EQUIVARIANT FORMAL GROUP LAWS.
MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Abstract. Motivated by complex oriented theories we define A-equivariant *
*formal group
laws for any abelian compact Lie group A, show there is a representing ri*
*ng for them, and
begin the investigation of it. We examine a number of topological cases, *
*including K-theory
in some detail.
JPCG thanks Hal Sadofsky for many useful conversations, the University of Chi*
*cago and
the University of Oregon for their hospitality during May 1996, when this work *
*was begun,
and the Nuffield Foundation for its support.
1. Introduction
The purpose of this article is to formulate and study the notion of equivaria*
*nt formal
group law appropriate for understanding orientable complex stable equivariant c*
*ohomology
theories E*A(.), at least when the compact Lie group A of equivariance is abeli*
*an.
The aim is to understand cohomology theories which behave in a simple way on *
*complex
vector bundles, and hence give rise to a good theory of characteristic classes.*
* In particular we
want to understand tom Dieck's homotopical equivariant complex bordism [7]. Bor*
*dism is
universal amongst such theories in a topological sense [16, 5], and non-equivar*
*iantly Quillen
showed its coefficient ring is the Lazard ring, which is universal in the algeb*
*raic sense. This
possibility is more seductive equivariantly, since so little is known about the*
* coefficient ring
of equivariant bordism. In [10] an equivariant version of Quillen's theorem is*
* proved, but
without making the structure of the equivariant Lazard ring explicit.
Since we work with an abelian group A, all complex representations are one di*
*mensional,
and it is enough to consider line bundles. Line bundles are classified by the A*
*-space CP 1
of lines in a complete A-universe, so we study E*A(CP 1), endowed with all the *
*structure
inherited from CP 1. Since CP 1 is a space, E*A(CP 1) is a ring, the tensor pr*
*oduct of
line bundles makes CP 1 into an abelian group object in the homotopy category, *
*so (when
we have a K"unneth isomorphism) E*A(CP 1) is a cogroup object, and finally, sin*
*ce one can
tensor a line bundle with any one dimensional representation, CP 1 has an actio*
*n of the dual
group A*, giving rise to a coaction of A* on E*A(CP 1).
To ensure the additive structure is reasonable, we restrict attention to orie*
*nted complex
stable theories. Cole has shown [3] that this condition ensures that E*A(CP (V*
* )) is well
behaved. In the non-equivariant case the ring structure is then necessarily tha*
*t of a truncated
polynomial ring, but equivariantly this is too much to ask. However the Splitti*
*ng Theorem
[3, 4] shows that for any complete flag V 1 V 2 V 3 . . .V n= V in V the corres*
*ponding
filtration of E*A(CP (V )) splits additively, as a direct sum of copies of E*A,*
* with basis elements
1; y(V 1); y(V 2); : :;:y(V n-1) constructed from the orientation, using the A**
*-action and the
ring structure.
Thus an A-equivariant formal group law over a commutative ring k is a topolog*
*ical Hopf k-
algebra R together with a coaction of A* and special elements y(ff) (related by*
* the coaction as
1
2 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
the notation suggests) so that R=(y(ff)) = k for all ff, and so that R is compl*
*ete with respect
to the ideal (ffy(ff)). This is a considerable refinement of the essentially no*
*n-equivariant,
classical notion of an Okonek equivariant formal group law [17]. In effect our*
* equivariant
formal group law is a very special `sub-cogroup of finite index' in an Okonek f*
*ormal group
law [17].
Returning to topology, any complete flag F in the universe gives an additive *
*topological
basis for E*A(CP 1), and we may express the ring structure, the cogroup structu*
*re, and the
A* action in terms of this basis using structure constants in E*A. Writing down*
* the formal
properties in terms of the basis we obtain a list of conditions which must be s*
*atisfied by
the structure constants, and this gives the notion of an (A; F )-equivariant fo*
*rmal group law,
essentially that given in [3] for cyclic groups. This description shows there i*
*s a representing
ring LA(F ) for such formal group laws. This process has an algebraic counterpa*
*rt, so that
an A-equivariant formal group law with a flag F gives an (A; F )-formal group l*
*aw, and the
latter give a convenient way of calculating with A-formal group laws. Reversing*
* the process,
we conclude that the notion of an (A; F )-formal group law is essentially indep*
*endent of the
flag F , and that the ring LA(F ) represents A-equivariant formal group laws.
There are a number of other points of view possible. Firstly, one can give a *
*coordinate free
description of the structure, which is done in geometric language in [9]. Ther*
*e is another
coordinate free description, which amounts to viewing an equivariant formal gro*
*up as a
certain type of deformation of a non-equivariant one: the structure studied by*
* Hopkins-
Kuhn-Ravenel [12] and Greenlees-Strickland [11] is equivalent to an equivariant*
* formal group
law over a suitably complete ring k.
The rest of the paper is layed out as follows. After summarizing the basic de*
*finitions and
the fundamental splitting theorem of [3, 4], our first task is to describe a go*
*od supply of
examples (Sections 6 to 10). We are then ready in Section 11 to give the defin*
*ition of an
equivariant formal group law. To make calculations we need to introduce the fra*
*mework of a
complete flag F giving the definition of an (A; F )-equivariant formal group la*
*w in Section 12.
After showing that an A-equivariant formal group law together with a flag is eq*
*uivalent to
an (A; F )-formal group law, it follows that the subsequent study of (A; F )-fo*
*rmal group laws
provides tools for calculation with equivariant formal group laws. In particula*
*r we show that
there is a representing ring for equivariant formal group laws. In appendices w*
*e illustrate
the theory by considering the special cases of additive and multiplicative form*
*al group laws,
and make formulae explicit for the group of order 2.
Contents
1. Introduction *
* 1
2. The classifying space for line bundles *
* 3
3. Complex stability and Euler classes. *
* 5
4. Orientations and the cohomology of CP 1. *
* 6
5. Orientations and Euler classes. *
* 7
6. Equivariant K-theory. *
* 8
7. The equivariant approach. *
* 9
8. Borel cohomology. *
* 10
9. Cohomology of the fixed point subspace. *
* 10
10. Bredon cohomology. *
*12
EQUIVARIANT FORMAL GROUP LAWS 3
11. The definition. *
* 12
12. The definition relative to a flag. *
* 13
13. Comparison 15
14. The representing ring *
* 16
15. Some relations in the representing ring: cosy warmup. *
* 18
16. Some relations in the representing ring: the general case *
* 20
Appendix A. The additive and multiplicative group laws. *
* 23
Appendix B. The coinverse. *
* 24
Appendix C. The group of order 2 *
* 26
References *
*29
Here is a summary of our notational conventions: all are introduced in more d*
*etail in the
text.
A is an abelian compact Lie group.
A* its dual group Hom (A; S1)
ff; fi; fl; : :a:re typical one dimensional complex representations.
ffl the trivial one dimensional representation.
V a complex representation.
T the circle group.
z the natural representation of T.
U a complete A universe.
F a complete flag in U.
V 1; V 2; V 3; : :t:he terms in F .
ff1; ff2; ff3; : :t:he subquotients of F .
lffleft multiplication by ff 2 A*.
(V ) the Thom class giving a complex stable structure.
O(V ) the pullback of (V ) along S0 -! SV .
e(V ) the strict Euler class defined by a complex orientation.
2.The classifying space for line bundles
For each complex representation V we may form the A-space CP (V ) of complex*
* lines
in V . It is sometimes useful to consider the representation V z of A x T, an*
*d then
CP (V ) = S(V z)=T.
Indeed CP defines a functor from the category of vector spaces and injective *
*maps to the
category of topological spaces and injective maps. Thus, in particular, if W V*
* we have
a pair (CP (V ); CP (W )). For example, if W is one dimensional, CP (W ) is a p*
*oint, so that
a one dimensional subspace of V specifies a basepoint of CP (V ): this is signi*
*ficant because
basepoints may lie in different components of the fixed point set. At the other*
* extreme, if ff
is one dimensional one may verify there is a cofibre sequence
2.1.
-1
CP (V ) -! CP (V ff) -! SV ff :
4 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
__
We remark that, as an A-space, V_ is isomorphic to V , so that the cofibre may *
*also be
described as the Thom space of V ff, as is often done when discussing Thom isom*
*orphisms.
The A-invariant complex lines are exactly the subrepresentations of V , so it*
* is easy to see
that
2.2.
a
CP (V )A = CP (Vff)
ff
where Vff= Hom A(ff; V ) is the ff-isotypical part of V . Note that if A is fin*
*ite and V = kCA
is a multiple of the regular representation, we have an isomorphism
~= A
A* x CP (kffl) -! CP (kCA)
given by (ff; W ) -! ff W . L L
For convenience we take U = k0 ff2A*ff as our complete A-universe and we*
* define
CP 1 = CP (U), with its topology as a colimitSof its subspaces CP (V ) with V f*
*inite dimen-
sional. It is also convenient to let CPW1 = k0 CP (kW ), so that CPf1flis an *
*A-fixed infinite
complex projective space.
*
* __
Lemma 2.3. The A-space CP 1 classifies line bundles. *
* |__|
The tensor product of line bundles is commutative and associative up to coher*
*ent isomor-
phism, and has ffl as a unit, and we shall constantly use the represented count*
*erpart.
Corollary 2.4. The A-space CP 1 is an abelian group object up to homotopy, and *
*the in-_
clusion of fixed points is a group homomorphism. *
* |__|
We have seen
a
(CP 1)A = CP (U)A = CP (Uff) ~=A* x CPf1fl:
ff
It is important to make explicit the map
i : A* x CPf1fl-! CP 1
(ff; W )7-! ff W:
This map of abelian groups is absolutely fundamental to our analysis.
Note also that since CPf1fis connected, there is a unique homotopy class A* -*
*! (CP 1)A
splitting the natural augmentation (CP 1)A -! A*, and it is a group homomorphis*
*m. In
particular A* acts on CP 1 throughLA-maps, by ff.L = ffL. To avoid confusion, w*
*e make it
explicit. Any vector v in U = ffUffcan be resolved into its components vff2 U*
*ff, and under
~=
the isomorphism ff : Uffl-!Uffit is best to view v as a function v : A* -! Uffl*
*. The action
of A* on U is then given by [ff . v](fi) = v(fiff-1), and the action of A by [a*
* . v](fi) = a(v(fi)).
