equivalent to completenessQwith res*
*pect to the
system of principal ideals generated by all finite products ffy(ff).
Theorem 1.3. [2, 13.2] If we choose a complete flag F = (V 1 V 2 . .).in a c*
*om-
plete A-universe, then an equivariant formal group law R has an additive topolo*
*gical k-basis __
1; y(V 1); y(V 2); : :w:here y(V n) = y(ff1)y(ff2) . .y.(ffn) if V = ff1 ff2 *
*. . .ffn. |__|
Remark 1.4. Note that if A is the trivial group this shows that Definition 1.*
*1 gives the
usual concept of a (non-equivariant, commutative, one dimensional) formal group*
* law.
In this note we consider equivariant formal group laws of a very simple form.
Definition 1.5. (i) An equivariant formal group law R is multiplicative if its *
*coproduct
takes the form
y(ffl) = 1 y(ffl) + y(ffl) 1 - vy(ffl) y(ffl)
for some v 2 k.
(ii) Given a multiplicative formal group law over k, we define a binary operati*
*on on k-modules
by x m y = x + y - vxy.
(iii) We also define a polynomial [n](x) in v and x inductively by [0](x) = 0 a*
*nd [n](x) =
([n - 1](x)) m x. Thus
[n](x) = (1 - (1 - vx)n)=v:
Remark 1.6. (i) Note that v is not required to be a unit. In particular, we *
*allow the
degenerate case v = 0, which is usually referred to as an additive law. If v is*
* a unit we say
the formal group law is strictly multiplicative.
(ii) If we assign gradings so that |v| + |x| = 0, then the polynomial [n](x) is*
* homogeneous,
and has the same degree as x. We shall give v degree 2.
(iii) The coefficient of y(ffl) y(ffl) is named to correspond to the Bott elem*
*ent in topological
K-theory.
(iv) The notion depends heavily on the orientation: it is a property of the for*
*mal group law
and not of its underlying formal group.
The purpose of the present note is to observe that there is a representing ri*
*ng for multi-
plicative formal group laws, to identify it explicitly, and to relate it to rep*
*resentation theory.
Readers used to equivariant formal group laws may be surprised by the simplicit*
*y of the
answer.
To prepare for the statement, we consider the complex representation ring R(A*
*). For any
complex representation V we may define the Euler classes O(V ) as the alternati*
*ng sum of
exterior powers of V . Thus if ff is one dimensional O(ff) = 1-ff, and O(V W ) *
*= O(V )O(W ).
Note that since A is abelian, the augmentation ideal J = ker(R(A) -! Z) is gene*
*rated by the
Euler classes O(ff) of one dimensional representations. Indeed, since 1-fffi = *
*(1-ff)+ff(1-fi)
it suffices to use O(ff) as ff runs through a set of generators for the dual gr*
*oup A*. The
Rees ring Rees(R(A); J) is the subring of R(A)[v; v-1] generated by v and the s*
*hifted Euler
classes e(V ) = v-|V |O(V ) of representations, where |V | denotes the complex *
*dimension of
the representation V . The Rees ring is thus R(A) in each positive even degree*
* and Jn in
degree -2n.
MULTIPLICATIVE EQUIVARIANT FORMAL GROUP LAWS. 3
Theorem 1.7. For any compact abelian group A there is a representing ring LmA*
*for equi-
variant formal group laws. The ring LmAis a Z[v]-algebra and may be described a*
*s follows.
(i) If A = B x C then
LmA= LmBZ[v]LmC
(ii) If A is a finite cyclic group of order n with dual group A* = then
LmA= Z[v; e]=([n](e)):
This becomes a graded ring if v has degree 2 and e = e(ff) is of degree -2.
(iii) If A is a circle group and A* = then
LmA= Z[v; f; f0]=(vff0- f - f0):
This becomes a graded ring if v has degree 2 and both f = e(z) and f0 = e(z-1) *
*have degree
-2.