It is easy to verify these commute. When A is finite, the action also restricts*
* to an action
on kCA for each k.
EQUIVARIANT FORMAL GROUP LAWS 5
3. Complex stability and Euler classes.
We have already seen that the fixed point spaces of interesting A-spaces are *
*disconnected,
so it is rare for there to be a preferred basepoint. This is one reason it is c*
*onvenient to work
throughout in the unbased context.
A genuine equivariant cohomology theory E*A(.) is an exact contravariant func*
*tor on A-
spaces, which admits an RO(G)-graded extension so that we have coherent suspens*
*ion iso-
morphisms
E"VA+n(SV ^ X) ~="EnA(X)
for all real representations V . Amongst these, the most familiar ones are tho*
*se with a
stronger stability property
"E|VA|+n(SV ^ X) ~="EnA(X)
when V is a complex representation, where |V | denotes the space V with trivi*
*al action.
This is very convenient: for most purposes we only need to look at the theory i*
*n integer
gradings. Following tom Dieck we call these theories complex stable. As examp*
*les, we
have the cohomology theory of the Borel construction, defined in terms of a non*
*equivariant
cohomology theory by X 7-! E*(EA xA X). A Serre spectral sequence argument sho*
*ws
this is complex stable, since A acts trivially on H|V(|SV ) when V is complex.*
* The other
examples we discuss below include complex equivariant K-theory.
Now let us suppose given a multiplicative, complex stable equivariant cohomol*
*ogy theory
E*A(.). For any complex representation V , complex stability provides an element
(V ) 2 "E|VA|(SV )
corresponding to the unit in E0A, and the E*A-module "E*(SV ) is free of rank 1*
* on this gener-
ator. All complex stability isomorphisms are given by multiplication by (V ), a*
*nd we have
(V W ) = (V )(W ). We then define the Euler class O(V ) = e*V((V )) 2 E|VA|, *
*where
eV : S0 -! SV is the inclusion. Thus we have O(V W ) = O(V )O(W ).
Note that for any based A-space X this gives
"EA*(X ^ S1V ) = "EA*(X)[1=O(V )]:
_ S
This leads to the localization theorem. In fact, if we take S1ae= V A=0SV , t*
*he inclusion
_ae' 1_ae
XA ^ S1 -! X ^ S
is an equivalence by obstruction theory. It follows that E"A*(XA ) -! E"A*(X) *
*becomes an
isomorphism if we invert all Euler classes O(V ) with V A = 0. By duality we d*
*educe the
localization theorem we need [6]: the finiteness assumption in the statement is*
* essential.
Lemma 3.1. If X is a based finite A-space then
E"*A(XA ) - E"*A(X)
*
* __
becomes an isomorphism if we invert all Euler classes O(V ) with V A= 0. *
* |__|
6 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
4. Orientations and the cohomology of CP 1.
Notice that when ffl ffl ff U we have
-1 1
*ffl= CP (ffl) CP (ffl ff) = Sff CP (U) = CP :
Definition 4.1. [3] We say that x(ffl) 2 E*A(CP 1; CP (ffl)) is an orientation *
*if for all one
dimensional representations ff 2 A*,
-1
resUfflffx(ffl) 2 E*A(CP (ffl ff); CP (ffl)) = "E*A(Sff )
*
* __
is a generator. *
* |__|
*
* -1
Remark 4.2. We do not require that x restricts to the standard generator (ff-*
*1) 2 "E*A(Sff ).
It is perhaps worth introducing the notation
resUfflffx(ffl) = uff-1(ff-1)
*
* __
for the unit concerned. *
* |__|
We may generate many other elements from an orientation. Firstly, pulling ba*
*ck along
the action
ff-1 : (CP 1; CP (ff)) -! (CP 1; CP (ffl))
we have x(ff) 2 E*A(CP 1; CP (ff)). To avoid confusion later, we write lff= (ff*
*-1)*; thus, in
particular, x(ff) = lffx(ffl). Taking external direct products, if V = ff1 . .*
* .ffn we obtain
x(V ) = x(ff1) * . .*.x(ffn) 2 E*A(CP 1; CP (V ))
Here the product * is defined by pulling back the external cup product along th*
*e map
__
: (CP (V W Z); CP (V W )) -! (CP (V Z); CP (V )) x (CP (W Z); CP (W ))
defined by (v : w : z) 7-! ((v : z); (w : z)). Forgetting the subspace, x(V ) *
*defines an
element y(V ) 2 E*A(CP 1) which restricts to zero on CP (V ). It turns out tha*
*t the pair
(CP (V W ); CP (V )) defines a short exact sequence
0 - E*A(CP (V )) - E*A(CP (V W )) - E*A(CP (V W ); CP (V )) - 0:
In particular y(ffl) is the image of x(ffl), and, since the restriction turns o*
*ut to be injective,
either one determines the other. It is clear that y(0) = 1, that y(V W ) = y(V*
* )y(W ), and
that (ff-1)*y(V ) = y(V ff). Thus all the elements y(V ) can be obtained from *
*y(ffl) using
the action of A* and the multiplication.
To obtain a topological additive basis of E*A(CP 1) we choose a complete flag
F = (V 0 V 1 V 2 . .).;
S
so that dimC(V i) = i, and i0V i= U . Associated to any such complete flag F *
*we have
the sequence ff1; ff2; : :o:f subquotients ffi = V i=V i-1, so that V n ~=ff1 *
*. . .ffn, and
y(V n) = y(ff1)y(ff2) . .y.(ffn). The basis only depends on the isomorphism cl*
*asses of the
sequence V n,*so the important structure is represented by the path ff1; ff2; f*
*f3; : :i:n the first
orthant, NA . We also allow the use of flags in other complete universes, since*
* none of the
relevant structure depends on our identification of universes. We sometimes spe*
*ak loosely
EQUIVARIANT FORMAL GROUP LAWS 7
as if the path were equivalent to the flag. The condition that F is a complete *
*flag is simply
that for each k, the path eventually enters the monoid-ideal generated by V = *
*(Vff)fffor
each finiteQdimensional representation V . We use the notation k{{yi|i 2 I}} to*
* denote the
product i2Ik where yiis the characteristic function of the ith factor.
The Splitting Theorem of [3, 4] is the appropriate substitute for the collaps*
*e of the Atiyah-
Hirzebruch spectral sequence in the non-equivariant case.
Theorem 4.3. (Cole [3, 4]) A complete flag F = (V 0 V 1 V 2. .).specifies a b*
*asis of
E*A(CP 1) as follows:
E*A(CP 1) = E*A{{y(V 0) = 1; y(V 1); y(V 2); : :}:}:
Similar results hold for products of copies of CP 1, in the sense that the K"un*
*neth theorem
holds with completed tensor products.
Proof: We argue by induction that E ^ (CP 1; CP (0)) splits as a wedge, and that
F ((CP 1; CP (0)); E) is the corresponding product. The cofibre sequence
-1 n+1 n 1 n+1 1 *
* n
(Sffn+1; *) = (CP (V ); CP (V )) -! (CP ; CP (V )) -! (CP ; CP (*
*V ))
gives a split exact sequence in homology or cohomology: the splitting is given *
*by x(V n+1).
This is equally true if CP 1 is replaced by CP (U) whenever V n+1 U. The result*
* follows_
by passage to limits. *
* |__|
Evidently this theorem gives a means for expressing the cup product, the map *
*induced by
tensor product, and the action induced by the action of A*, using collections o*
*f elements of
the coefficient ring E*A. This lets us describe the coarsest features of the ma*
*ps by identifying
the leading terms. We consider higher terms in Section 12 below.
5. Orientations and Euler classes.
The main point is that suitable restrictions of the orientation class are uni*
*t multiples of
the Euler classes.
Lemma 5.1. The restriction of y(ffl) to the point *ff= CP (ff) is a unit mult*
*iple of the Euler
class of ff:
resUff(y(ffl)) = uff-1O(ff-1)
where the unit uff-1is the one occurring in the definition of an orientation. I*
*n general
resUff(y(fi)) = uff-1fiO(ff-1fi):
Proof: Consider the diagram
(CP (ff Offl);OCP (ffl))//_(CPO1;OCP (ffl))
| |
| |
| |
-1 1
Sff ~=CPO(ffO ffl) CPOO
| |
| |
| = |
CP (ff)_____________//CP (ff)
*
* __
*
*|__|
8 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
It is convenient to introduce the notation e(fi) := ufiO(fi) for the elements*
*, and we re-
fer to them as strict Euler classes associated to a complex oriented theory. F*
*or a general
representation we define strict Euler classes so that e(V W ) = e(V )e(W ).
We can now read off the leading term in the expression for y(fi) in any basis.
Corollary 5.2. When working with a basis corresponding to a complete flag F wit*
*h first __
term ff, the coefficient of y(V 0) = 1 in the expression for y(fi) in the F -ba*
*sis is e(ff-1fi). |__|
Similarly, we may understand the coefficient of y(V j) in the expansion of y(*
*V i)y(V j).
Lemma 5.3. We have
y(V )y(V j) = e(ff-1j+1V )y(V j) + higher terms:
Proof: By induction it suffices to prove the special case when V = fi is one di*
*mensional. In __
this case y(fi) = e(ff-1j+1) mod y(ffj+1), and the result follows. *
* |__|
Of course if V contains ffj+1the displayed coefficient is zero. Later we cons*
*ider the higher
terms in some detail.
Our next task is to provide a good supply of examples (Sections 6 to 10). We *
*will then
give the formal definition and give tools for calculation.
6. Equivariant K-theory.
This is one of the few cases where calculations are easy. In fact, Bott perio*
*dicity shows K-
theory is complex stable, and that we may work entirely in degree 0. Thus the c*
*oefficient ring
KA = R(A) is the complex representation ring. There is a severe danger of confu*
*sion here:
the coefficient ring R(A) acts on KA(X) for any X. On the other hand, when X = *
*CP 1
we have an action of A* via ring homomorphisms: this is quite different from th*
*e action of
A* R(A). To minimize confusion, recall that we write lffx for the image of the*
* cohomology
class under the action of ff 2 A* so that lff= (ff-1)*.
This is certainly a case where an equivariant approach is appropriate, since
KAxT(S(V z)) = KA(CP (V )):
The point here is that the based cofibre sequence S(V )+ -! D(V )+ -! SV gives *
*rise to
Gysin sequence. By Bott periodicity, it takes the form
O(V z) -1
0 - KAxT(S(V z)) - R(A)[z; z-1] - R(A)[z; z ] - . .:.
Furthermore, if V = ff1 . . .ffn, then
O(V z) = (1 - ff1z)(1 - ff2z) . .(.1 - ffnz);
which is a regular element. Thus
KA(CP (V )) = KAxT(S(V z)) = R(A)[z; z-1]=O(V z):
Next, note that z is already invertible in R(A)[z]=O(V z); indeed
1 - O(V z) = z . (V + higher terms):
EQUIVARIANT FORMAL GROUP LAWS 9
Either by the completion theorem, or simply by passage to inverse limits, we *
*see that
KA(CPV1) = R(A)[z]^O(V z):
Now observe that y = 1 - z is an orientation; K-theory is unusual in that this *
*has finite
degree in z. To verify it is indeed an orientation, we note that 1 - z makes s*
*ense as an
element of KA(CP (V )) for any V , and that 1 - z visibly generates the kernel *
*of
KA(CP (ffl ff)) = R(A)[z]=(1 - z)(1 - ffz) -! R(A)[z]=(1 - z) = KA(CP (f*
*fl)):
The element z, regarded as an element of KA(CP (V )), is the canonical line b*
*undle over
CP (V ), so it is easy to identify the A* action: lffz = ffz. Since the action *
*is through ring
homomorphisms, y(ff) = lff(1 - z) = 1 - ffz.
Next, we specializeQto case A is finite and V is the regular representation,*
* and we let
= O(CA z) = ff(1 - zff). There is a straightforward and standard way to ada*
*pt the
discussion to an arbitrary abelian compact Lie group A. The inclusion i : A* x *
*CPf1fl-!
CP 1 induces a map
Y
i* : R(A)[z]^ -! R(A)[z]^(1-ffz):
ff
We have chosen coordinates so that the ffth component is induced by completing *
*the identity
map of R(A)[z] with respect to in the domain and (1-ffz) in the codomain, as i*
*s legitimate
since (1-ffz) divides . Note in particular that i* is injective, since the same*
* primes contain
the product and the intersection of the ideals (1 - ffz). It is also not hard t*
*o see that if we
invert all the Euler classes O(ff) = 1 - ff that the ideals become coprime. Thu*
*s if we invert
the Euler classes before completion, we obtain an isomorphism by the Chinese Re*
*mainder
Theorem.
Working modulo all Euler classes, we see that 1 - ffz = 1 - z, so that i* is *
*the diag-
onal inclusion. If we complete at the augmentation ideal the components of i* *
*are each
isomorphisms.
7. The equivariant approach.
This is a class of examples generalizing equivariant K-theory, and certainly *
*including
equivariant complex bordism and related theories.
It often happens that there is an A x T-equivariant form of the cohomology th*
*eory, so
that E*AxT(X) = E*A(X=T) when X is T-free. In this case we have
E*A(CP (V )) = E*AxT(S(V z));
just as for K-theory. We also have a completion theorem in this context, stating
E*A(CPV1) = (EAxT)^O(V z):
Because we are only considering the cohomology of the single infinite sphere S(*
*1(V z)),
the completion theorem only requires complex stability, and not highly structur*
*ed ring and
module technology. The most naive form of the statement would involve local ho*
*mology,
but the calculation 4.3 shows this reduces to the classical completion.
We still find that if V = ff1 . . .ffn, then
O(V z) = O(ff1 z)O(ff2 z) . .O.(ffn z):
10 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Thus, (with the same adaptions to the infinite case as for K-theory) the inclus*
*ion i : A* x
CPf1fl-! CP 1 induces a map
Y
i* : (E*AxT)^ -! (E*AxT)^O(ffz);
ff
Q
where = ffO(ffz). The map i* is again injective, and the analogue in which E*
*uler classes
O(V ) with V A= 0 are inverted before completion, is an isomorphism.
We have an action of A* on CP 1, and hence on the completion of E*AxT. It can*
* be rather
useful to know the action exists before completion. In fact the group A* acts o*
*n the group
A x T by the formula
ff . (a; z) = (a; ff(a)z):
This induces an action of A* on the cohomology theory E*AxT(.). Since composit*
*ion with
ff : U z -! U z takes a subspace V z of U to V ff z, we see that the action*
* on
E*AxT(S(U z)) is the same as that on E*A(CP 1) discussed earlier.
8.Borel cohomology.
If E is any non-equivariant complex oriented theory we may consider the assoc*
*iated Borel
cohomology:
b(E)*A(X) = E*(EA xA X):
Since A acts trivially on H*(SV ) if V is a complex representation, a Serre spe*
*ctral sequence
argument shows that this is a complex stable theory, and a complex stable struc*
*ture is
*
provided by choosing Thom classes. Any class in ]b(E)A(CP 1) restricting to a g*
*enerator of
]b(E)*(S2) is automatically a complex orientation.
A
These examples are special because b(E)*Ais already complete at the ideal I g*
*enerated by
the Euler classes. This means as expected that the associated formal group law *
*is essentially
the nonequivariant formal group law with its base extended.
We may consider ordinary Borel cohomology with integer coefficients. From the*
* equivari-
ant point of view of Section 7 or otherwise, we see that b(H)*A(CP 1) = H*(A) *
*H*(BS1).
Choosing an orientation c we find this is H*(A)[c]. Furthermore H2(BA) ~=A*, an*
*d we have
e(ff) = ff. It turns out that this gives an additive formal group law. We shall*
* see that the
fact that Euler classes are Z-torsion for finite A is a necessary consequence.
9. Cohomology of the fixed point subspace.
The other extreme case is given by cohomology theories in which all Euler cla*
*sses are
invertible. Thus for this section we suppose E*A(.) is a complex stable theory *
*in which all
Euler classes O(ff) (ff 6= ffl) are invertible. We shall see that such a theory*
* is always complex
orientable. Theories of this type give Okonek formal groups [17].
Since Euler classes are invertible, we have an equivalence
_ae
E = E ^ S0 -'! E ^ S1 ;
_ S
where S1ae= V A=0SV as before. It follows that for any A-space X, the inclusi*
*on XA -! X
is a cohomology isomorphism (since the cohomology of any non-fixed cell is zero*
*). This
EQUIVARIANT FORMAL GROUP LAWS 11
applies to X = CP 1, with (CP 1)A = A* x CPf1fl, where the multiplication comes*
* from the
group structure in A* and CPf1fl. Thus
Y *
E*A(CP 1) ~= E*A(CPf1fl) = E*A(CPf1fl)A :
ff
We*should explain the notation: if k isQa cogroup in the category of rings, we *
*use the notation
kA for the object which is a product ffk as a ring, but where the coproduct *
*combines that
of k with the group operation on A*. This should not be confused with the group*
* ring k[A].
More explicitly, if we choose a coordinate yffl
Y
E*A(CP 1) = E*A[[yff]]
ff
and the coproduct is determined by the group structure of A* and its effect on *
*the generator
yffl.
More generally, 2.2 shows that
Y
E*A(CP (V )) = E*A[yff]=(y|Vff|ff);
ff
so it is easy to understand orientations, x = x(ffl) = (xff)ff. An element (fff*
*(yff)) 2 E*A(CP 1)
gives an element of E*A(CP 1; CP (ffl)) if fffl(yffl) has zero constant term. T*
*o give a generator
of E*A(CP (ffl ffl); CP (ffl)) the coefficient of yfflmust be a unit. Finally,*
* to give a generator of
E*A(CP (ffl ff); CP (ffl)) when ff 6= ffl, the constant term in fff(yff) must *
*be a unit. Thus, if
Euler classes are given, an orientation is specified by (i) a classical orienta*
*tion of the classical
formal group law E*A(CPf1fl) and (ii) for each ff 6= ffl, an arbitrary power se*
*ries gff(yff) so that
fff(yff) = e(ff) + yffgff(yff).
The simplest orientation is x = (yffl; (O(ff))ff6=ffl). We use this orientati*
*on unless otherwise
stated, and thus we have
y(fi) = (yfi; (O(fiff-1))ff6=fi);
(i.e. the fi'th component of y(fi) is yfi, and the ff'th component is O(fiff-1)*
* if ff 6= fi).
Example 9.1. It is perhaps worth a short calculation with A of order 2 to kill*
* certain
preconceptions. Let j denote the nontrivial one dimensional representation, and*
* O = O(j).
We thus have
E*A(CP 1) = E*A[[yffl]] x E*A[[yj]];
with E*Aacting diagonally on the factors, and A* exchanging the factors. Note t*
*hat
y(ffl) = (yffl; O)
and
y(j) = (O; yj):
We can thus write y(j) in terms of the ffl; j; ffl; j; ffl; : :b:asis
__
y(j) = O . 1 + -1 . y(ffl) + O-1 . y(ffl)y(j): |__|
12 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
10. Bredon cohomology.
Guided by the non-equivariant case we might have expected to find additive fo*
*rmal groups
associated to ordinary cohomology. We show here that this is misguided: readers*
* without
prior prejudice should omit this section.
First we show that ordinary cohomology is hardly ever complex stable. Recall*
* that an
ordinary cohomology is one satisfying the dimension axiom, and it is thus deter*
*mined by its
values M(B) := E0A(A=B): From the suspension isomorphisms, the functor A=B 7-! *
*M(B)
is an additive functor on the stable orbit category: a Mackey functor. Bredon h*
*as shown how
to construct the ordinary cohomology theory H*A(.; M) associated to M [2], and *
*it extends
to an RO(G)-graded theory if and only if M is a Mackey functor [13].
We shall see that Bredon cohomology is very rarely orientable. However, even *
*when the
theory is orientable, it is obvious that all Euler classes of non-zero represen*
*tations are trivial,
since they lie in the zero group.