In the course of proving the theorem we will obtain a rather complete underst*
*anding of
multiplicative formal group laws themselves. The rest of this section is devote*
*d to deducing
a number of consequences of the theorem.
If we suppose A = B x C where B is finite and C is a d-dimensional torus, we *
*have the
presentation
A* =
of the dual group for suitable integers n1; n2; : :;:nr. We write ei = e(fii), *
*and fj = e(zj)
and f0j= e(z-1j).
Corollary 1.8. With the above notation
LmA= Z[v; e1; e2; : :;:er; f1; f01; f2; f02; : :;:fd; f0d]=a
where the ideal of relations is
a = ([n1](e1); [n2](e2); : :;:[nr](er); vf1f01= f1 + f01; vf2f02= f2 + f02; : *
*:;:vfdf0d= fd + f0d):
Corollary 1.9. There is a natural map
: LmA-! R(A)[v; v-1];
with image equal to the Rees ring.
Proof: Corollary 1.8 states that LmAis generated by v, and Euler classes of gen*
*erating one
dimensional representations. The defining relations also hold in R(A)[v; v-1],*
* and so the
map exists.
Since the Rees ring is generated by 1 and v together with e's, f's and f0's t*
*he_image_is as
claimed. *
* |__|
The above description of LmAdepends strongly on the chosen presentation of th*
*e group A.
If A is (topologically) cyclic we have a more satisfactory description.
Proposition 1.10. (i) If A is cyclic then the map of 1.9 induces an isomorphism
LmA~=Rees(R(A); J):
(ii) For any abelian group A, is the localization away from v:
LmA[v-1] ~=R(A)[v; v-1]:
4 J.P.C.GREENLEES
(iii) If A is not cyclic then LmAcontains Z-torsion and v-torsion.
Proof: In view of Part (iii), it is appropriate to give the proofs of Parts (i)*
* and (ii) in some
detail.
We begin by proving Part (ii). If A is cyclic of order n the relation [n](e) *
*= 0 is equivalent to
(1-ve)n = 1 once v is inverted, so Part (ii) follows from the presentation R(A)*
* = Z[ff]=ffn =
1. If A is the circle group the relation vff0 = f +f0becomes equivalent to (1-v*
*f)(1-vf0) = 1
once v is inverted, so Part (ii) follows from the presentation R(A) = Z[z; z0]=*
*zz0= 1. Part
(ii) now follows in general, since R(A)[v; v-1] is flat over Z[v].
When A is a circle or a finite cyclic group, it is easy to check LmAhas no v-*
*torsion, and
therefore Part (i) follows from Part (ii) in this case. An arbitrary cyclic gro*
*up A is of the
form B xC with B finite cyclic and C a torus. From Part (ii) and Theorem 1.7 (i*
*), it follows
that BxC = B Z[v]LmC. Part (i) now follows in general from the less obvious f*
*act that
Z[v; f; f0]=(vff0- f - f0) is flat over Z[v] ([5, 22.6]).
Now if A is not cyclic there are independent elements ff; fi 2 A* of order p *
*for some prime
p. Furthermore, we may suppose they lie in a subgroup of form B* = Cpax Cpbfor *
*some
a; b 1 generating a retract of A*. Thus A = B x C, and so LmA= LmBZ[v]LmCby 1.*
*7 (i);
since LmAis augmented over Z[v] it follows that LmBis a Z[v]-subalgebra of LmA,*
* and we may
thus suppose A* = Cpax Cpb.
Let e = e(ff) and f = e(fi). We thus have pe = ve2s(e) and pf = vf2s(f) for a*
* polynomial
s(x) 2 Z[v][x] of degree p - 2. Hence t = fe2s(e) - ef2s(f) is v-torsion and th*
*erefore et and
ft are p-torsion. To see that t; et and ft are themselves non-zero it suffices*
* to check this
mod p. Working mod p we find [p](x) = -vp-1xp, so the relevant ring is
a pb
LmA=p = Z=p[v; x; y]=((vx)p =v; (vy) =v)
a-1 pb-1 p-2 p p *
* __
with e = (vx)p =v and f = (vy) =v, and t = v (ef - e f). *
* |__|
Corollary 1.11. If A is cyclic, the representing ring LmA for multiplicative A-*
*equivariant
formal group laws, may be identified with the Rees ring
LmA= Rees(R(A); J):
In any case, the representing ring for strictly multiplicative A-equivariant fo*
*rmal group laws
LsmA= R(A)[v; v-1]:
Remark 1.12. (1) If we set v = 1 we recover the observation of [2] that the u*
*niversal ring
for multiplicative formal group laws of the form
y(ffl) = 1 y(ffl) + y(ffl) 1 - y(ffl) y(ffl)
is the representation ring R(A).
(2) The ring LsmA= LmA[v-1] = R(A)[v; v-1] is the coefficient ring of equivaria*
*nt K-theory.
(3) It is shown in [3] that if A is of prime order then the coefficient ring of*
* equivariant
connective K-theory is LmA. However it is also remarked that this cannot be tr*
*ue for all
abelian groups. Indeed, the completion of the coefficient ring of A-equivarian*
*t connective
K-theory must be ku*(BA), and this usually has non-zero groups in odd degrees (*
*for exam-
ple if A is elementary abelian of rank 3). It would be very interesting to hav*
*e a purely __
algebraic prediction for the coefficient ring of equivariant connective K-theor*
*y in general. |__|
MULTIPLICATIVE EQUIVARIANT FORMAL GROUP LAWS. 5
Finally, we record the corresponding results for additive formal group laws, *
*which follow
by setting v = 0.
Corollary 1.13. There is a universal ring LaAfor additive A-equivariant formal *
*group laws.
It is the free commutative ring on the abelian group A*, and with the above not*
*ation for A*,
it has the presentation
*
* __
LaA= Z[e1; e2; : :;:er; f1; f2; : :;:fd]=(n1e1; n2e2; : :;:nrer): *
* |__|
2. Euler classes and group schemes.
To begin with, we recall that the equivariance of the coproduct allows us to *
*deduce the
action of A* from the coproduct.
Lemma 2.1. [2, 16.7] For any one dimensional representation ff
__
y(ff) = e(ff) + (1 - ve(ff))y(ffl): |__|
Because there are no higher terms in the expression for y(ff) the ring R is m*
*uch simpler
than for a general equivariant formal group law.
*
* __
Corollary 2.2. The orientation y(ffl) is a topological generator of R. *
* |__|
The topological k-algebra R represents a formal scheme
G(l) = Hom cts(R; l):
The coproduct on R gives G(l) the structure of an abelian group.
Lemma 2.3. There is a natural identification
G(l) = A-nil(l)
where A-nil(l) is the ideal of l defined by
A-nil(l) = {x 2 l | ff(e(ff) - (1 - ve(ff))x) is topologically nilpot*
*ent}:
Under this identification the group operation is given by
x m y = x + y - vxy:
If A is infinite, the statement about topological nilpotence is to be interpret*
*ed as stating
that a sequence of products of elements (e(ff) - (1 - ve(ff))x) tends to zero p*
*rovided each
representation ff occurs infinitely often.
Proof: Because y(ffl) generates R, a map R -! l is determined by its image, and*
* we may
view G(l) as a subset of l. However there are two differences from a classical *
*formal group
law. Firstly, the element y(ffl) is not topologically nilpotent in general, an*
*d secondly it is
not a free generator. Because the complete universe contains the trivial repres*
*entation in-
finitely often, y(ffl) is transcendental over k, and R is the completion of k[y*
*(ffl)] with_respect
to ffy(ff). Applying 2.1 we deduce the given description of A-nil(l). *
* |__|
6 J.P.C.GREENLEES
Because G(l) can be viewed as an ideal in l as in the classical situation we *
*define the
polynomial [n](x) inductively by [0](x) = 0 and [n](x) = ([n - 1](x)) m x. Thus
[n](x) = (1 - (1 - vx)n)=v:
Next we record the fact that the coproduct describes the Euler classes of tenso*
*r products.