Let us consider the special case when A is cyclic. First, consider the case M*
*(B) = B for an
A-module . If V is A-free then H*A(S(V ); M) = H*(S(V )=A; ). Choosing V to be *
*of large
dimension we see this is the group cohomology H*(A; ) in a range of degrees. No*
*w consider
the cofibre sequence S(V )+ -! S0 -! SV , and deduce that H*A(SV ; ) = H*(A; ) *
*from
degree 2 up to to the dimension of V . Thus complex stability implies H*(A; ) i*
*s zero in
positive degrees. The main examples occur when the order of A is invertible in *
* and when
is projective over ZA. Both these cases are relatively dull.
Now suppose that A is an arbitrary finite abelian group. If V is a non-trivia*
*l one dimen-
sional complex representation of A it has a kernel K so that A=K is cyclic, gen*
*erated by c
say. We then have two cofibre sequences
A=K+ 1-c-!A=K+ -! S(V )+ and S(V )+ -! S0 -! SV
The first gives an exact sequence
0 - H1A(S(V ); M) - M(K) 1-c-M(K) - H0(S(V ); M) - 0;
and the second shows H1A(S(V ); M) ~=H2A(SV ; M) and that there is an exact seq*
*uence
0 - H1A(SV ; M) - H0A(S(V ); M) - M(A) - H"0A(SV ; M) - 0
Suppose that H*A(.; M) is complex stable, so that H*A(SV ) = H*A(S2). There is *
*thus an exact
sequence
resAK
0 - M(A) - M(K) 1-c-M(K) - M(A) - 0;
showing that M(A) = M(K)A=K and that restriction gives an isomorphism M(A) ~=M(*
*K)A=K.
Applying a similar argument to subgroups, we see that M is the Mackey functor a*
*ssociated
to an A-module by M(B) = B. Furthermore, the discussion of the case above sho*
*ws
that it too is very restricted.
11.The definition.
We are now ready to give the definition of an A-equivariant formal group law.*
* As usual, the
word law refers to the fact that the definition is relative to a particular cho*
*ice of coordinate
specified by the orientation.
The definition is motivated by the fact that CP 1 is an abelian group`object *
*(so that
E*A(CP 1) has a product and a coproduct) and the inclusion of A* = ffCP (ff) *
*-! CP 1
EQUIVARIANT FORMAL GROUP LAWS 13
is a group homomorphism. There is also an axiom encoding some of the special pr*
*operties
of the orientation element.
Definition 11.1. If A is a finite abelian group, an A-equivariant formal group *
*law over a
commutative ring k is a k-algebra R together with an ideal at which it is comp*
*lete together
with
Afgl1: a comultiplication
: R -! R ^R
which is a map of k-algebras, cocommutative, coassociative and counital,
Afgl2: a continuous map *
: R -! kA
of k-algebras compatible with the coproduct
Afgl3: A system of elements y(ff) 2 R for ff 2 A* so that
i.y(ff) is regular for each ff
ii.(ff) induces an isomorphism R=(y(ff)) ~=k; for each ff
iii. = (ffy(ff)) and
iv.y(ff) = ((ff) id)(y(ffl)) for ff 2 A*
If A is a general compact abelian group the ideal is replaced by the system of*
* all finite
product ideals (iy(ffi)).
Remark 11.2. (i) The element y(ffl) is called the orientation of the formal g*
*roup law. More
generally, any element y0(ffl) defines elements y0(ff) by Afgl3 (iv), and if th*
*is system of elements
satisfies Condition 3 we say y0(ffl) is an orientation of R.
(ii) In view of both the topological motivation and the terminology, the reader*
* may have
expected a coinverse as part of the structure. We shall show in Appendix B tha*
*t just as
for graded connected bialgebras, the existence of a unique coinverse is automat*
*ic. Thus
Condition Afgl1 may be replaced by the requirement that R is a topological Hopf*
* k-algebra,
and Condition Afgl2 by the requirement that is a map of topological k-algebras.
By contrast with the classical case, the underlying ring R is not always a po*
*wer series ring.
We shall show in Section 13 that a choice of a complete flag in our universe gi*
*ves rise to an
additive basis of R. Once we have such a basis, we may make calculations, and w*
*e develop
the necessary machinery for this. The abstract fruit of this is that there is a*
* universal ring
for equivariant formal group laws.
Example 11.3. (i) If E*A(.) is a complex oriented theory, we may obtain an A-e*
*quivariant
formal group law by taking k = E*A, R = E*A(CP 1) and y(ff) to be constructed f*
*rom the
orientation.
(ii) A 1-equivariant formal group law is a formal group law in the classical se*
*nse.
(iii) Any formal group law over k gives an A-equivariant formal group*law over *
*k by taking
the map to be the composite of the counit and the diagonal k -! kA .
12. The definition relative to a flag.
The definition is motivated by the topological case, in which we may choose a*
* particular
flag and express all available structure with respect to the resulting basis. T*
*his idea comes
from the first author's thesis [3], where it is applied to the case of a cyclic*
* group, and the
obvious type of periodic complete flag. We find it essential to have the flexib*
*ility to discuss
14 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
bases arising from different flags, and have used completely different notati*
*on from [3] to
avoid confusion, since our indexing conventions are different. In particular*
* we view y(V i)
as having a superscript, and use the summation convention to determine the po*
*sition of
decorations in other notation.
Thus the product structure takes the form
X
y(V i)y(V j) = bi;jsy(V s);
s0
the action of A* takes the form
X
lffy(V i) = d(ff)isy(V s);
s0
and the tensor product is expressed by
X
t*y(V i) = fis;ty(V s) y(V t):
s;t0
This structure is required to be continuous so that we have the following.
Continuity conditions:
1. for fixed i; s the coefficients bi;jsare zero for j sufficiently large, *
*and similarly with i and
j exchanged,
2. for fixed ff; s the coefficients d(ff)isare zero for i sufficiently larg*
*e, and
3. for fixed s; t the coefficients fis;tare zero for i sufficiently large.
One may go on to identify leading terms, but first we list the formal prope*
*rties. The
continuity conditions are necessary in some of them to ensure the sums are fi*
*nite. We have
resisted the temptation to write them out explicitly, but the reader is encou*
*raged to do this
at least once, and we have listed the properties separately to assist with th*
*is.
(R) that the ring is
1.commutative,
2.associative, and
3.unital;
(A) that the action is
1.through ring homomorphisms,
2.associative, and
3.unital;
(T) that the tensor product is
1.through ring homomorphisms,
2.equivariant in the sense that t O lfffi= (lff^lfi) O t,
3.commutative,
4.associative and
5.unital.
There are also two normalization conditions
(Flag) y(ffj+1)y(V j) = y(V j+1).
(Ideal)For each i, the ideal (y(V i)) has additive topological basis y(V i); y(*
*V i+1); y(V i+2); : :.:
For the abstract definition, some more notation is convenient.
EQUIVARIANT FORMAL GROUP LAWS 15
Notation 12.1. Given a ring k and a complete flag F we write
k{{F }} = k{{1; y(V 1); y(V 2); . .}.}:
Definition 12.2. An (A; F )-formal group law over a commutative ring k is a top*
*ological
ring k{{F }} with a continuous coproduct and a continuous action of A* satisfyi*
*ng Conditions
(R), (A), (T), (Flag), and (Ideal). The topology is defined the ideals (y(V n)).
Example 12.3. (i) It is immediate from our motivation that any complex oriente*
*d coho-
mology theory E*A(.) gives rise to an (A; F )-equivariant formal group law over*
* E*Afor any
complete flag F .
(ii) If A is the trivial group, an equivariant formal group law is simply a cla*
*ssical one di-
mensional commutative formal group law, specified by a coproduct on the power s*
*eries ring
k[[y]].
Remark 12.4. As in the non-equivariant case one may also give a fully coordin*
*ate free
definition (i.e. a definition without specifying the orientation y(ffl)): this *
*gives the notion of
an equivariant formal group. We defer further discussion to [9].
13. Comparison
We shall show that an A-equivariant formal group law (as defined in 11.1) tog*
*ether with
a flag F is equivalent to an (A; F )-formal group law (as defined in 12.2). Thi*
*s gives a means
for calculation with A-equivariant formal group laws. It also proves that an (A*
*; F )-formal
group law is essentially independent of the flag F ; later we give formulae sho*
*wing how the
structure constants for an (A; F )-formal group law are related to those of an *
*(A; F 0)-formal
group law.
Lemma 13.1. An (A; F )-formal group law k{{F }} is an A-equivariant formal gr*
*oup law
Proof: For simplicity assume F begins with ffl. We take R = k{{F }}, and use th*
*e elements
y(ff) = lffy(ffl) as the notation suggests. Now define by Afgl3 (iii). Since F*
* is a complete
flag, k{{F }} is -complete. The comultiplication is explicitly present in our *
*formulation,
and required to have the properties stated in Afgl1.
First observe that k{{F }}=(y(ffl)) ~= k: it is clear that y(V 1); y(V 2); :*
* :l:ie in the ideal
(y(ffl)), and our Condition (Ideal) shows that they also span it. Condition Afg*
*l3 (ii) follows
for arbitrary ff by applying lff.
We show y(ffl) is regular. Note that 1; y(V 2=ffl); y(V 3=ffl); : :g:ives a *
*splitting of the map
y(ffl)
k{{F }} -! (y(ffl)), so that it suffices to show that k{{F }} is topologically*
* spanned as a k-
module by these elements. This can be done by adapting the proof of Lemma 13.2 *
*below.
Condition 3 (i) follows for arbitrary*ff by applying lff.
We define : k{{F }} -! kA by taking (r)(ff) to be the constant coefficient *
*in lff(r).
This is a continuous map of rings since lffis. Condition Afgl2 follows from the*
* equivariance of
the coproduct. Finally, Condition Afgl3 (iv) is given by taking the coefficient*
*s of 1 k{{F }} __
in (lff 1) O t*y(ffl) = t*lffy(ffl): it is y(ff) by the counital condition. *
* |__|
Lemma 13.2. If R is an A-formal group law we obtain an (A; F )-formal group l*
*aw by
defining y(V ) = y(ff1)y(ff2) . .y.(ffn) where V = ff1 ff2 . . .ffn.
16 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Proof: The main point is to show that if we choose a complete flag F , the elem*
*ents
1; y(V 1); y(V 2); : :f:orm a topological basis. Indeed, the elements y(V i) d*
*efine a contin-
uous function : k{{F }} -! R of k-modules, and we claim it is an isomorphism. *
* This
will immediately define the ring structure satisfying (R), and the coproduct st*
*ructure. The
action lffis given by the composite
* ssff1
R -! R ^R -! kA R -! {ff} R:
It is easy to verify that this is an action and that the coproduct is equivaria*
*nt.
Surjectivity of follows by approximating elements of R by a convergent serie*
*s in the
image of . Indeed, by Condition Afgl3 (ii), given s 2 R we may choose r0 2 k s*
*o that
s - r0 2 (y(ff1)), say s = r0+ y(ff1)s1. Similarly, s1 = r1+ y(ff2)s2 and so fo*
*rth. To establish
injectivity suppose (iriy(V i)) = 0, and suppose there is a first nonzero coeff*
*icient ri0.
Thus
(iriy(V i)) = y(V i0)(ii0riy(V i=V i0)):
Since y(V i0) is regular this means (iriy(V i=V i0)) = 0, but this is a contrad*
*iction since
reducing mod y(ffi0+1) recovers ri0.
Condition (Flag) is built into the definition, as are the continuity conditio*
*ns. To prove
Condition (Ideal) it suffices to show y(ff)y(V i) has zero coefficients of y(V *
*j) with j < i. If
not
y(ff)y(V i) = y(ff)y(V j)y(V i=V j) = y(V j)z
with V i=V j6= 0 and z = z0 + z1y(V j+1=V j) + . .w.ith z0 6= 0. Since y(V j) i*
*s regular by
Condition Afgl3 (i), y(ff)y(V i=V j) = z. Now reduce mod y(ffj+1), and by Condi*
*tion Afgl3_
(ii), we contradict the fact z0 6= 0. *
* |__|
This shows that k{{F }} does not depend in an essential way on the flag F , s*
*o we allow
ourselves to write k{{F }} = k{{U}}. We extend this notation to finite dimensio*
*nal subspaces
V U by taking k{{V }} = k{{U}}=(y(V )). Finally, we may extend it to infinite *
*dimensional
subspaces U0 U by taking k{{U0}} to be the inverse limit of the rings k{{V 0}}*
* with the
inverse limit topology. There is then an induced map resUU0: k{{U}} -! k{{U0}}.*
* We also
have a commutative square
13.3.
k{{U}} -! k{{U0ffi}}
lffi# # lffi
k{{U}} -! k{{U0}}:
14.The representing ring
It is immediate that an (A; F )-formal group law is uniquely specified by str*
*ucture constants
bi;js; d(ff)isand fis;tsatisfying the continuity conditions, and so that k{{F }*
*} satisfies Conditions
(R), (A), (T), (Flag), and (Ideal). We now want to form the representing ring *
*LA(F ) for
(A; F )-formal group laws, as the Z-algebra with generators bi;js; d(ff)isand f*
*is;tsubject to the
relations implied by these conditions. The remaining obstacle is that we must *
*show the
continuity conditions can be given by uniform formulae.
By way of motivation, consider the topological case. Various vanishing condit*
*ions result
from the fact that y(V ) generates the ideal of elements restricting to zero on*
* CP (V ).
EQUIVARIANT FORMAL GROUP LAWS 17
o (VRs) The product y(V 0)y(V 00) is zero on restriction to CP (V 0) or CP (*
*V 00)
o (VAs) lffy(V ) = y(V ff) vanishes on restriction to CP (V ff)
o (VTs) The coproduct t*y(V ) is zero on restriction to CP (W1) x CP (W2) if*
* CP (W1) x
CP (W2) maps into CP (V ) up to homotopy. A sufficient condition for this*
* is given
in terms of the dimensions of the fixed point sets by the CW-approximation*
* theorem.
We require that the inequality dim CP (W1)B + dimCP (W2)B dimCP (V )B hol*
*ds for
all subgroups B A. It is easy but unilluminating to express this in term*
*s of the
representations V; W1 and W2.
The reader should now make explicit the vanishing of structure constants thes*
*e condi-
tions imply in the topological case. For bi;jsand d(ff)isthe answer is precise*
*ly as in the
following proposition. For the tensor product it states that the coefficient f*
*is;tis zero if
dim CP (V s+1)B +dim CP (V t+1)B dimCP (V i)B holds for all subgroups B A. Si*
*nce any
fixed estimate suffices for our purpose we shall be satisfied with a cruder one.
Proposition 14.1. For any (A; F )-formal group over k we have the following exp*
*licit van-
ishing conditions:
o (VR)
bi;js= 0 ifs < i or s < j
o (VA)
d(ff)is= 0 ifV i ff V s+1
o (VT)
fis;t= 0 ifV i ff-11V s+1 V t+1:
Remark 14.2. Note that because the flag F exhausts U, the proposition gives e*
*xplicit forms
of the continuity conditions of Section 12.
Proof: Note that (VR) is equivalent to (Ideal): indeed, it is clear by (Flag) t*
*hat all elements
y(V j) with j i lie in the ideal (y(V i)). Both conditions are equivalent to r*
*equiring that
for all j, y(V i)y(V j) has zero coefficient of y(V k) for all k < i.
Next we note that (VA) follows. Indeed, if ffV i V s+1then lffy(V i) = y(ffV *
*i) is a multiple
of y(V s+1) by (Flag). We have just observed that (VR) shows that any such ele*
*ment has
zero coefficient of y(V k) with k < s + 1.
For (VT) we use the counit conditionPand the fact that t* is a ring map, toge*
*ther with
(Ideal). For an element u = p;qap;qy(V p) y(V q) we say that u vanishes up *
*to (s; t) if
ap;q= 0 if p s and q t. We shall find an i0 = i0(s; t) so that t*y(V i) vanis*
*hes up to (s; t)
whenever i i0: we do not attempt to find the best possible i0.
Note that by (Ideal), if u vanishes up to (s; t) then any product uv vanishes*
* up to (s; t).
Furthermore, we may ensure vanishing up to (s+1; t) by using suitable elements *
*v. Again by
(Ideal), the only terms in u which can contribute to non-vanishing up to (s + 1*
*; t) in uv are
as+1;jy(V s+1) y(V j) for j t. Thus multiplying by v1 = y(ffs+2) z or v2 = z*
* y(V t+1)
ensures vanishing up to (s + 1; t).
The counit condition states
t*(y(V )) = 1 y(V ) + y(V ) 1 mod (y(V 1) y(V 1)):
In particular t*(y(V t+1)) vanishes up to (0; t), proving (VT) for s = 0; we pr*
*ove the general
case by induction on s.
18 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
For the inductive step we apply equivariance (lff 1) O t* = t*O lffto the cou*
*nit condition,
and find
t*(y(ffV )) = 1 y(V ) + y(ffV ) 1 mod (y(ffV 1) y(V 1)):
Thus, taking ff = ffs+2ff-11and W = ffV we see that if u vanishes up to (s; t) *
*then t*(y(W_))u_
vanishes up to (s + 1; t) provided W contains ffs+2ff-11V t+1. *
* |__|
We may now proceed to form the representing ring.
Corollary 14.3. There is a representing ring LA(F ) for (A; F )-formal group la*
*ws, con-
structed as the Z-algebra with generators bi;js; d(ff)isand fis;tsubject to the*
* relations_implied
by (R), (A), (T), (Flag), (VR), (VA) and (VT). *
* |__|
Remark 14.4. Since 13.1 gives a canonical way to view an (A; F )-formal group*
* law as an
A-formal group law and 13.2 gives a canonical way to view this as an (A; F 0)-f*
*ormal group
~= *
* __
law, there is a canonical isomorphism LA(F 0) -! LA(F ). *
* |__|
Note that there is massive redundancy in the generating set: we shall see in *
*the next few
sections that the ring is generated by the elements d(ff)10and f1s;t.
Finally we comment briefly on the representing ring for objects analogous to *
*orientable
complex stable theories: A-equivariant formal group laws over k with specified *
*Euler classes.
The Euler classes are specified by a function O : A* -! k and we let LstrictA(F*
*; k; O) denote the
representing ring when the strict Euler classes are required to agree with O. N*
*ote that this
will not necessarily contain k: for example the condition O(ffl) = 0 is imposed*
*. Understanding
A-equivariant formal group laws where the strict Euler classes are unit multipl*
*es of specified
Euler classes (ie orientability and orientations of complex stable theories) se*
*ems more subtle.
15. Some relations in the representing ring: cosy warmup.
After Section 13 we have enough structure to work in the algebraic setup just*
* as if it arose
from topology. The purpose of this section and the next is to establish relati*
*ons amongst
the structure constants for equivariant formal group laws. Because the leading *
*terms are the
most significant, and in view of the confusing forest of superscripts and subsc*
*ripts, we have
decided to present discussion of the first few terms separately, as a motivatio*
*n for the general
case. The reader may also find it helpful to refer to Appendix C where some cal*
*culations
are done when A is of order 2. As a matter of logic the present section may be *
*omitted.
We shall need to discuss various different flags, so when necessary we write *
*b(F )i;jsor (bF)i;js
to emphasize we are working with the flag F , and similarly for the d's and f's*
*. The essential
difficulty is in the form of the product, so we comment on y(V i)y(V j) a littl*
*e further. We
show that to express it in terms of the flag basis, it is sufficient to underst*
*and the action
of A* in that basis. The idea is that the path ffj; ffj+1; ffj+2; : :a:lso def*
*ines a complete
flag F=V j, and the corresponding basis is y(V j=V j) = 1; y(V j+1=V j); y(V j+*
*2=V j); : :.:The
point of this is that y(V j)y(V j+n=V j) = y(V j+n), which is an element of the*
* original basis.
_i;j
Thus we simply express y(V i) in terms of the new basis as y(V i) = kbj+ky(V j+*
*k=V j) and
EQUIVARIANT FORMAL GROUP LAWS 19
_i;j _i;j
then y(V i)y(V j) = kbj+ky(V j+k). Thus bs = bi;js, explaining the notation. *
*In practice
we do this one step at a time, since y(V i) = y(ff1) . .y.(ffi). It therefore s*
*uffices to identify
y(ff)y(V j), for all ff and j. For this we note y(ff) = lffff-1j+1y(ffj+1), and*
* work with respect to
the F=V jbasis.
We also write F ffi for the flag (V 0ffi V 1ffi V 2ffi . .).whose associat*
*ed path is
ff1ffi; ff2ffi; ff3ffi; : :.: We refer to F ffi as the rotated flag. It is cle*
*ar by applying lffito the
formula defining dF that
dFffi(ffffi)is= dF(ff)is:
It is thus no loss of generality to work with flags beginning with V 1= ffl, *
*and in this
section we assume the flag path begins ffl; ff; fi; fl; : :.:We show that for a*
*ny complete flag F 0
the action coefficients dF0(*)1*can be deduced from the coefficients dF(*)1*. I*
*t then follows
that all product coefficients (bF)*;**can be deduced, and all the higher dF's. *
* Since we are
just dealing with the first few terms, we simplify notation; we already know th*
*at for r 2 A*
we have e(r) = d(r)10, and we define (r) = d(r)11; (r) = d(r)12; (r) = d(r)13; *
*: :.:Thus by
definition,
y(r) = e(r) + (r)y(ffl) + (r)y(ffl)y(ff) + (r)y(ffl)y(ff)y(fi) + . *
*.:.
The first observation is that the coefficient dF(r)1ionly depends on the flag a*
*s far as V i+1.
This is because the coefficient can be recovered after reduction modulo y(V i+1*
*) =
y(ffl)y(ff2) . .y.(ffi+1). Accordingly we write e(r); ff(r); ff;fi(r); : :t:o e*
*mphasize this.
We also need the coefficients enF(r); nF(r); nF(r); : :i:n lry(V n), and we s*
*how by induction
on n that these can be deduced from the coefficients e; ff; ff;fi; : :.: Indeed*
* y(V n+1) =
y(V n)y(ffn+1), so we have
lry(V n+1) = enF(r)y(ffn+1r) + nF(r)y(V 1)y(ffn+1r) + nF(r)y(V 2)y(ffn+1r)*
* + . .:.
The trick for calculating the i'th term is to express y(ffn+1r) in the F=V ibas*
*is. Of course this
uses the coefficients eF=V i; F=V i; F=V i; : :.:This looks dangerously close t*
*o being circular,
so we show explicitly that it is not.
Lemma 15.1. The leading term is given by
enF(r) = e(V nr):
Proof: The proof here and in the following two lemmas is to expand the right ha*
*nd side of
the equation
lry(V n+1) = lr[y(V n)]y(ffn+1r):
To do this we first expand lr[y(V n)] to obtain
enF(r)y(ffn+1r) + nF(r)y(ffl)y(ffn+1r) + nF(r)y(ffl)y(ff)y(ffn+1r) +*
* . .;.
Now expand y(ffn+1r) with respect to the ffl; ff; fi; : :b:asis in the first te*
*rm, with respect to
the ff; fi; fl; : :b:asis in the second term, with respect to the fi; fl; ffi; *
*: :b:asis_in the third
term, and so forth. *
* |__|
One important consequence is that if r-1 occurs in V nthis leading term vanis*
*hes. Since
F is a complete flag we thus see enF(r) = 0 if n is sufficiently large. Note al*
*so that for n 2
this coefficient depends on more than the first subquotient of F .
20 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
For the higher terms we cannot expect a closed formula, but a recursive algor*
*ithm is quite
sufficient. The proofs are precisely like those for Lemma 15.1 above, and the g*
*eneral case is
presented in detail in the next section.
Lemma 15.2. The first term is given recursively in terms of Euler classes and*
* ff(r) by the
formula
__
n+1F(r) = e(V nr)ff(ffn+1r) + nF(r)e(ff-1ffn+1r): |__|
We note here that once n is large enough that e(V nr) = 0 the recursion state*
*s n+1F(r) =
nF(r)e(ff-1ffn+1r): Thus if ffn+1 = ffr-1, we obtain zero. Once again we see th*
*at nF(r) = 0
if n is sufficiently large. Finally, we note that if n 2 then nF(r) = 0 modulo*
* Euler classes.
It is instructive to record one further instance explicitly.
Lemma 15.3. The second term is given recursively in terms of Euler classes an*
*d ff(r) by
the formula
*
* __
n+1F(r) = e(V nr)ff;fi(ffn+1r) + nF(r)ff-1fi(ff-1ffn+1r) + nF(r)e(fi-1ffn+*
*1r): |__|
The new feature here is that the coefficient is associated to a different fl*
*ag. We had
therefore better ensure that we can deduce it before we need to apply this recu*
*rsive formula.
Suppose then that F 0begins ffl; i; : :.:Thus y(r) = e(r)+i(r)y(ffl) modulo y*
*(ffl)y(i). Now,
find the first occurence of i in F , and express y(r) in terms of the F basis:
y(r) = e(r) + ff(r)y(ffl) + ff;fi(r)y(ffl)y(ff) + . .;.
it is sufficient to work modulo y(ffl)y(ff) . .y.(i). Now we expand y(ff); y(ff*
*)y(fi); y(ff)y(fi)y(fl); : : :
in terms of the F 0=ffl basis. In the present case we only seek the coefficient*
* of y(ffl) so we only
need the constant term in the expansions, and this is easily calculated. Thus w*
*e obtain
i(r) = ff(r) + ff;fi(r)e(ffi-1) + ff;fi;fl(r)e(ffi-1)e(fii-1) + . .*
*;.
and the sum is finite, since e(ii-1) = 0.
Note in particular that this shows that ff(r) = i(r) modulo Euler classes, so*
* taking
i = r we see ff(r) = 1 modulo Euler classes.
16. Some relations in the representing ring: the general case
We extend the calculations of the previous section to higher coefficients. Th*
*e discussion
is directed towards understanding the universal ring LF(A). Our first task is *
*to give a
reasonably efficient set of generators. The reader may also find it helpful to *
*refer to Appendix
C where some calculations are done when A is of order 2.
Theorem 16.1. The representing ring LF(A) is generated as an algebra by the E*
*uler classes
e(ff) and the coefficients f(F )1j;k. The representing ring LstrictA(k; F; O) i*
*s generated as a k-
algebra by the elements f1s;t.
The strategy of proof is as follows, where we write x - (y; z) to mean that *
*x can be
expressed in terms of y and z, together with some self-explanatory abbreviation*
*s.
Process 0. dF0(*)1n+1- (dF0(*)1n; d1F) (see 16.4).
EQUIVARIANT FORMAL GROUP LAWS 21
Process 1. fn+1 - (fn; d*) (from the fact that t*(y(V n+1)) = t*(y(V n))t*(y(f*
*fn+1)).
Process 2. dn+1 - (dn ; d1) (see 16.3).
Process 3. dn - (fn; e) (see 16.7).
Here, Process 0 has been used to avoid specifying the flags used for the d1's*
* in Process
2. Using Process 3 we obtain the generators d1 from f1's and e's. Then, using*
* Process 2
recursively, we obtain all generators dn. Finally Process 1 can be used recursi*
*vely to obtain
all the coefficients fn. We have already seen how the coefficients b can be obt*
*ained from the
d's. We now turn to the detailed implementation of the strategy.
First, let us write out the definition of the coefficients dF in longhand:
y(r) = dF(r)10+ dF(r)11y(V 1) + dF(r)12y(V 2) + dF(r)13y(V 3) + . .*
*:.
The first observation is that the coefficient dF(r)1ionly depends on the flag a*
*s far as V i+1.
This is because the coefficient is determined by reducing modulo y(V i+1) = y(f*
*fl)y(ff2) . .y.(ffi+1).
Accordingly we write dV 1(r)0; dV 2(r)1; dV 3(r)2; : :t:o emphasize this; note *
*that we have also
omitted the superscript 1. By contrast with the previous section we have not no*
*rmalized the
flag to begin with ffl, so the start of the flag is an essential piece of infor*
*mation.
We begin by summarizing the results of the previous section in the general no*
*tation: note
that we have rotated the flags to the natural position.
Lemma 16.2. (i) The leading term is given by
dF(r)n0= e(ff-11V nr):
This is zero if n is sufficiently large.
(ii) The first term is given recursively in terms of Euler classes and dV 2(*)1*
**by the formula
dF(r)n+11= dF(r)n0dV 2(ff-11ffn+1r)1 + dF(r)n1dV 2=V(1ff-12ffn+1r)*
*0:
This is zero if n is sufficiently large, and if n 2 then d(r)n1= 0 modulo Eule*
*r classes.
(iii) The second term is given recursively in terms of the coefficients dF(*)1**
*by the formula
*
* __
dF(r)n+12= dF(r)n0dV 3(ff-11ffn+1r)2+dF(r)n1dV 3=V(1ff-12ffn+1r)1+dF(r)n2dV 3=V*
*(2ff-13ffn+1r)0: |__|
It is not hard to write down the general recursion, and it should now be poss*
*ible to
understand what it means.
Lemma 16.3.
dF(r)n+1k= dF(r)n0dV k+1(ff-11ffn+1r)k + dF(r)n1dV k+1=V(1ff-12ffn+1r)k-1+
dF(r)n2dV k+1=V(2ff-13ffn+1r)k-2+. .+.dF(r)nk-1dV k+1=V k-1(ff-1kffn+1r)1+dF(r)*
*nkdV k+1=V(kff-1k+1ffn+1r)0
Proof: First note that lry(V n+1) = [lry(V n)]y(ffn+1r). When calculating the t*
*erm
dF(r)niy(V i)y(ffn+1r), we need to express y(ffn+1r) in terms of the F=V iflag,*
* so the jth __
coefficient is dF=V i(ff-1i+1ffn+1r)1j= dV i+j+1=V(iff-1i+1ffn+1r)j. *
* |__|
Now, suppose by induction that all coefficients dF0(*)1lwith l < k and all co*
*efficients
dF(*)*lwith l k can be expressed in terms of the coefficients dF(*)1*. Noting*
* that the
22 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
only occurence of d(*)1kon the left hand side uses the F basis, the lemma shows*
* that the
coefficients dF(*)n+1kcan also be so expressed.
It thus follows by induction that if all coefficients dF0(*)1lwith l < k can *
*be expressed in
terms of the coefficients dF(*)1lwith l k then all coefficients dF(*)*lwith l *
* k can be so
expressed.
The following lemma completes the justification of Process 0.
Lemma 16.4. The coefficient dF0(r)1kcan be expressed in terms of the coeffici*
*ents dF0(*)1l
with l < k and the coefficients dF(*)1*.
Proof: By rotation we may suppose that F 0also begins with ff1. Now dF0(r)1k= d*
*(V 0)k+1(r)k
is the coefficient of y((V 0)k) in the F 0expansion of y(r). We will give a wa*
*yPof calculat-
ing this coefficient in terms of the dF(*)1*'s. Indeed, we also have y(r) = *
*idF(r)1iy(V i).
Furthermore y(V i) = y(ff1)y(V i=V 1), and we may express y(V i=V 1) in terms o*
*f the F 0=V 1
basis, and the coefficient of y((V 0)k=V 1) will give us the contribution to th*
*e coefficient in
question. Furthermore, the contribution from y(V i) is zero once V icontains (V*
* 0)k+1, since
y(V i) is then already zero modulo y((V 0)k+1); thus the number of terms is fin*
*ite. It thus_
suffices to apply the following lemma with W = V i=V 1and F 00= F 0=V 1. *
* |__|
P
Lemma 16.5. If we write y(W ) in the F 00basis as y(W ) = l[F 00]Wiy((V 00)*
*i) then the
coefficients [F 00]Wl may be expressed in terms of coefficients dV(*)m with m *
*l.
Proof: The proof is by induction on the dimension of W . If W = 0 the result *
*is triv-
ial.P The inductive step is to write W = W 0 fi. Then we have y(W ) = y(W 0)y(*
*fi) =
00W0 00i 0*
*0 00i
i[F ]i y((V ) )y(fi), and in the ith term we write y(fi) in terms of the F =*
*(V ) basis,
using the coefficients dF00=(V 00)i(fi(ff00i+1)-1)1j. The contributing coeffici*
*ents_have i + j = l, so
both i; j l. *
* |__|
Corollary 16.6. Modulo Euler classes we have dF(r)11= 1, and dF(r)1k= 0 if k 6=*
* 1.
Proof: We have given explicit formulae when k = 0 or 1 in the previous section.*
* In principle
the previous two lemmas also give an explicit formula in terms of Euler classes*
*, but it is
perhaps worth giving a less cluttered proof.
Suppose then that k 2, and that the result has been proved for l < k for all*
* flags.
For the rest of the proof we work modulo Euler classes without comment. We prov*
*e that
dF(r)1k= dF0(r)1kfor any flag F 0, and hence by taking F 0to begin with r, that*
* both vanish.
First note by induction, in any basis F 00
y(r) = y(s) + dF00(sr-1)1ky((V 00)k) + higher terms;
so that y(V ) = y(W ) if dim(V ) = dim(W ) < k. Now take any two complete flags*
* F , and
F 0. By rotation we assume that both F and F 0begin with ffl. Expanding y(r) in*
* the F 0basis
we have
y(r) = y(ffl) + dF0(r)1ky((V 0)k) + higher terms
Similarly, in the F basis we have
y(r) = y(ffl) + dF(r)1ky(V k) + higher terms
EQUIVARIANT FORMAL GROUP LAWS 23
However y(V k) = y(ffl)y(V k=ffl), and by the observation, y(V k=ffl) = y((V 0)*
*k=ffl) modulo
y((V 0)k+1=ffl). Similarly y(V k+i) = y(ffl)y(V k=ffl)y(V k+i=V k) = 0, modulo *
*y((V 0)k+1=ffl)._Thus
dF(r)1k= dF0(r)1kas required. *
* |__|
Finally we see that the equivariance of t* allows us to deduce the coefficien*
*ts dF(ff)1*from
the coproduct and the Euler classes.
Lemma 16.7. The following formula gives the action coefficients in terms of t*
*he cogroup
coefficients and the Euler classes
X
dF(fi)ik= fij;ke(fiV 1 V j):
j
Note that the j = 0 term is fi0;k= ffiik.
Proof: Firstly we see from equivariance of t* that
lfi= (resUfi-1^1)t*:
Applying this to y(V i), the coefficient of y(V k) on the left is dF(fi)ik. To *
*identify the coefficient
of y(V k) on the right we recall that by definition resUfi-1= resUVl1fiV,1and c*
*alculate
P
(resUfi-1^1)t*y(V=i)(resUVl1fiV^11) j;kfij;ky(V j) y(V k)
P P j
= (resUV^11) j;kfij;k ld(fiV 1)ly(V l) y(V k)
P j __
= j;kfij;kd(fiV 1)0y(V k) |__|
*
* __
This completes the proof of Theorem 16.1. *
* |__|
Appendix A. The additive and multiplicative group laws.
Let us consider two special cases: the additive and multiplicative laws. It i*
*s easy to pick
these out since we do not need to say much about which flag we are considering.*
* The additive
law is given by
t*a(y(ffl)) = y0(ffl) + y00(ffl)
and the multiplicative law by
t*m(y(ffl)) = y0(ffl) + y00(ffl) - y0(ffl)y00(ffl)
The instructive thing here is how this imposes restrictions on the Euler classe*
*s. In particular
the additive law implies the Euler classes are all Z-torsion when A is finite; *
*this shows that
we cannot expect to use logarithms in the same way as the non-equivariant case,*
* since Euler
classes are not generally of this form. However, results of tom Dieck [8] sugge*
*st that (at least
for cyclic groups) we may hope to use the multiplicative logarithm as in [1, I.*
*6.7]
For convenience of calculation we shall assume the flag begins with ffl. Thus*
* the additive
case has f11;0= f10;1= 1 and f1j;k= 0 otherwise. Thus, from 16.7 we see
y(fi) = e(fi) + y(ffl):
24 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
Rotating by ff it follows that e(fffi) = e(ff) + e(fi), ie that e : A* -! k is *
*a group homomor-
phism, and in particular if fin = ffl we see that ne(fi) = 0. This identifies t*
*he universal ring
for additive equivariant formal group laws.
Proposition A.1. The free commutative ring Symm(A*) on the abelian group A* is*
* uni-
versal for additive formal group laws. If A is a torus, this is the coefficient*
* ring_of ordinary
Borel cohomology. *
* |__|
The multiplicative case has f11;0= f10;1= 1; f11;1= -1 and f1j;k= 0 otherwise*
*. Thus, from
16.7 we see
y(fi) = e(fi) + [1 - e(fi)]y(ffl):
Rotating by ff shows that (1 - e(fffi)) = (1 - e(ff))(1 - e(fi)), ie that 1 - e*
* : A* -! kx is a
group homomorphism. This identifies the universal ring for multiplicative group*
* laws.
Proposition A.2. The group ring Z[A*] is the universal ring for multiplicative*
* group laws.
Since A is abelian, Z[A*] ~=R(A), and the coefficient ring of K-theory is unive*
*rsal for_mul-
tiplicative formal group laws. *
* |__|
Appendix B. The coinverse.
We show here that any A-equivariant formal group law has a unique coinverse, *
*and is thus a
Hopf algebra. This is exactly analogous to the fact that a connected bialgebra *
*has a coinverse,
and the proof is analogous to that of [15, 8.2], although substantially more co*
*mplicated. The
idea is that the formal group law is equipped*with a filtration: the subquotien*
*ts are controlled
by the well understood Hopf algebra kA and the filtration is complete.
Proposition B.1. Given an A-formal group law (R; ; ; {y(ff)}ff2A*), there exis*
*ts a unique
algebra homomorphism fl :R ! R such that the following diagram commutes.
_____ 1fl_//_
R //Rb kR Rb kR
(ffl)|| |m|
fflffl| fflffl|
k ________j________//R:
Moreover,
1.fl O fl = 1R.
2.(fl fl) O = O fl.
3.lffO fl = fl O lff-1,
4.(ff) O fl = (ff-1).
Proof: Let 0 V 1 V 2 . .b.e a complete flag in U with V i+1= V i ffi+1as usual*
*. We
assume for convenience that ff1 = V 1= ffl. If fl exists we may express it in t*
*erms of the flag
basis P
fl(y(V i)) = 1j=0cijy(V j):
and if it is to be continuous, we require that for any fixed j we have cij= 0 f*
*or large i.
EQUIVARIANT FORMAL GROUP LAWS 25
We shall construct fl by finding the coefficients ci0; ci1; ci2; : :i:n turn.*
* We therefore take
P n
fln(y(V i)) = j=0cijy(V;j)
and suppose inductively that fln : R -! R is a k-module homomorphism with the p*
*roperty
that
m O (1 fln) O (x) j O (ffl)(x) mod (y(V n+1))
for any x 2 R. To start the induction, set fl0 = (ffl).
Now suppose fln has been constructed: we wish to find coefficients cin+12 k s*
*o as to define
a map fln+1 that has the inverse property modulo (y(V n+2)). For any x 2 R we d*
*efine rx 2 k
by
m O (1 fln) O (x) j O (ffl)(x) + rxy(V n+1) mod (y(V n+2)):
We need to choose the coefficients cin+1in such a way that
m O (1 (fln+1 - fln)) O (x) -rxy(V n+1) mod (y(V n+2))
for any x 2 R. It suffices to establish this as x runs through the topological *
*basis y(ffn+2) =
y(V 1ffn+2); y(V 2ffn+2); y(V 3ffn+2); : :.:We write the coproduct : R ! Rb kR *
*as
P
(y(V i)) = fis;ty(V s) y(V t):
Recall that for fixed s; t, fis;t= 0 for large i and that the unital condition *
*of the A-equivariant
formal group law implies (
1 if s = i,
fis;0= fi0;s=
0 if s 6= i.
Using the A*-equivariant property of , we see that
P
(y(V iffn+2)) = fis;ty(V sffn+2) y(V t):
Thus P
m O (1 (fln+1 - fln)) O (y(V iffn+2)) = fis;tctn+1y(V sffn+2)y(V n*
*+1):
Since ff1 = ffl we see that y(V n+2) divides y(V sffn+2)y(V n+1) if s 1. Since*
* fi0;tis 1 when
t = i and is 0 otherwise we find
m O (1 (fln+1 - fln)) O (y(V iffn+2)) cin+1y(V n+1) mod (y(V n+2)):
Thus we define cin+1= -ry(V iffn+2). It remains only to check the continuity co*
*ndition that
these coefficients must vanish for large i. This follows since m O (1 fln+1) *
*O gives the
continuous homomorphism j O (ffl) : R ! R=(y(V n+2)).
So far we have proved the existence of a k-module homomorphism fl :R ! R that*
* makes
the diagram commute. It is clear from the way we constructed it that fl is uniq*
*ue with this
property, but in any case this follows from an easy formal argument: if fl and *
*fl0 are both
inverses then the composite
fl1fl0 m
R -! Rb kRb kR -! Rb kRb kR -! R
must agree simultaneously with fl and fl0.
To show that fl is a k-algebra homomorphism and to establish the other claims*
* we argue
similarly by uniqueness of inverse. Let S denote Rb kR. Note that S has a cop*
*roduct
S :S ! S bkS defined by the composite
Rb kR -! Rb kRb kRb kR 1o1-!Rb kRb kRb kR:
26 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
where o denotes the twist map.*The ring S has a unit map jS :k ! S defined by j*
*S = j j
and an augmentation S :S ! kA defined as the composite
* A* m A*
Rb kR -! kA b kk -! k :
In fact, we may regard S as a two dimensional A-formal group over k. Now by di*
*agram
chasing it can be shown that m O (fl fl) and fl O m are both inverse to the mu*
*ltiplication
m: S ! R in the sense that if OE denotes either of m O (fl fl) or fl O m, then*
* the diagram
S mOE
S_____//S bkS___//_Rb kR
S(ffl)|| |m|
fflffl| fflffl|
k________j________//R
commutes. It follows formally that m O (fl fl) = fl O m and hence that fl is *
*an algebra
homomorphism. Dually, (fl fl) O and O fl must agree since a diagram chase sh*
*ows that
they are both inverse to . Similarly, fl O fl and 1 agree since they are both i*
*nverse to fl. For
claim (3) we use the equivariant property of and observe that
m O (1 (lffO fl O lff))=Om O ((lffO lff-1) (lffO fl O lff)) O
= m O (lff lff) O (1 fl) O (lff-1 lff) O
= lffO m O (1 fl) O
= lffO j O (ffl)
= j O (ffl):
Thus lffO fl O lff= fl since they are both inverse to 1. One can check directl*
*y from our
construction of fl that fl(y(ffl)) 2 (y(ffl)). Hence (ffl) O fl = (ffl). It fol*
*lows that
(ff) O fl= (ffl) O lffO fl = (ffl) O fl O lff-1= (ffl) O lff-1
= (ff-1):
*
* __
*
*|__|
Appendix C. The group of order 2
In this appendix we make some of the formulae explicit for the group A = C2 o*
*f order 2.
We use the flag with alternating subquotients ffl; j; ffl; j; ffl; j; ffl; j; :*
* :.:Since there is only one
non-trivial element of A* we write simply dij= d(j)ij, and continue with the co*
*nvention that
we drop i if i = 1. It is convenient to declare dij= 0 if j < 0. As we give for*
*mulae it will
become ever more apparent that there is a sense in which an equivariant formal *
*group is a
deformation of a non-equivariant formal group with the Euler class e = d0 as de*
*formation
parameter [9].
First note that since
y(j) = d0 + d1y(ffl) + d2y(ffl)y(j) + . . .
by rotation we have
y(ffl) = d0 + d1y(j) + d2y(j)y(ffl) + . .;.
EQUIVARIANT FORMAL GROUP LAWS 27
and thus, premultiplying both sides by y(ffl) we have an expression for y(ffl)2*
* in the standard
basis:
b1;1k= dk-1:
Thus
k{{U}} = k[[x]][y]=(y2 = p(x) + yq(x))
where y = y(ffl), x = y(ffl)y(j), and where p(x) = d1x + d3x2 + d5x3 + . . .and*
* q(x) =
e + d2x + d4x2 + . ...Now consider y(V i)y(V j); if i or j is even, it is clear*
*ly y(V i+j), and
if both are odd it is y(V i-1)y(V j-1)y(ffl)2 = y(V i+j-2)y(ffl)2. Combining th*
*ese into a single
formula we have
bi;jk= d(jij)k-i-j+1:
For dninote that if n is even, ljy(V n) = y(V n), so that dni= ffini: The sam*
*e observation
when n is odd gives dnk= dk-n+1. Also, the fact that lj = (resj 1) O t* gives
dk = ffi1k+ ef11;k:
Altogether we have ae
ffink if n is even
dnk= ffin 1
k + ef1;k-n+1if n is odd
In a single formula
dnk= ffink+ e(jn)f11;k-n+1;
and
bi;jk= ffii+jk+ e(jij)f11;k-i-j+1:
We can obtain useful relations from the statement ljlj = 1 (or alternatively *
*by using the
recursive formula for dn+1). Thus
y(j) = (d0 + d1y(ffl)) + y(V 2)(d2 + d3y(ffl)) + y(V 4)(d4 + d5y(ffl)*
*) + . .;.
and applying lj again
y(ffl)= (d0 + d1d0)y(V 0) + d1d1y(V 1) + d1d2y(V 2) + d1d3y(V 3) + . *
*. .
+ (d2 + d3d0)y(V 2) + d3d1y(V 3) + d3d2y(V 4) + d3d3y(V 5) + . *
*. .
+ (d4 + d5d0)y(V 4) + d5d1y(V 5) + d5d2y(V 6) + d5d3y(V 7) + . *
*. .
+ (d6 + d7d0)y(V 6) + d7d1y(V 7) + d7d2y(V 8) + d7d3y(V 9) + . *
*. .
Comparing coefficients of y(V 0) and y(V 1) we find
d0 + d1d0 = 0
and
d1d1 = 1:
Similarly comparing coefficients of y(V 2n) and y(V 2n+1) for n 1 we find
d1d2n+ d3d2n-2+ . .+.d2n+1d0 + d2n = 0
and
d1d2n+1+ d3d2n-1+ . .+.d2n+1d1 = 0:
Writing the first two in terms of f11;*we obtain
e(2 + ef11;1) = 0
and
(1 + ef11;1)2 = 1:
28 MICHAEL COLE, J.P.C.GREENLEES, AND I.KRIZ
The first of these allows us to omit e(2+ef11;1)f11;2nfrom the subsequent even *
*terms to obtain
1 1 1 1 1 1
e2 f1;3f1;2n-2+ f1;5f1;2n-4+ . .+.f1;2n+1f1;0= 0
and
2e(1 + ef11;1)f11;2n+1+ e2 f11;3f11;2n-1+ f11;5f11;2n-3+ . .+.f11;2n-1*
*f11;3= 0:
We really want to examine the consequences of these relations e-adically and wi*
*th e inverted.
As a first step, note that mod e2 the first states 2e = 0, and the rest are all*
* consequences.
Mod 2 we the first reads e2f11;1= 0, and the second follows from it. The remai*
*ning even
terms read
e2 f11;3f11;2n-2+ f11;5f11;2n-4+ . .+.f11;2n+1f11;0= 0
and the remaining odd terms are trivial if 2n + 1 = 3 mod 4 and gives
e2(f11;2m+1)2 = 0
in the remaining case (where 2n + 1 = 4m + 1). If we instead invert e, the fir*
*st relation
states that ef11;1= -2, the second is a consequence. The relation from y(V 2) g*
*ives f11;3= 0,
and the successive relations from y(V 4); y(V 6); : :a:llow us to deduce f11;2n*
*+1= 0 for n 1.
The relation from y(V 2n+1) is automatically satisfied.
To obtain the recursive formula for fn+1 we use t*y(V n+1) = t*y(V n)t*y(ffn+*
*1), and there
are then two cases, since ffn+1 is ffl or j depending on whether n + 1 is odd o*
*r even. The first
case is straightforward, but the second uses t*(y(j)) = (1 lj)(t*(y(ffl)); tog*
*ether these give
the formulae
aeP
fni;jf1k;ld(jik)s-i-k+1d(jjl)t-j-l+1if n + 1 is odd
fn+1s;t= P i;j;k;l n 1 ik l jm
i;j;k;l;mfi;jfk;ld(j )s-i-k+1dm d(j )t-j-m+1if n + 1 is *
*even
Combining this with our expressions for the d's we obtain
X
fn+1s;t= fni;jf1k;lffii+ks+ e(jik)f11;s-i-k+1ffilm+ e(jn)f11;m-l+1ffij+mt+*
* e(jjm)f11;t-j-m+1:
i;j;k;l;m
It is perhaps worth making explicit the structure of the additive and multipl*
*icative laws
in this case. For an additive law, the action is given by
ljy(V n) = y(V n) + e(jn)y(V n-1)
and the product is given by
y(V i)y(V j) = y(V i+j) + e(jij)y(V i+j-1):
Let x = y(V 2) = y(ffl)y(j), and y = y(ffl). The ring k{{F }} is the x-adic com*
*pletion of the
ring k[x; y]=(y2 = x + ey) = k[y]. The action by A* is given by y 7-! e + y, wh*
*ich is of order
2 since e is. Of course x = y(e + y), so the completion is the one we are used *
*to.
For a multiplicative law, the action is given by
ljy(V n) = [1 - e(jn)]y(V n) + e(jn)y(V n-1)
and the product is given by
ij i+j ij i+j-1
y(V i)y(V j) = 1 - e(j )y(V ) + e(j )y(V ):
Let x = y(V 2) = y(ffl)y(j), and y = y(ffl). The ring k{{F }} is the x-adic co*
*mpletion of
k[x; y]=(y2 = (1 - e)x + ey). The action by A* is given by y 7-! e + (1 - e)y, *
*which is of
order 2 since (1 - e)2 = 1. Since (1 - e) is a unit, the ring is k[y] again. Th*
*e completion
is with respect to x = y(e + (1 - e)y). We may let z = 1 - y, and it is reasona*
*ble to view
EQUIVARIANT FORMAL GROUP LAWS 29
1 - e as the image of j in k, under the classifying map R(A) -! k so that we ob*
*tain the
completed ring k[z]^(1-z)(1-jz)in the form familiar from K-theory.
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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-0001
E-mail address: mmcole@math.lsa.umich.edu
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003
E-mail address: ikriz@math.lsa.umich.edu