Lemma 2.4.
e(fffi) = e(ff) m e(fi)
and therefore
e(ffn) = [n](e(ff))
and
e(ffn) = e(ffn-1) + e(ff)(1 - ve(ff))n-1:
Proof: The first formula follows from the fact that the structure map is a map*
* of Hopf __
algebras. The resulting equation on y(ffl) gives the formula when evaluated at *
*(ff-1; fi-1). |__|
Note that if ffn = ffl then we have [n](e(ff)) = 0. This is slightly stronger*
* than the statement
that (1 - ve(ff))n = 1.
Corollary 2.5. For any one dimensional representation ff, the element 1 - ve(ff*
*) is a unit
with inverse 1 - ve(ff-1).
Proof: We have
(1 - ve(ff))(1 - ve(fi)) = 1 - ve(ff) m e(fi) = 1 - ve(fffi);
*
* __
so that 1 - ve(ff-1) is inverse to 1 - ve(ff). *
* |__|
3. Decoupling and its consequences.
The purpose of this section is to show that for multiplicative formal group l*
*aws the co-
product and Euler classes can be largely separated. This then allows us to give*
* the formal
reduction of the main theorem to the cases of the finite cyclic groups and the *
*circle.
An equivariant formal group is a more complicated object than a non-equivaria*
*nt one. In
the non-equivariant case, an orientation gives an isomorphism R = k[y] and the *
*coproduct is
defined relative to that ring structure. However, in general the ring structure*
* on R depends
on the structure map , and the formulation of the condition that is a Hopf map*
* requires
recursive use of itself. Fortunately, things are simpler in the multiplicative*
* case. First note
that the multiplicative coproduct is only polynomial and restricts to a coprod*
*uct on k[y].
This allows us to prove the following key result separating the two parts of th*
*e structure for
multiplicative group laws.
Proposition*3.1. (Decoupling of coproduct and Euler classes) Any Hopf map 0: k[*
*y] -!
kA from a multiplicative Hopf algebra determines a unique A-equivariant multip*
*licative for-
mal group law whose structure map extends 0.
MULTIPLICATIVE EQUIVARIANT FORMAL GROUP LAWS. 7
Proof: First note that we may define a topology on k[y] by taking y(ff) = e(ff)*
*+(1-ve(ff))y
in line with 2.1. Next we claim that 0is continuous for the topology. For this *
*it suffices to
note that
0(y(ff))(fi)= 0(e(ff) + (1 - ve(ff))y)(fi)
= e(ff) + (1 - ve(ff))e(fi-1)
= e(ff) m e(fi-1)
= e(fffi-1)
so that 0(y(ff)) vanishes in the ff'th coordinate.
We may now let R be the completion of k[y] for this topology. It is is clear*
* that the
multiplicative coproduct extends to R, and continuity of 0ensures that it exten*
*ds to a map
. Finally we need to verify Conditions (i) and (ii) of Definition 1.1. For (i),*
* suppose that a
sequence (yfn(y))n of polynomials tend to zero, so that any finite product of y*
*(ff)'s divides
some yfn(y). Since y is regular on k[y] it follows that the sequence (fn(y))n a*
*lso tends to zero.
For (ii), we know 0(y)(ffl) = 0, so we need only note that if fn(y) is a conver*
*gent sequence
then, since k is discrete, (fn(y))(ffl) is ultimately constant. However (fn(y))*
*(ffl) = fn(0)._
Finally, uniqueness of the formal group law follows since k[y] is always dens*
*e by 2.2. |__|
Corollary 3.2. A multiplicative A-equivariant*formal group law over k is given *
*by an_ele-
ment v 2 k and a map 0: k[y] -! kA of Hopf algebras. *
* |__|
We may now easily explain how the proof of the main theorem may be reduced to*
* the
special cases when A is the circle or a finite cyclic group. Note first that a*
*n arbitrary
abelian compact Lie group is a product of these special groups: this product de*
*composition
propogates through the entire structure.
For the following two well known lemmas, think of Hopf algebras as group obje*
*cts in the
category of cocommutative coalgebras (so in particular they are cocommutative).
Lemma 3.3. If H1 and H2 are Hopf algebras then H1 H2 is also a Hopf algebra, *
*and it is
the categorical product.
Proof: It is a formality that the forgetful map from group objects in a categor*
*y to all objects
creates products. It therefore suffices to check that the tensor product of two*
* coalgebras_is
their categorical product. *
* |__|
Lemma 3.4. For discrete abelian groups B0; C0 there is a natural isomorphism
0xC0 B0 C0
kB ~=k k
0xC0 *
* __
expressing kB as a categorical product of Hopf algebras using the group proj*
*ections. |__|
The proof of Part (i) of Theorem 1.7 is now a formality.
Corollary 3.5. If A = B x C then LmA= LmBZ[v]LmC:
Proof: We saw in 3.2 that an equivariant*formal group law is specified by v 2 k*
* together
with a Hopf map : k[y] -! kA .
8 J.P.C.GREENLEES
* * *
Fix v, and note that since k(BxC) is the Hopf algebra product of kB and kC ,
* B* C*
Hopf(k[y]; kA ) = Hopf(k[y]; k ) x Hopf(k[y]; k ):
*
* __
It follows that the representing ring is the coproduct of LmBand LmC. *
* |__|
4. Proof of the main theorem
After Section 3, it remains only to prove Theorem 1.7 Parts (ii) and (iii). *
*Let "LmA=
Z[v; e]=[n](e) if A is cyclic of order n or Z[v; f; f0]=vff0 = f + f0 if A is t*
*he circle. Since
the specified relation holds in all multiplicative formal group laws, we have a*
* natural map
"LmA-! LmA, and we must show it is an isomorphism.
We have seen that if R is a multiplicative equivariant formal group law, then*
* its structure
is determined by the elements v; e if A is finite or v; f; f0if A is the circle*
*. This shows the map
is a surjective. To complete the proof it suffices to show that there is an A-e*
*quivariant formal
group law over "LmAfor which the structure constants are as implied by the nome*
*nclature of
the generators of "LmA. Since A is cyclic, we have shown in 1.10 that "LmAis a*
* subring of
"LmA[v-1] = R(A)[v; v-1], and it suffices to show there is such an A-equivarian*
*t formal group
law over k = R(A)[v; v-1]. Since v is not a zero divisor in k, Euler classes ar*
*e determined by
ve(ff) = 1 - ff, and it remains only to check that the corresponding map is a *
*map of Hopf
algebras. *
By 3.1, it suffices to consider the restriction 0: k[y] -! kA , defined by 0(*
*y)(ff) = e(ff-1).
The fact that the resulting map 0is a map of Hopf algebras may be verified by e*
*valuation
on y, and this is the calculation
ve(fffi) = 1 - fffi = (1 - ff) + (1 - fi) - (1 - ff)(1 - fi) = ve(ff) + ve(f*
*i) - ve(ff)ve(fi):
*
* __
*
*|__|
Remark 4.1. Topologists will note that the existence of the appropriate equiv*
*ariant formal
group law over R(A)[v; v-1] also follows from the fact that equivariant K-theor*
*y is a complex
oriented theory. However, this relies on equivariant Bott periodicity, and is t*
*herefore much
less elementary.
*
* __
*
*|__|
5.The structure of multiplicative equivariant formal group laws.
Note that 2.2 shows that for any A the underlying ring R of an equivariant fo*
*rmal group
lawQcan be described as a completion of the polynomial ring k[y] at the finite *
*products
ffy(ff). Collecting together the results of the Section 2 we are able to give*
* a more explicit
description when A is finite. This makes the geometry of the situation a little*
* clearer.
For the rest of the section we assume A is finite and adopt the abbreviations*
* y = y(ffl) and
x = ffy(ff).
Proposition 5.1. If A is a group of order N, then
R = k[[x]][y]=(yN = ux + yr(y))
MULTIPLICATIVE EQUIVARIANT FORMAL GROUP LAWS. 9
for some polynomial r(y) of degree N - 2 and some unit u.
Remark 5.2. The proof will show that the polynomial r(y) and the unit u are e*
*ssentially
independent of R. More precisely, the element u and the coefficients of r(y) ca*
*n be expressed
as elements of Z[v; e1; e2; : :;:er] where ei= e(fii).
Indeed, the proof will give an algorithm for finding u and r(y) explicitly. F*
*or instance, if
A is cyclic we may choose a generator ff of A* and take e = e(ff) to obtain
k[[x]][y]=(y2 = (1 - ev)x + ey) if A is of order 2
and
k[[x]][y]=(y3 = x + ey(y(3 - ev) - e(2 - ev))) if A is of order 3
Proof: Certainly there is a natural map k[x; y] -! R, determined by our choice *
*of orienta-
tion, and this extends to the completion k[[x]][y]. The map is surjective by 2.*
*2.
Now choose a periodic complete flag with ffi = ffi+N and V kN = kCA. By 1.3,*
* we
know 1; y = y(V 1); y(V 2); : :;:x = y(V N); xy = y(V N+1); xy(V 2) = y(V N+2);*
* : :;:x2 =
y(V 2N); x2y = y(V 2N+1); : :a:re topologically independent over k. It therefo*
*re suffices to
establish the relation yN = ux + yr(y) in R, and we prove something a little mo*
*re general,
which applies whether A is finite or not.
Lemma 5.3. For any n 1 there is a relation yn = uny(ff1)y(ff2) . .y.(ffn) + *
*yrn(y) in R
where un is a unit and rn(y) is of degree n - 2. The element un and the coeffi*
*cients of
rn(y) can be expressed as elements of Z[v; e1; e2; : :;:es] where the elements *
*e1; e2; : :;:es are
Euler classess of monoid generators of A*. We have the recursive formulae u1 = *
*1; r1(y) = 0
and for n 2,
un+1 = un(1 - ve(ff-1n+1))
and
rn+1(y) = rn(y)[y + e(ffn+1)(1 - ve(ff-1n+1))] - yn-1e(ffn+1)(1 - ve(f*
*f-1n+1))
Proof: We prove this by induction on n, noting it is trivial for n = 1. For the*
* inductive step
we suppose the result is true as stated and note that y(ffn+1) = e(ffn+1) + (1 *
*- ve(ffn+1))y
by 2.1. Since (1 - ve(ffn+1)) is a unit by 2.5 we obtain
yn+1 = uny(ff1)y(ff2) . .y.(ffn)(1 - ve(ffn+1))-1(y(ffn+1) - e(ffn+1)) +*
* y2rn(y):
Noting that any Euler class can be expressed as a polynomial in e1; e2; : :;:er*
* by 2.4,_this
gives an equation of the required form. *
* |__|
*
* __
*
*|__|
References
[1]R.R.Bruner and J.P.C.Greenlees "Varieties and local cohomology for the conn*
*ective K-theory of finite
groups." (In preparation)
[2]M.M.Cole, J.P.C.Greenlees and I.Kriz "Equivariant formal group laws." (Subm*
*itted for publication).
[3]J.P.C.Greenlees "Equivariant forms of connective K-theory." Topology (to ap*
*pear)
[4]J.P.C.Greenlees "The coefficient ring of equivariant bordism is the univers*
*al ring for equivariant formal
group laws." Preprint (1998)
[5]H. Matsumura "Commutative ring theory" Cambridge University Press (1986)
10 J.P.C.GREENLEES
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk