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 ring 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 Chicago 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 equivariant formal group law appropriate for understanding orientable complex stable equivariant cohomology theories E_{A}^{∗}(·), at least when the compact Lie group A of equivariance is abelian.

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 [5]. Bordism is universal amongst such theories in a topological sense [13], and non-equivariantly Quillen showed its coefficient ring is the Lazard ring, which is universal in the algebraic sense. This possibility is more seductive equivariantly, since so little is known about the coefficient ring of equivariant bordism. In [8] 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 dimensional, and it is enough to consider line bundles. Line bundles are classified by the A-space ℂP^{∞} of lines in a complete A-universe, so we study E_{A}^{∗}(ℂP^{∞}), endowed with all the structure inherited from ℂP^{∞}. Since ℂP^{∞} is a space, E_{A}^{∗}(ℂP^{∞}) is a ring, the tensor product of line bundles makes ℂP^{∞} into an abelian group object in the homotopy category, so (when we have a Künneth isomorphism) E_{A}^{∗}(ℂP^{∞}) is a cogroup object, and finally, since one can tensor a line bundle with any one dimensional representation, ℂP^{∞} has an action of the dual group A^{∗}, giving rise to an action of A^{∗} on E_{A}^{∗}(ℂP^{∞}).

To ensure the additive structure is reasonable, we restrict attention to oriented 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 that of a truncated polynomial ring, but equivariantly this is too much to ask. However the Splitting Theorem [3, 4] shows that for any complete flag V^{1} ⊂ V^{2} ⊂ V^{3} ⊂ ⋯ ⊂ V^{n} = V in V the corresponding 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 topological k-Hopf algebra R together with a coaction of A^{∗} and special elements y(α) (related by the coaction as the notation suggests) so that R/(y(α)) = k for all α, and so that R is complete with respect to Π_{α}y(α). This is a considerable refinement of the essentially non-equivariant, classical notion of an Okonek equivariant formal group law [14]. In effect our equivariant formal group law is a very special ‘sub-cogroup of finite index’ in an Okonek formal group law [14].

Returning to topology, any complete flag F in the universe gives an additive topological basis for E_{A}^{∗}(ℂP^{∞}), and we may express the ring structure, the cogroup structure, 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 satisfied by the structure constants, and this gives the notion of an (A, F)-equivariant formal group law, essentially that given in [3] for cyclic groups. This description shows there is a representing ring L_{A}(F) for such formal group laws. This process has an algebraic counterpart, so that an A-equivariant formal group law with a flag F gives an (A, F)-formal group law, 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 independent of the flag F, and that the ring L_{A}(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 [7]. There is another coordinate free description, which amounts to viewing an equivariant formal group as a certain type of deformation of a non-equivariant one: the structure studied by Hopkins-Kuhn-Ravenel [10] and Greenlees-Strickland [9] 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 definitions and the fundamental splitting theorem of [3, 4], our first task is to describe a good supply of examples (Sections 6 to 10). We are then ready in Section 11 to give the definition of an equivariant formal group law. To make calculations we need to introduce the framework of a complete flag F giving the definition of an (A, F)-equivariant formal group law in Section 12. After showing that an A-equivariant formal group law together with a flag is equivalent to an (A, F)-formal group law, it follows that the subsequent study of (A, F)-formal group law provides tools for calculation with equivariant formal group laws. In particular we show that there is a representing ring for equivariant formal group laws. In appendices we illustrate the theory by considering the special cases of additive and multiplicative formal 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^{∞}. 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 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 group of order 2 24 References 26

The following summarizes certain notation introduced in the text.

A is an abelian compact Lie group. A^{∗} its dual group Hom(A, S^{1}) α, β, γ, … are typical one dimensional complex representations. ϵ 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}, … the terms in F. α_{1}, α_{2}, α_{3}, … the subquotients of F. l_{α} left multiplication by α ∈ A^{∗}. λ(V) the Thom class giving a complex stable structure. χ(V) the pullback of λ(V) along S^{0} ⟶ S^{V}. 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 × 𝕋, and 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 point, so that a one dimensional subspace of V specifies a basepoint of ℂP (V): this is significant because basepoints may lie in different components of the fixed point set. At the other extreme, if α is one dimensional one may verify there is a cofibre sequence

2.1.

ℂP (V) ⟶ ℂP (V ⊕ α) ⟶ S^{V ⊗α^{−1}}.

We remark that, as an A-space, V is isomorphic to \overline{V}, so that the cofibre may also be described as the Thom space of \overline{V} ⊕α, as is often done when discussing Thom isomorphisms.

The A-invariant complex lines are exactly the subrepresentations of V, so it is easy to see that

2.2.

ℂP (V)^{A} = ∐_{α} ℂP (V_{α})

where V_{α} = Hom_{A}(α, V) is the α-isotypical part of V. Note that if A is finite and V = kℂA is a multiple of the regular representation, we have an isomorphism

A^{∗} × ℂP (kϵ) ⟶^{≅} ℂP (kℂA)^{A}

given by (α, W) ⟶ α ⊗ W.

For convenience we take 𝒰 = ⨁_{k≥0} ⨁_{α∈A^{∗}} α as our complete A-universe and we define ℂP^{∞} = ℂP (𝒰), with its topology as a colimit of its subspaces ℂP (V) with V finite dimensional. It is also convenient to let ℂP_{W}^{∞} = ⋃_{k≥0} ℂP (kW), so that ℂP_{ϵ}^{∞} is an A-fixed infinite complex projective space.

Lemma 2.3. The A-space ℂP^{∞} classifies line bundles.

Tensor product of line bundles is commutative and associative up to coherent isomorphism, and has ϵ as a unit, and we shall constantly use the represented counterpart.

Corollary 2.4. The A-space ℂP^{∞} is an abelian group object up to homotopy, and the inclusion of fixed points is a group homomorphism.

We have seen

(ℂP^{∞})^{A} = ℂP (𝒰)^{A} = ∐_{α} ℂP (𝒰_{α}) ≅ A^{∗} × ℂP_{ϵ}^{∞}.

It is important to make explicit the map

\begin{matrix}i : A^{∗} × ℂP_{ϵ}^{∞} & ⟶ & ℂP^{∞} \\ (α, W) & ⟼ & α ⊗ W. \end{matrix}

This map of abelian groups is absolutely fundamental to our analysis.

Note also that since ℂP_{α}^{∞} is connected, there is a unique homotopy class A^{∗} ⟶ (ℂP^{∞})^{A} splitting the natural augmentation (ℂP^{∞})^{A} ⟶ A^{∗}, and it is a group homomorphism. In particular A^{∗} acts on ℂP^{∞} through A-maps, by α·L = α⊗L. To avoid confusion, we make it explicit. Any vector v in 𝒰 = ⨁_{α} 𝒰_{α} can be resolved into its components v_{α} ∈ 𝒰_{α}, and under the isomorphism α⊗ : 𝒰_{ϵ} ⟶^{≅} 𝒰_{α} it is best to view v as a function v : A^{∗} ⟶ 𝒰_{ϵ}. The action of A^{∗} on 𝒰 is then given by [α · v](β) = v(βα^{−1}), and the action of A by [a · v](β) = a(v(β)). It is easy to verify these commute. When A is finite, the action also restricts to an action on kℂA for each k.

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 convenient to work throughout in the unbased context.

A genuine equivariant cohomology theory E_{A}^{∗}(·) is an exact contravariant functor on Aspaces, which admits an RO(G)-graded extension so that we have coherent suspension isomorphisms

Ẽ_{A}^{V +n}(S^{V} ∧ X) ≅ Ẽ_{A}^{n}(X)

for all real representations V. Amongst these, the most familiar ones are those with a stronger stability property

Ẽ_{A}^{|V |+n}(S^{V} ∧ X) ≅ Ẽ_{A}^{n}(X)

when V is a complex representation, where |V | denotes the space V with trivial action. This is very convenient: for most purposes we only need to look at the theory in integer gradings. Following tom Dieck we call these theories complex stable. As examples, we have the cohomology theory of the Borel construction, defined in terms of a nonequivariant cohomology theory by X ⟼ E^{∗}(EG ×_{G} X). A Serre spectral sequence argument shows this is complex stable, since A acts trivially on H^{|V |}(S^{V}) 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 cohomology theory E_{A}^{∗}(·). For any complex representation V, complex stability provides an element

λ(V) ∈ Ẽ_{A}^{|V |}(S^{V})

corresponding to the unit in E_{A}^{0}, and the E_{A}^{∗}-module Ẽ^{∗}(S^{V}) is free of rank 1 on this generator. All complex stability isomorphisms are given by multiplication by λ(V), and we have λ(V ⊕ W) = λ(V)λ(W). We then define the Euler class χ(V) = e_{V}^{∗} (λ(V)) ∈ E_{A}^{|V |}, where e_{V} : S^{0} ⟶ S^{V} is the inclusion. Thus we have χ(V ⊕ W) = χ(V)χ(W).

Note that for any based A-space X this gives

Ẽ_{∗}^{A}(X ∧ S^{∞V}) = Ẽ_{∗}^{A}(X)[1/χ(V)].

This leads to the localization theorem. In fact, if we take S^{∞\overline{ρ}} = ⋃_{V^{A}=0} S^{V}, the inclusion

X^{A} ∧ S^{∞\overline{ρ}} ⟶^{≃} X ∧ S^{∞\overline{ρ}}

is an equivalence by obstruction theory. It follows that Ẽ_{∗}^{A}(X^{A}) ⟶ Ẽ_{∗}^{A}(X) becomes an isomorphism if we invert all Euler classes χ(V) with V^{A} = 0. By duality we deduce the localization theorem we need: the finiteness assumption in the statement is essential.

Lemma 3.1. If X is a based finite A-space then

Ẽ_{A}^{∗}(X^{A}) ⟵ Ẽ_{A}^{∗}(X)

becomes an isomorphism if we invert all Euler classes χ(V) with V^{A} = 0.

4. Orientations and the cohomology of ℂP^{∞}.

Notice that when ϵ ⊆ ϵ ⊕ α ⊆ 𝒰 we have

∗_{ϵ} = ℂP (ϵ) ⊆ ℂP (ϵ ⊕ α) = S^{α^{−1}} ⊆ ℂP (𝒰) = ℂP^{∞}.

Definition 4.1. [3] We say that x(ϵ) ∈ E_{A}^{∗}(ℂP^{∞}, ℂP (ϵ)) is an orientation if for all one dimensional representations α ∈ A^{∗},

res_{ϵ⊕α}^{𝒰}x(ϵ) ∈ E_{A}^{∗}(ℂP (ϵ ⊕ α), ℂP (ϵ)) = Ẽ_{A}^{∗}(S^{α^{−1}})

is a generator.

Remark 4.2. We do not require that x restricts to the standard generator λ(α^{−1}) ∈ Ẽ_{A}^{∗}(S^{α^{−1}}). It is perhaps worth introducing the notation

res_{ϵ⊕α}^{𝒰}x(ϵ) = u_{α^{−1}}λ(α^{−1})

for the unit concerned.

We may generate many other elements from an orientation. Firstly, pulling back along the action

α^{−1} : (ℂP^{∞}, ℂP (α)) ⟶ (ℂP^{∞}, ℂP (ϵ))

we have x(α) ∈ E_{A}^{∗}(ℂP^{∞}, ℂP (α)). To avoid confusion later, we write l_{α} = (α^{−1})^{∗}; thus, in particular, x(α) = l_{α}x(ϵ). Taking external direct products, if V = α_{1} ⊕ ⋯ ⊕ α_{n} we obtain

x(V) = x(α_{1}) ∗ ⋯ ∗ x(α_{n}) ∈ E_{A}^{∗}(ℂP^{∞}, ℂP (V))

Here the product ∗ is defined by pulling back the external cup product along the map

\overline{Δ} : (ℂP (V ⊕ W ⊕ Z), ℂP (V ⊕ W)) ⟶ (ℂP (V ⊕ Z), ℂP (V)) × (ℂP (W ⊕ Z), ℂP (W))

defined by (v : w : z) ⟼ ((v : z), (w : z)). Forgetting the subspace, x(V) defines an element y(V) ∈ E_{A}^{∗}(ℂP^{∞}) which restricts to zero on ℂP (V). It turns out that 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(ϵ) is the image of x(ϵ), and, since the restriction turns out 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 (α^{−1})^{∗}y(V) = y(V ⊗ α). Thus all the elements y(V) can be obtained from y(ϵ) using the action of A^{∗} and the multiplication.

To obtain a topological additive basis of E_{A}^{∗}(ℂP^{∞}) we choose a complete flag

F = (V^{0} ⊂ V^{1} ⊂ V^{2} ⊂ ⋯),

so that dim_{ℂ}(V^{i}) = i, and ⋃_{i≥0} V^{i} = 𝒰 . Associated to any such complete flag F we have the sequence α_{1}, α_{2}, … of subquotients α_{i} = V^{i}/V^{i−1}, so that V^{n} ≅ α_{1} ⊕ ⋯ ⊕ α_{n}, and y(V^{n}) = y(α_{1})y(α_{2}) ⋯ y(α_{n}). The basis only depends on the isomorphism classes of the sequence V^{n}, so the important structure is represented by the path α_{1}, α_{2}, α_{3}, … in the first orthant, ℕ^{A^{∗}}. 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 speak loosely 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 = (V_{α})_{α} for each finite dimensional representation V. We use the notation k{{y_{i}|i ∈ I}} to denote the product ∏_{i∈I} k where y_{i} is the characteristic function of the ith factor.

The Splitting Theorem of [3, 4] is the appropriate substitute for the collapse 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 basis of E_{A}^{∗}(ℂP^{∞}) as follows:

E_{A}^{∗}(ℂP^{∞}) = E_{A}^{∗}{{y(V^{0}) = 1, y(V^{1}), y(V^{2}), …}}.

Similar results hold for products of copies of ℂP^{∞}, in the sense that the Künneth theorem holds with completed tensor products.

Proof: We argue by induction that E ∧ (ℂP^{∞}, ℂP (0)) splits as a wedge, and that F ((ℂP^{∞}, ℂP (0)), E) is the corresponding product. The cofibre sequence

(S^{α_{n+1}^{−1}}, ∗) = (ℂP (V^{n+1}), ℂP (V^{n})) ⟶ (ℂP^{∞}, ℂP (V^{n+1})) ⟶ (ℂP^{∞}, ℂP (V^{n}))

gives a split exact sequence in homology or cohomology: the splitting is given by x(V^{n+1}). This is equally true if ℂP^{∞} 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 of elements of the coefficient ring E_{A}^{∗}. This lets us describe the coarsest features of the maps 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 unit multiples of the Euler classes.

Lemma 5.1. The restriction of y(ϵ) to the point ∗_{α} = ℂP (α) is a unit multiple of the Euler class of α:

res_{α}^{𝒰}(y(ϵ)) = u_{α^{−1}}χ(α^{−1})

where the unit u_{α^{−1}} is the one occurring in the definition of an orientation. In general

res_{α}^{𝒰}(y(β)) = u_{α^{−1}β}χ(α^{−1}β).

Proof: Consider the diagram

(ℂP (α ⊕ ϵ), ℂP (ϵ)) ⟶ (ℂP^{∞}, ℂP (ϵ))
↑  ↑
S^{α^{−1}} ≅ ℂP (α ⊕ ϵ)  ℂP^{∞}
↑  ↑
ℂP (α)  ⟶^{=} ℂP (α)

It is convenient to introduce the notation e(β) := u_{β}χ(β) for the elements, and we refer to them as strict Euler classes associated to a complex oriented theory. For 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(β) in any basis.

Corollary 5.2. When working with a basis corresponding to a complete flag F with first term α, the coefficient of y(V^{0}) = 1 in the expression for y(β) in the F -basis is e(α^{−1}β).

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(α_{j+1}^{−1}V)y(V^{j}) + higher terms .

Proof: By induction it suffices to prove the special case when V = β is one dimensional. In this case y(β) = e(α_{j+1}^{−1}) mod y(α_{j+1}), and the result follows.

Of course if V contains α_{j+1} the displayed coefficient is zero. Later we consider 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 periodicity shows Ktheory is complex stable, and that we may work entirely in degree 0. Thus the coefficient ring K_{A} = R(A) is the complex representation ring. There is a severe danger of confusion here: the coefficient ring R(A) acts on K_{A}(X) for any X. On the other hand, when X = ℂP^{∞} we have an action of A^{∗} via ring homomorphisms: this is quite different from the action of A^{∗} ⊆ R(A). To minimize confusion, recall that we write l_{α}x for the image of the cohomology class under the action of α ∈ A^{∗} so that l_{α} = (α^{−1})^{∗}.

This is certainly a case where an equivariant approach is appropriate, since

K_{A×𝕋}(S(V ⊗ z)) = K_{A}(ℂP (V)).

The point here is that the based cofibre sequence S(V)_{+} ⟶ D(V)_{+} ⟶ S^{V} gives rise to Gysin sequence. By Bott periodicity, it takes the form

0 ⟵ K_{A×𝕋}(S(V ⊗ z)) ⟵ R(A)[z, z^{−1}] ⟵^{χ(V ⊗z)} R(A)[z, z^{−1}] ⟵ · · · .

Furthermore, if V = α_{1} ⊕ ⋯ ⊕ α_{n}, then

χ(V ⊗ z) = (1 − α_{1}z)(1 − α_{2}z) ⋯ (1 − α_{n}z),

which is a regular element. Thus

K_{A}(ℂP (V)) = K_{A×𝕋}(S(V ⊗ z)) = R(A)[z, z^{−1}]/χ(V ⊗ z).

Next, note that z is already invertible in R(A)[z]/χ(V ⊗ z); indeed

1 − χ(V ⊗ z) = z · (V + higher terms).

Either by the completion theorem, or simply by passage to inverse limits, we see that

K_{A}(ℂP_{V}^{∞}) = R(A)[z]_{χ(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 sense as an element of K_{A}(ℂP (V)) for any V, and that 1 − z visibly generates the kernel of

K_{A}(ℂP (ϵ ⊕ α)) = R(A)[z]/(1 − z)(1 − αz) ⟶ R(A)[z]/(1 − z) = K_{A}(ℂP (ϵ)).

The element z, regarded as an element of K_{A}(ℂP (V)), is the canonical line bundle over ℂP (V), so it is easy to identify the A^{∗} action: l_{α}z = αz. Since the action is through ring homomorphisms, y(α) = l_{α}(1 − z) = 1 − αz.

Next, we specialize to case A is finite and V is the regular representation, and we let Π = χ(ℂA ⊗ z) = ∏_{α}(1 − zα). There is a straightforward and standard way to adapt the discussion to an arbitrary abelian compact Lie group A. The inclusion i : A^{∗} × ℂP_{ϵ}^{∞} ⟶ ℂP^{∞} induces a map

i^{∗} : R(A)[z]_{Π}^{∧} ⟶ ∏_{α} R(A)[z]_{(1−αz)}^{∧}.

We have chosen coordinates so that the αth component is induced by completing the identity map of R(A)[z] with respect to Π in the domain and (1−αz) in the codomain, as is legitimate since (1−αz) divides Π. Note in particular that i^{∗} is injective, since the same primes contain the product and the intersection of the ideals (1 − αz). It is also not hard to see that if we invert all the Euler classes χ(α) = 1 − α that the ideals become coprime. Thus if we invert the Euler classes before completion, we obtain an isomorphism by the Chinese Remainder Theorem.

Working modulo all Euler classes, we see that 1 − αz = 1 − z, so that i^{∗} is the diagonal 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 × 𝕋-equivariant form of the cohomology theory, so that E_{A×𝕋}^{∗}(X) = E_{A}^{∗}(X/𝕋) when X is 𝕋-free. In this case we have

E_{A}^{∗}(ℂP (V)) = E_{A×𝕋}^{∗}(S(V ⊗ z)),

just as for K-theory. We also have a completion theorem in this context, stating

E_{A}^{∗}(ℂP_{V}^{∞}) = (E_{A×𝕋})_{χ(V ⊗z)}^{∧}.

Because we are only considering the cohomology of the single infinite sphere S(∞(V ⊗ z)), the completion theorem only requires complex stability, and not highly structured ring and module technology. The most naive form of the statement would involve local homology, but the calculation 4.3 shows this reduces to the classical completion.

We still find that if V = α_{1} ⊕ ⋯ ⊕ α_{n}, then

χ(V ⊗ z) = χ(α_{1} ⊗ z)χ(α_{2} ⊗ z) ⋯ χ(α_{n} ⊗ z).

Thus, (with the same adaptions to the infinite case as for K-theory) the inclusion i : A^{∗} × ℂP_{ϵ}^{∞} ⟶ ℂP^{∞} induces a map

i^{∗} : (E_{A×𝕋}^{∗})_{Π}^{∧} ⟶ ∏_{α}(E_{A×𝕋}^{∗})_{χ(αz)}^{∧},

where Π = ∏_{α} χ(αz). The map i^{∗} is again injective, and the analogue in which Euler classes χ(V) with V^{A} = 0 are inverted before completion, is an isomorphism.

We have an action of A^{∗} on ℂP^{∞}, and hence on the completion of E_{A×𝕋}^{∗}. It can be rather useful to know the action exists before completion. In fact the group A^{∗} acts on the group A × 𝕋 by the formula

α · (a, z) = (a, α(a)z).

This induces an action of A^{∗} on the cohomology theory E_{A×𝕋}^{∗}(·). Since composition with α : 𝒰 ⊗ z ⟶ 𝒰 ⊗ z takes a subspace V ⊗ z of 𝒰 to V α ⊗ z, we see that the action on E_{A×𝕋}^{∗}(S(𝒰 ⊗ z)) is the same as that on E_{A}^{∗}(ℂP^{∞}) discussed earlier.

8. Borel cohomology.

If E is any non-equivariant complex oriented theory we may consider the associated Borel cohomology:

b(E)_{A}^{∗}(X) = E^{∗}(EA ×_{G} X).

Since A acts trivially on H^{∗}(S^{V}) if V is a complex representation, a Serre spectral sequence argument shows that this is a complex stable theory, and a complex stable structure is provided by choosing Thom classes. Any class in b̃(E)_{A}^{∗}(ℂP^{∞}) restricting to a generator of b̃(E)_{A}^{∗}(S^{2}) is automatically a complex orientation.

These examples are special because E_{A}^{∗} is already complete at the ideal I generated 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 equivariant point of view of Section 7 or otherwise, we see that b(H)_{A}^{∗}(ℂP^{∞}) = H^{∗}(A) ⊗ H^{∗}(BS^{1}). Choosing an orientation c we find this is H^{∗}(A)[c]. Furthermore H^{2}(BA) ≅ A^{∗}, and we have e(α) = α. 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 classes are invertible. Thus for this section we suppose E_{A}^{∗}(·) is a complex stable theory in which all Euler classes χ(α) (α ≠ ϵ) are invertible. We shall see that such a theory is always complex orientable. Theories of this type give Okonek formal groups [14].

Since Euler classes are invertible, we have an equivalence

E = E ∧ S^{0} ⟶^{≃} E ∧ S^{∞\overline{ρ}},

where S^{∞\overline{ρ}} = ⋃_{V^{A}=0} S^{V} as before. It follows that for any A-space X, the inclusion X^{A} ⟶ X is a cohomology isomorphism (since the cohomology of any non-fixed cell is zero). This applies to X = ℂP^{∞}, with (ℂP^{∞})^{A} = A^{∗} × ℂP_{ϵ}^{∞}, where the multiplication comes from the group structure in A^{∗} and ℂP_{ϵ}^{∞}. Thus

E_{A}^{∗}(ℂP^{∞}) ≅ ∏_{α} E_{A}^{∗}(ℂP_{ϵ}^{∞}) = E_{A}^{∗}(ℂP_{ϵ}^{∞})^{A^{∗}}.

We should explain the notation: if k is a cogroup in the category of rings, we use the notation k^{A^{∗}} for the object which is a product ∏_{α} k 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 y_{ϵ}

E_{A}^{∗}(ℂP^{∞}) = ∏_{α} E_{A}^{∗}[[y_{α}]]

and the coproduct is determined by the group structure of A^{∗} and its effect on the generator y_{ϵ}.

More generally, 2.2 shows that

E_{A}^{∗}(ℂP (V)) = ∏_{α} E_{A}^{∗}[y_{α}]/(y_{α}^{|V_{α}|}),

so it is easy to understand orientations, x = x(ϵ) = (x_{α})_{α}. An element (f_{α}(y_{α})) ∈ E_{A}^{∗}(ℂP^{∞}) gives an element of E_{A}^{∗}(ℂP^{∞}, ℂP (ϵ)) if f_{ϵ}(y_{ϵ}) has zero constant term. To give a generator of E_{A}^{∗}(ℂP (ϵ ⊕ ϵ), ℂP (ϵ)) the coefficient of y_{ϵ} must be a unit. Finally, to give a generator of E_{A}^{∗}(ℂP (ϵ ⊕ α), ℂP (ϵ)) the constant term in f_{α}(y_{α}) must be a unit. Thus, if Euler classes are given, an orientation is specified by (i) a classical orientation of the classical formal group law E_{A}^{∗}(ℂP_{ϵ}^{∞}) and (ii) for each α ≠ ϵ, an arbitrary power series g_{α}(y_{α}) so that f_{α}(y_{α}) = e(α) + y_{α}g_{α}(y_{α}).

The simplest orientation is x = (y_{ϵ}, (χ(α))_{α≠ϵ}). We use this orientation unless otherwise stated, and thus we have

y(β) = (y_{β}, (χ(βα^{−1}))_{α≠β}),

(i.e. the β’th component of y(β) is y_{β}, and the α’th component is χ(βα^{−1}) if α ≠ β).

Example 9.1. It is perhaps worth a short calculation with A of order 2 to kill certain preconceptions. Let η denote the nontrivial one dimensional representation, and χ = χ(η). We thus have

E_{A}^{∗}(ℂP^{∞}) = E_{A}^{∗}[[y_{ϵ}]] × E_{A}^{∗}[[y_{η}]],

with E_{A}^{∗} acting diagonally on the factors. Note that

y(ϵ) = (y_{ϵ}, χ)

and

y(η) = (χ, y_{η}).

We can thus write y(η) in terms of the ϵ, η, ϵ, η, ϵ, … basis

y(η) = χ · 1 + −1 · y(ϵ) + χ^{−1} · y(ϵ)y(η).

10. Bredon cohomology.

Guided by the non-equivariant case we might have expected to find additive formal 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 determined by its values M(B) := E_{A}^{0}(A/B). From the suspension isomorphisms, the functor A/B ⟼ M(B) is an additive functor on the stable orbit category: a Mackey functor. Bredon has 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 [11].

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 representations 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) = Λ^{B} for an A-module Λ. If V is A-free then H_{A}^{∗}(S(V); M) = H^{∗}(S(V)/A; Λ). Choosing V to be of large dimension we see this is the group cohomology H^{∗}(A; Λ) in a range of degrees. Now consider the cofibre sequence S(V)_{+} ⟶ S^{0} ⟶ S^{V}, and deduce that H_{A}^{∗}(S^{V}; Λ) = ΣH^{∗}(A; Λ) from degree 2 up to to the dimension of V. Thus complex stability implies H^{∗}(A; Λ) is zero in positive degrees. The main examples occur when the order of A is invertible in Λ and when Λ 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-trivial one dimensional complex representation of A it has a kernel K so that A/K is cyclic, generated by c say. We then have two cofibre sequences

A/K_{+} ⟶^{1−c} A/K_{+} ⟶ S(V)_{+} and S(V)_{+} ⟶ S^{0} ⟶ S^{V}

The first gives an exact sequence

0 ⟵ H_{A}^{1}(S(V); M) ⟵ M(K) ⟵^{1−c} M(K) ⟵ H^{0}(S(V); M) ⟵ 0,

and the second shows H_{A}^{1}(S(V); M) ≅ H_{A}^{2}(S^{V}; M) and that there is an exact sequence

0 ⟵ H_{A}^{1}(S^{V}; M) ⟵ H_{A}^{0}(S(V); M) ⟵ M(A) ⟵ H̃_{A}^{0}(S^{V}; M) ⟵ 0

Suppose that H_{A}^{∗}(·; M) is complex stable, so that H_{A}^{∗}(S^{V}) = H_{A}^{∗}(S^{2}). There is thus an exact sequence

0 ⟵ M(A) ⟵ M(K) ⟵^{1−c} M(K) ⟵^{res_{K}^{A}} 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 associated to an A-module Λ by M(B) = Λ^{B}. Furthermore, the discussion of the case Λ above shows 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 orientation.

The definition is motivated by the fact that ℂP^{∞} is an abelian group object (so that E_{A}^{∗}(ℂP^{∞}) has a product and a coproduct) and the inclusion of A^{∗} = ∐_{α} ℂP (α) ⟶ ℂP^{∞} is a group homomorphism. There is also an axiom encoding some of the special properties 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 Π at which it is complete together with

1. a comultiplication

Δ : R ⟶ R ˆ⊗R

which is cocommutative, coassociative and counital,

2. a continuous map

θ : R ⟶ k^{A^{∗}}

of rings compatible with the coproduct

3. A system of elements y(α) ∈ R for α ∈ A^{∗} so that (i) y(α) is regular for each α (ii) R/(y(α)) ≅ k, for each α (iii) Π = (Π_{α}y(α)) and (iv) y(α) = (θ(α) ⊗ id)Δ(y(ϵ)) for α ∈ A^{∗}

If A is a general compact abelian group the ideal Π is replaced by the system of all finite product ideals (Π_{i}y(α_{i})).

By contrast with the classical case, the underlying ring R is not always a power series ring. We shall show in Section 13 that a choice of a complete flag in our universe gives rise to an additive basis of R. Once we have such a basis, we may make calculations, and we develop the necessary machinery for this. The abstract fruit of this is that there is a universal ring for equivariant formal group laws.

Example 11.2. (i) If E_{A}^{∗}(·) is a complex oriented theory, we may obtain an A-equivariant formal group law by taking k = E_{A}^{∗}, R = E_{A}^{∗}(ℂP^{∞}) and y(α) to be constructed from the orientation. (ii) A 1-equivariant formal group law is a formal group law in the classical sense. (iii) Any formal group law over k gives an A-equivariant formal group law over k by taking the map θ to be the composite of the counit and the diagonal k ⟶ k^{A^{∗}}.

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. This 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 flexibility to discuss bases arising from different flags, and have used completely different notation 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 position of decorations in other notation.

Thus the product structure takes the form

y(V^{i})y(V^{j}) = ∑_{s≥0} b_{s}^{i,j}y(V^{s}),

the action of A^{∗} takes the form

l_{α}y(V^{i}) = ∑_{s≥0} d(α)_{s}^{i}y(V^{s}),

and the tensor product is expressed by

t^{∗}y(V^{i}) = ∑_{s,t≥0} f_{s,t}^{i}y(V^{s}) ⊗ y(V^{t}).

This structure is required to be continuous so that we have the following.

Continuity conditions:

1. for fixed i, s the coefficients b_{s}^{i,j} are zero for j sufficiently large, and similarly with i and j exchanged,

2. for fixed α, s the coefficients d(α)_{s}^{i} are zero for i sufficiently large, and

3. for fixed s, t the coefficients f_{s,t}^{i} are zero for i sufficiently large.

One may go on to identify leading terms, but first we list the formal properties. The continuity conditions are necessary in some of them to ensure the sums are finite. We have resisted the temptation to write them out explicitly, but the reader is encouraged to do this at least once. (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 ∘ l_{αβ} = (l_{α} ˆ⊗l_{β}) ∘ t,

3. commutative,

4. associative and

5. unital. There are also two normalization conditions

(Flag) y(α_{j+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.

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 topological ring k{{F }} with a coproduct and a continuous action of A^{∗} satisfying Conditions (R), (A), (T), (Flag), and (Ideal). We also refer to this as an A-equivariant formal group law relative to the flag F.

Example 12.3. (i) It is immediate from our motivation that any complex oriented cohomology theory E_{A}^{∗}(·) gives rise to an (A, F)-equivariant formal group law over E_{A}^{∗} for any complete flag F. (ii) If A is the trivial group, an equivariant formal group law is simply a classical one dimensional commutative formal group law, specified by a coproduct on the power series ring k[[y]].

Remark 12.4. As in the non-equivariant case one may also give a fully coordinate free definition (i.e. a definition without specifying the orientation y(ϵ)): this gives the notion of an equivariant formal group. We defer further discussion to [7].

13. Comparison

We shall show that an A-equivariant formal group law together with a flag F is equivalent to an (A, F)-formal group law. This 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 showing how the structure constants for an (A, F)-formal group law are related to those of an (A, F′)-formal group law.

Lemma 13.1. An (A, F)-formal group law k{{F }} is an A-equivariant formal group law

Proof: For simplicity assume F begins with ϵ. We take R = k{{F }}, and use the elements y(α) = l_{α}y(ϵ) as the notation suggests. Now define Π by Formula 3 (iii) of 11.1. Since F is a complete flag, k{{F }} is Π-complete. The comultiplication is explicitly present in our formulation, and required to have the properties stated in Part 1 of 11.1.

First observe that k{{F }}/(y(ϵ)) ≅ k: it is clear that y(V^{1}), y(V^{2}), … lie in the ideal (y(ϵ)), and our Condition (Ideal) shows that they also span it. Condition 3 (ii) of 11.1 follows for arbitrary α by applying l_{α}.

We show y(ϵ) is regular. Note that 1, y(V^{2}/ϵ), y(V^{3}/ϵ), … gives a splitting of the map

k{{F }} ⟶^{y(ϵ)} (y(ϵ)), so that it suffices to show that k{{F }} is topologically spanned as a kmodule by these elements. This can be done by adapting the proof of Lemma 13.2 below. Condition 3 (i) follows for arbitrary α by applying l_{α}.

We define θ : k{{F }} ⟶ k^{A^{∗}} by taking θ(r)(α) to be the constant coefficient in l_{α}(r). This is a continuous map of rings since l_{α} is. Condition 2 of 11.1 follows from the equivariance of the coproduct. Finally, Condition 3 (iv) is given by taking the coefficients of 1 ⊗ k{{F }} in (l_{α} ⊗ 1) ∘ t^{∗}y(ϵ) = t^{∗}l_{α}y(ϵ): it is y(α) by the counital condition.

Lemma 13.2. If R is an A-formal group law we obtain an (A, F)-formal group law by defining y(V) = y(α_{1})y(α_{2}) ⋯ y(α_{n}) where V = α_{1} ⊕ α_{2} ⊕ ⋯ ⊕ α_{n}.

Proof: The main point is to show that if we choose a complete flag F, the elements 1, y(V^{1}), y(V^{2}), … form a topological basis. Indeed, the elements y(V^{i}) define a continuous function λ : 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 structure. The action l_{α} is given by the composite

R ⟶^{Δ} R ˆ⊗R ⟶^{θ} k^{A^{∗}} ⊗ R ⟶^{π_{α}⊗1} {α} ⊗ R.

It is easy to verify that this is an action and that the coproduct is equivariant.

Surjectivity of λ follows by approximating elements of R by a convergent series in the image of λ. Indeed, by Condition 3 (ii), given s ∈ R we may choose r_{0} ∈ k so that s − r_{0} ∈ (y(α_{1})), say s = r_{0} + y(α_{1})s_{1}. Similarly, s_{1} = r_{1} + y(α_{2})s_{2} and so forth. To establish injectivity suppose λ(Σ_{i}r_{i}y(V^{i})) = 0, and suppose there is a first nonzero coefficient r_{i_{0}}. Thus

λ(Σ_{i}r_{i}y(V^{i})) = y(V^{i_{0}})λ(Σ_{i≥i_{0}}r_{i}y(V^{i}/V^{i_{0}})).

Since y(V^{i_{0}}) is regular this means λ(Σ_{i}r_{i}y(V^{i}/V^{i_{0}})) = 0, but this is a contradiction since reducing mod y(α_{i_{0}+1}) recovers r_{i_{0}}.

Condition (Flag) is built into the definition, as are the continuity conditions. To prove Condition (Ideal) it suffices to show y(α)y(V^{i}) has zero coefficients of y(V^{j}) with j < i. If not

y(α)y(V^{i}) = y(α)y(V^{j})y(V^{i}/V^{j}) = y(V^{j})z

with V^{i}/V^{j} ≠ 0 and z = z_{0} + z_{1}y(V^{j+1}/V^{j}) + ⋯ with z_{0} ≠ 0. Since y(V^{j}) is regular by Condition 3 (i), y(α)y(V^{i}/V^{j}) = z. Now reduce mod y(α_{j+1}), and by Condition 3 (ii), we contradict the fact z_{0} ≠ 0.

This shows that k{{F }} does not depend in an essential way on the flag F, so we allow ourselves to write k{{F }} = k{{𝒰}}. We extend this notation to finite dimensional subspaces V ⊂ 𝒰 by taking k{{V }} = k{{𝒰}}/(y(V)). Finally, we may extend it to infinite dimensional subspaces 𝒰′ ⊆ 𝒰 by taking k{{𝒰′}} to be the inverse limit of the rings k{{V′}} with the inverse limit topology. There is then an induced map res_{𝒰′}^{𝒰} : k{{𝒰}} ⟶ k{{𝒰′}}. We also have a commutative square

13.3.

k{{𝒰}} ⟶ k{{𝒰′δ}}
l_{δ} ↓  ↓ l_{δ}
k{{𝒰}} ⟶ k{{𝒰′}}.

14. The representing ring

It is immediate that an (A, F)-formal group law is uniquely specified by structure constants b_{s}^{i,j}, d(α)_{s}^{i} and f_{s,t}^{i} satisfying the continuity conditions, and so that k{{F }} satisfies Conditions (R), (A), (T), (Flag), and (Ideal). We now want to form the representing ring L_{A}(F) for (A, F)-formal group laws, as the ℤ-algebra with generators b_{s}^{i,j}, d(α)_{s}^{i} and f_{s,t}^{i} subject 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 conditions result from the fact that y(V) generates the ideal of elements restricting to zero on ℂP (V).

• (VRs) The product y(V′)y(V′′) is zero on restriction to ℂP (V′) or ℂP (V′′)

• (VAs) l_{α}y(V) = y(V ⊗ α) vanishes on restriction to ℂP (V ⊗ α)

• (VTs) The coproduct t^{∗}y(V) is zero on restriction to ℂP (W_{1}) × ℂP (W_{2}) if ℂP (W_{1}) × ℂP (W_{2}) 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 (W_{1})^{B} + dim ℂP (W_{2})^{B} ≤ dim ℂP (V)^{B} holds for all subgroups B ⊆ A. It is easy but unilluminating to express this in terms of the representations V, W_{1} and W_{2}.

The reader should now make explicit the vanishing of structure constants these conditions imply in the topological case. For b_{s}^{i,j} and d(α)_{s}^{i} the answer is precisely as in the following proposition. For the tensor product it states that the coefficient f_{s,t}^{i} is zero if dim ℂP (V^{s+1})^{B} +dim ℂP (V^{t+1})^{B} ≤ dim ℂP (V^{i})^{B} holds for all subgroups B ⊆ A. Since 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 explicit vanishing conditions:

• (VR)

b_{s}^{i,j} = 0 if s < i or s < j

• (VA)

d(α)_{s}^{i} = 0 if V^{i} ⊗ α ≥ V^{s+1}

• (VT)

f_{s,t}^{i} = 0 if V^{i} ≥ α_{1}^{−1}V^{s+1} ⊗ V^{t+1}.

Remark 14.2. Note that because the flag F exhausts 𝒰, the proposition gives explicit forms of the continuity conditions of Section 12.

Proof: Note that (VR) is equivalent to (Ideal): indeed, it is clear by (Flag) that all elements y(V^{j}) with j ≥ i lie in the ideal (y(V^{i})). Both conditions are equivalent to requiring 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 αV^{i} ≥ V^{s+1} then l_{α}y(V^{i}) = y(αV^{i}) is a multiple of y(V^{s+1}) by (Flag). We have just observed that (VR) shows that any such element has zero coefficient of y(V^{k}) with k < s + 1.

For (VT) we use the counit condition and the fact that t^{∗} is a ring map, together with (Ideal). For an element u = ∑_{p,q} a_{p,q}y(V^{p}) ⊗ y(V^{q}) we say that u vanishes up to (s, t) if a_{p,q} = 0 if p ≤ s and q ≤ t. We shall find an i_{0} = i_{0}(s, t) so that t^{∗}y(V^{i}) vanishes up to (s, t) whenever i ≥ i_{0}: we do not attempt to find the best possible i_{0}.

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 a_{s+1,j}y(V^{s+1}) ⊗ y(V^{j}) for j ≤ t. Thus multiplying by v_{1} = y(α_{s+2}) ⊗ z or v_{2} = 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 prove the general case by induction on s.

For the inductive step we apply equivariance (l_{α} ⊗ 1) ∘ t^{∗} = t^{∗} ∘ l_{α} to the counit condition, and find

t^{∗}(y(αV)) = 1 ⊗ y(V) + y(αV) ⊗ 1 mod (y(αV^{1}) ⊗ y(V^{1})).

Thus, taking α = α_{s+2}α_{1}^{−1} and W = αV we see that if u vanishes up to (s, t) then t^{∗}(y(W))u vanishes up to (s + 1, t) provided W contains α_{s+2}α_{1}^{−1}V^{t+1}.

We may now proceed to form the representing ring.

Corollary 14.3. There is a representing ring L_{A}(F) for (A, F)-formal group laws, constructed as the ℤ-algebra with generators b_{s}^{i,j}, d(α)_{s}^{i} and f_{s,t}^{i} subject 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′)-formal group law, there is a canonical isomorphism L_{A}(F′) ⟶^{≅} L_{A}(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(α)_{0}^{1} and f_{s,t}^{1}.

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 χ : A^{∗} ⟶ k and we let L_{A}^{strict}(F, k, χ) denote the representing ring when the strict Euler classes are required to agree with χ. Note that this will not necessarily contain k: for example the condition χ(ϵ) = 0 is imposed. Understanding A-equivariant formal group laws where the strict Euler classes are unit multiples of specified Euler classes (ie orientability and orientations of complex stable theories) seems 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 relations 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 subscripts, we have decided to present discussion of the first few terms separately, as a motivation for the general case. The reader may also find it helpful to refer to Appendix B where some calculations 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)_{s}^{i,j} or (b_{F})_{s}^{i,j} 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 little further. We show that to express it in terms of the flag basis, it is sufficient to understand the action of A^{∗} in that basis. The idea is that the path α_{j}, α_{j+1}, α_{j+2}, … also defines 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. Thus we simply express y(V^{i}) in terms of the new basis as y(V^{i}) = Σ_{k}\overline{b}_{j+k}^{i,j}y(V^{j+k}/V^{j}) and then y(V^{i})y(V^{j}) = Σ_{k}\overline{b}_{j+k}^{i,j}y(V^{j+k}). Thus \overline{b}_{s}^{i,j} = b_{s}^{i,j}, explaining the notation. In practice we do this one step at a time, since y(V^{i}) = y(α_{1}) ⋯ y(α_{i}). It therefore suffices to identify y(α)y(V^{j}), for all α and j. For this we note y(α) = l_{αα_{j+1}^{−1}}y(α_{j+1}), and work with respect to the F/V^{j} basis.

We also write F δ for the flag (V^{0}δ ⊂ V^{1}δ ⊂ V^{2}δ ⊂ ⋯) whose associated path is α_{1}δ, α_{2}δ, α_{3}δ, . . .. We refer to F δ as the rotated flag. It is clear by applying l_{δ} to the formula defining d_{F} that

d_{F δ}(αδ)_{s}^{i} = d_{F} (α)_{s}^{i}.

It is thus no loss of generality to work with flags beginning with V^{1} = ϵ, and in this section we assume the flag path begins ϵ, α, β, γ, . . .. We show that for any complete flag F′ the action coefficients d_{F′}(∗)_{∗}^{1} can be deduced from the coefficients d_{F} (∗)_{∗}^{1}. It then follows that all product coefficients (b_{F})_{∗}^{∗,∗} can be deduced, and all the higher d_{F} ’s. Since we are just dealing with the first few terms, we simplify notation; we already know that for r ∈ A^{∗} we have e(r) = d(r)_{0}^{1}, and we define λ(r) = d(r)_{1}^{1}, μ(r) = d(r)_{2}^{1}, ν(r) = d(r)_{3}^{1}, . . .. Thus by definition,

y(r) = e(r) + λ(r)y(ϵ) + μ(r)y(ϵ)y(α) + ν(r)y(ϵ)y(α)y(β) + · · · .

The first observation is that the coefficient d_{F} (r)_{i}^{1} only depends on the flag as far as V^{i+1}. This is because the coefficient can be recovered after reduction modulo y(V^{i+1}) = y(ϵ)y(α_{2}) ⋯ y(α_{i+1}). Accordingly we write e(r), λ_{α}(r), μ_{α,β}(r), … to emphasize this.

We also need the coefficients e_{F}^{n} (r), λ_{F}^{n} (r), μ_{F}^{n} (r), … in l_{r}y(V^{n}), and we show by induction on n that these can be deduced from the coefficients e, λ_{α}, μ_{α,β}, . . .. Indeed y(V^{n+1}) = y(V^{n})y(α_{n+1}), so we have

l_{r}y(V^{n+1}) = e_{F}^{n} (r)y(α_{n+1}r) + λ_{F}^{n} (r)y(V^{1})y(α_{n+1}r) + μ_{F}^{n} (r)y(V^{2})y(α_{n+1}r) + · · · .

The trick for calculating the i’th term is to express y(α_{n+1}r) in the F/V^{i} basis. Of course this uses the coefficients e_{F/V^{i}}, λ_{F/V^{i}}, μ_{F/V^{i}}, . . .. This looks dangerously close to being circular, so we show explicitly that it is not.

Lemma 15.1. The leading term is given by

e_{F}^{n} (r) = e(V^{n}r).

Proof: The proof here and in the following two lemmas is to expand the right hand side of the equation

l_{r}y(V^{n+1}) = l_{r}[y(V^{n})]y(α_{n+1}r).

To do this we first expand l_{r}[y(V^{n})] to obtain

e_{F}^{n} (r)y(α_{n+1}r) + λ_{F}^{n} (r)y(ϵ)y(α_{n+1}r) + μ_{F}^{n} (r)y(ϵ)y(α)y(α_{n+1}r) + ⋯,

Now expand y(α_{n+1}r) with respect to the ϵ, α, β, … basis in the first term, with respect to the α, β, γ, … basis in the second term, with respect to the β, γ, δ, … basis in the third term, and so forth.

One important consequence is that if r^{−1} occurs in V^{n} this leading term vanishes. Since F is a complete flag we thus see e_{F}^{n} (r) = 0 if n is sufficiently large. Note also that for n ≥ 2 this coefficient depends on more than the first subquotient of F.

For the higher terms we cannot expect a closed formula, but a recursive algorithm is quite sufficient. The proofs are precisely like those for Lemma 15.1 above, and the general case is presented in detail in the next section.

Lemma 15.2. The first term is given recursively in terms of Euler classes and λ_{α}(r) by the formula

λ_{F}^{n+1}(r) = e(V^{n}r)λ_{α}(α_{n+1}r) + λ_{F}^{n} (r)e(α^{−1}α_{n+1}r).

We note here that once n is large enough that e(V^{n}r) = 0 the recursion states λ_{F}^{n+1}(r) = λ_{F}^{n} (r)e(α^{−1}α_{n+1}r). Thus if α_{n+1} = αr^{−1}, we obtain zero. Once again we see that λ_{F}^{n} (r) = 0 if n is sufficiently large. Finally, we note that if n ≥ 2 then λ_{F}^{n} (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 and λ_{α}(r) by the formula

μ_{F}^{n+1}(r) = e(V^{n}r)μ_{α,β}(α_{n+1}r) + λ_{F}^{n} (r)λ_{α^{−1}β}(α^{−1}α_{n+1}r) + μ_{F}^{n} (r)e(β^{−1}α_{n+1}r).

The new feature here is that the coefficient λ is associated to a different flag. We had therefore better ensure that we can deduce it before we need to apply this recursive formula.

Suppose then that F′ begins ϵ, ζ, . . .. Thus y(r) = e(r)+λ_{ζ}(r)y(ϵ) modulo y(ϵ)y(ζ). Now, find the first occurence of ζ in F, and express y(r) in terms of the F basis:

y(r) = e(r) + λ_{α}(r)y(ϵ) + μ_{α,β}(r)y(ϵ)y(α) + ⋯;

it is sufficient to work modulo y(ϵ)y(α) ⋯ y(ζ). Now we expand y(α), y(α)y(β), y(α)y(β)y(γ), … in terms of the F′/ϵ basis. In the present case we only seek the coefficient of y(ϵ) so we only need the constant term in the expansions, and this is easily calculated. Thus we obtain

λ_{ζ}(r) = λ_{α}(r) + μ_{α,β}(r)e(αζ^{−1}) + ν_{α,β,γ}(r)e(αζ^{−1})e(βζ^{−1}) + ⋯,

and the sum is finite, since e(ζζ^{−1}) = 0.

Note in particular that this shows that λ_{α}(r) = λ_{ζ}(r) modulo Euler classes, so taking ζ = r we see λ_{α}(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. The discussion is directed towards understanding the universal ring L_{F} (A). Our first task is to give a reasonably efficient set of generators. The reader may also find it helpful to refer to Appendix B where some calculations are done when A is of order 2.

Theorem 16.1. The representing ring L_{F} (A) is generated as an algebra by the Euler classes e(α) and the coefficients f(F)_{j,k}^{1}. The representing ring L_{A}^{strict}(k, F, χ) is generated as a kalgebra by the elements f_{s,t}^{1}.

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 abbreviations.

Process 0. d_{F′}(∗)_{n+1}^{1} ⟵ (d_{F′}(∗)_{n}^{1}, d_{F}^{1} ) (see 16.4). Process 1. f^{n+1} ⟵ (f^{n}, d^{∗}) (from the fact that t^{∗}(y(V^{n+1})) = t^{∗}(y(V^{n}))t^{∗}(y(α_{n+1})). Process 2. d^{n+1} ⟵ (d^{≤n}, d^{1}) (see 16.3). Process 3. d^{n} ⟵ (f^{n}, e) (see 16.7).

Here, Process 0 has been used to avoid specifying the flags used for the d^{1}’s in Process 2. Using Process 3 we obtain the generators d^{1} from f^{1}’s and e’s. Then, using Process 2 recursively, we obtain all generators d^{n}. Finally Process 1 can be used recursively to obtain all the coefficients f^{n}. We have already seen how the coefficients b can be obtained from the d’s. We now turn to the detailed implementation of the strategy.

First, let us write out the definition of the coefficients d_{F} in longhand:

y(r) = d_{F} (r)_{0}^{1} + d_{F} (r)_{1}^{1}y(V^{1}) + d_{F} (r)_{2}^{1}y(V^{2}) + d_{F} (r)_{3}^{1}y(V^{3}) + · · · .

The first observation is that the coefficient d_{F} (r)_{i}^{1} only depends on the flag as far as V^{i+1}. This is because the coefficient is determined by reducing modulo y(V^{i+1}) = y(ϵ)y(α_{2}) ⋯ y(α_{i+1}). Accordingly we write d_{V^{1}}(r)_{0}, d_{V^{2}}(r)_{1}, d_{V^{3}}(r)_{2}, … to emphasize this; note that we have also omitted the superscript 1. By contrast with the previous section we have not normalized the flag to begin with ϵ, so the start of the flag is an essential piece of information.

We begin by summarizing the results of the previous section in the general notation: note that we have rotated the flags to the natural position.

Lemma 16.2. (i) The leading term is given by

d_{F} (r)_{0}^{n} = e(α_{1}^{−1}V^{n}r).

This is zero if n is sufficiently large. (ii) The first term is given recursively in terms of Euler classes and d_{V^{2}}(∗)_{∗}^{1} by the formula

d_{F} (r)_{1}^{n+1} = d_{F} (r)_{0}^{n}d_{V^{2}}(α_{1}^{−1}α_{n+1}r)_{1} + d_{F} (r)_{1}^{n}d_{V^{2}/V^{1}}(α_{2}^{−1}α_{n+1}r)_{0}.

This is zero if n is sufficiently large, and if n ≥ 2 then d(r)_{1}^{n} = 0 modulo Euler classes. (iii) The second term is given recursively in terms of the coefficients d_{F} (∗)_{∗}^{1} by the formula

d_{F} (r)_{2}^{n+1} = d_{F} (r)_{0}^{n}d_{V^{3}}(α_{1}^{−1}α_{n+1}r)_{2}+d_{F} (r)_{1}^{n}d_{V^{3}/V^{1}}(α_{2}^{−1}α_{n+1}r)_{1}+d_{F} (r)_{2}^{n}d_{V^{3}/V^{2}}(α_{3}^{−1}α_{n+1}r)_{0}.

It is not hard to write down the general recursion, and it should now be possible to understand what it means.

Lemma 16.3.

d_{F} (r)_{k}^{n+1} = d_{F} (r)_{0}^{n}d_{V^{k+1}}(α_{1}^{−1}α_{n+1}r)_{k} + d_{F} (r)_{1}^{n}d_{V^{k+1}/V^{1}}(α_{2}^{−1}α_{n+1}r)_{k−1} +

d_{F} (r)_{2}^{n}d_{V^{k+1}/V^{2}}(α_{3}^{−1}α_{n+1}r)_{k−2}+⋯+d_{F} (r)_{k−1}^{n}d_{V^{k+1}/V^{k−1}}(α_{k}^{−1}α_{n+1}r)_{1}+d_{F} (r)_{k}^{n}d_{V^{k+1}/V^{k}}(α_{k+1}^{−1}α_{n+1}r)_{0}

Proof: First note that l_{r}y(V^{n+1}) = [l_{r}y(V^{n})]y(α_{n+1}r). When calculating the term d_{F} (r)_{i}^{n}y(V^{i})y(α_{n+1}r), we need to express y(α_{n+1}r) in terms of the F/V^{i} flag, so the jth coefficient is d_{F/V^{i}}(α_{i+1}^{−1}α_{n+1}r)_{j}^{1} = d_{V^{i+j+1}/V^{i}}(α_{i+1}^{−1}α_{n+1}r)_{j}.

Now, suppose by induction that all coefficients d_{F′}(∗)_{l}^{1} with l < k and all coefficients d_{F} (∗)_{l}^{∗} with l ≤ k can be expressed in terms of the coefficients d_{F} (∗)_{∗}^{1}. Noting that the only occurence of d(∗)_{k}^{1} on the left hand side uses the F basis, the lemma shows that the coefficients d_{F} (∗)_{k}^{n+1} can also be so expressed.

It thus follows by induction that if all coefficients d_{F′}(∗)_{l}^{1} with l < k can be expressed in terms of the coefficients d_{F} (∗)_{l}^{1} with l ≤ k then all coefficients d_{F} (∗)_{l}^{∗} with l ≤ k can be so expressed.

The following lemma completes the justification of Process 0.

Lemma 16.4. The coefficient d_{F′}(r)_{k}^{1} can be expressed in terms of the coefficients d_{F′}(∗)_{l}^{1} with l < k and the coefficients d_{F} (∗)_{∗}^{1}.

Proof: By rotation we may suppose that F′ also begins with α_{1}. Now d_{F′}(r)_{k}^{1} = d_{(V′)^{k+1}}(r)_{k} is the coefficient of y((V′)^{k}) in the F′ expansion of y(r). We will give a way of calculating this coefficient in terms of the d_{F} (∗)_{∗}^{1}’s. Indeed, we also have y(r) = ∑_{i} d_{F} (r)_{i}^{1}y(V^{i}). Furthermore y(V^{i}) = y(α_{1})y(V^{i}/V^{1}), and we may express y(V^{i}/V^{1}) in terms of the F′/V^{1} basis, and the coefficient of y((V′)^{k}/V^{1}) will give us the contribution to the coefficient in question. Furthermore, the contribution from y(V^{i}) is zero once V^{i} contains (V′)^{k+1}, since y(V^{i}) is then already zero modulo y((V′)^{k+1}); thus the number of terms is finite. It thus suffices to apply the following lemma with W = V^{i}/V^{1} and F′′ = F′/V^{1}.

Lemma 16.5. If we write y(W) in the F′′ basis as y(W) = ∑_{l}[F′′]_{i}^{W} y((V′′)^{i}) then the coefficients [F′′]_{l}^{W} may be expressed in terms of coefficients d_{V} (∗)_{m} with m ≤ l.

Proof: The proof is by induction on the dimension of W. If W = 0 the result is trivial. The inductive step is to write W = W′ ⊕ β. Then we have y(W) = y(W′)y(β) = ∑_{i}[F′′]_{i}^{W′}y((V′′)^{i})y(β), and in the ith term we write y(β) in terms of the F′′/(V′′)^{i} basis, using the coefficients d_{F′′/(V′′)^{i}}(β(α′′_{i+1})^{−1})_{j}^{1}. The contributing coefficients have i + j = l, so both i, j ≤ l.

Corollary 16.6. Modulo Euler classes we have d_{F} (r)_{1}^{1} = 1, and d_{F} (r)_{k}^{1} = 0 if k ≠ 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 prove that d_{F} (r)_{k}^{1} = d_{F′}(r)_{k}^{1} for any flag F′, and hence by taking F′ to begin with r, that both vanish.

First note by induction, in any basis F′′

y(r) = y(s) + d_{F′′}(sr^{−1})_{k}^{1}y((V′′)^{k}) + higher terms,

so that y(V) = y(W) if dim(V) = dim(W) < k. Now take any two complete flags F, and F′. By rotation we assume that both F and F′ begin with ϵ. Expanding y(r) in the F′ basis we have

y(r) = y(ϵ) + d_{F′}(r)_{k}^{1}y((V′)^{k}) + higher terms

Similarly, in the F basis we have

y(r) = y(ϵ) + d_{F} (r)_{k}^{1}y(V^{k}) + higher terms

However y(V^{k}) = y(ϵ)y(V^{k}/ϵ), and by the observation, y(V^{k}/ϵ) = y((V′)^{k}/ϵ) modulo y((V′)^{k+1}/ϵ). Similarly y(V^{k+i}) = y(ϵ)y(V^{k}/ϵ)y(V^{k+i}/V^{k}) = 0, modulo y((V′)^{k+1}/ϵ). Thus d_{F} (r)_{k}^{1} = d_{F′}(r)_{k}^{1} as required.

Finally we see that the equivariance of t^{∗} allows us to deduce the coefficients d_{F} (α)_{∗}^{1} from the coproduct and the Euler classes.

Lemma 16.7. The following formula gives the action coefficients in terms of the cogroup coefficients and the Euler classes

d_{F} (β)_{k}^{i} = ∑_{j} f_{j,k}^{i}e(βV^{1} ⊗ V^{j}).

Note that the j = 0 term is f_{0,k}^{i} = δ_{k}^{i}.

Proof: Firstly we see from equivariance of t^{∗} that

l_{β} = (res_{β^{−1}}^{𝒰} ˆ⊗1)t^{∗}.

Applying this to y(V^{i}), the coefficient of y(V^{k}) on the left is d_{F} (β)_{k}^{i}. To identify the coefficient of y(V^{k}) on the right we recall that by definition res_{β^{−1}}^{𝒰} = res_{V^{1}}^{𝒰}l_{βV^{1}}, and calculate

(res_{β^{−1}}^{𝒰} ˆ⊗1)t^{∗}y(V^{i}) = (res_{V^{1}}^{𝒰}l_{βV^{1}} ˆ⊗1) ∑_{j,k} f_{j,k}^{i}y(V^{j}) ⊗ y(V^{k})
= (res_{V^{1}}^{𝒰} ˆ⊗1) ∑_{j,k} f_{j,k}^{i} ∑_{l} d(βV^{1})_{l}^{j}y(V^{l}) ⊗ y(V^{k})
= ∑_{j,k} f_{j,k}^{i}d(βV^{1})_{0}^{j}y(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 is 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(ϵ)) = y′(ϵ) + y′′(ϵ)

and the multiplicative law by

t_{m}^{∗}(y(ϵ)) = y′(ϵ) + y′′(ϵ) − y′(ϵ)y′′(ϵ)

The instructive thing here is how this imposes restrictions on the Euler classes. 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 [6] suggest 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 ϵ. Thus the additive case has f_{1,0}^{1} = f_{0,1}^{1} = 1 and f_{j,k}^{1} = 0 otherwise. Thus, from 16.7 we see

y(β) = e(β) + y(ϵ).

Rotating by α it follows that e(αβ) = e(α) + e(β), ie that e : A^{∗} ⟶ k is a group homomorphism, and in particular if β^{n} = ϵ we see that ne(β) = 0. This identifies the universal ring for additive equivariant formal group laws.

Proposition A.1. The free commutative ring Symm(A^{∗}) on the abelian group A^{∗} is universal for additive formal group laws. If A is a torus, this is the coefficient ring of ordinary Borel cohomology.

The multiplicative case has f_{1,0}^{1} = f_{0,1}^{1} = 1, f_{1,1}^{1} = −1 and f_{j,k}^{1} = 0 otherwise. Thus, from 16.7 we see

y(β) = e(β) + [1 − e(β)] y(ϵ).

Rotating by α shows that (1 − e(αβ)) = (1 − e(α))(1 − e(β)), ie that 1 − e : A^{∗} ⟶ k^{×} 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 universal for multiplicative formal group laws.

Appendix B. The group of order 2

In this appendix we make some of the formulae explicit for the group A = C_{2} of order 2. We use the flag with alternating subquotients ϵ, η, ϵ, η, ϵ, η, ϵ, η, . . .. Since there is only one non-trivial element of A^{∗} we write simply d_{j}^{i} = d(η)_{j}^{i}, and continue with the convention that we drop i if i = 1. It is convenient to declare d_{j}^{i} = 0 if j < 0. As we give formulae 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 = d_{0} as deformation parameter [7].

First note that since

y(η) = d_{0} + d_{1}y(ϵ) + d_{2}y(ϵ)y(η) + ⋯

by rotation we have

y(ϵ) = d_{0} + d_{1}y(η) + d_{2}y(η)y(ϵ) + ⋯,

and thus, premultiplying both sides by y(ϵ) we have an expression for y(ϵ)^{2} in the standard basis:

b_{k}^{1,1} = d_{k−1}.

Thus

k{{𝒰}} = k[[x]][y]/(y^{2} = p(x) + yq(x))

where y = y(ϵ), x = y(ϵ)y(η), and where p(x) = d_{1}x + d_{3}x^{2} + d_{5}x^{3} + ⋯ and q(x) = e + d_{2}x + d_{4}x^{2} + · · ·. Now consider y(V^{i})y(V^{j}); if i or j is even, it is clearly y(V^{i+j}), and if both are odd it is y(V^{i−1})y(V^{j−1})y(ϵ)^{2} = y(V^{i+j−2})y(ϵ)^{2}. Combining these into a single formula we have

b_{k}^{i,j} = d(η^{ij})_{k−i−j+1}.

For d_{i}^{n} note that if n is even, l_{η}y(V^{n}) = y(V^{n}), so that d_{i}^{n} = δ_{i}^{n}. The same observation when n is odd gives d_{k}^{n} = d_{k−n+1}. Also, the fact that l_{η} = (res_{η} ⊗ 1) ∘ t^{∗} gives

d_{k} = δ_{k}^{1} + ef_{1,k}^{1}.

Altogether we have

d_{k}^{n} = \{ \begin{matrix}δ_{k}^{n} & if n is even \\ δ_{k}^{n} + ef_{1,k−n+1}^{1} & if n is odd \end{matrix}

In a single formula

d_{k}^{n} = δ_{k}^{n} + e(η^{n})f_{1,k−n+1}^{1},

and

b_{k}^{i,j} = δ_{k}^{i+j} + e(η^{ij})f_{1,k−i−j+1}^{1}.

We can obtain useful relations from the statement l_{η}l_{η} = 1 (or alternatively by using the recursive formula for d^{n+1}). Thus

y(η) = (d_{0} + d_{1}y(ϵ)) + y(V^{2})(d_{2} + d_{3}y(ϵ)) + y(V^{4})(d_{4} + d_{5}y(ϵ)) + ⋯,

and applying l_{η} again

y(ϵ) = (d_{0} + d_{1}d_{0})y(V^{0}) + d_{1}d_{1}y(V^{1}) + d_{1}d_{2}y(V^{2}) + d_{1}d_{3}y(V^{3}) + ⋯
+ (d_{2} + d_{3}d_{0})y(V^{2}) + d_{3}d_{1}y(V^{3}) + d_{3}d_{2}y(V^{4}) + d_{3}d_{3}y(V^{5}) + ⋯
+ (d_{4} + d_{5}d_{0})y(V^{4}) + d_{5}d_{1}y(V^{5}) + d_{5}d_{2}y(V^{6}) + d_{5}d_{3}y(V^{7}) + ⋯
+ (d_{6} + d_{7}d_{0})y(V^{6}) + d_{7}d_{1}y(V^{7}) + d_{7}d_{2}y(V^{8}) + d_{7}d_{3}y(V^{9}) + ⋯

Comparing coefficients of y(V^{0}) and y(V^{1}) we find

d_{0} + d_{1}d_{0} = 0

and

d_{1}d_{1} = 1.

Similarly comparing coefficients of y(V^{2n}) and y(V^{2n+1}) for n ≥ 1 we find

d_{1}d_{2n} + d_{3}d_{2n−2} + ⋯ + d_{2n+1}d_{0} + d_{2n} = 0

and

d_{1}d_{2n+1} + d_{3}d_{2n−1} + ⋯ + d_{2n+1}d_{1} = 0.

Writing the first two in terms of f_{1,∗}^{1} we obtain

e(2 + ef_{1,1}^{1}) = 0

and

(1 + ef_{1,1}^{1})^{2} = 1.

The first of these allows us to omit e(2+ef_{1,1}^{1})f_{1,2n}^{1} from the subsequent even terms to obtain

e^{2} [f_{1,3}^{1}f_{1,2n−2}^{1} + f_{1,5}^{1}f_{1,2n−4}^{1} + ⋯ + f_{1,2n+1}^{1}f_{1,0}^{1}] = 0

and

2e(1 + ef_{1,1}^{1})f_{1,2n+1}^{1} + e^{2} [f_{1,3}^{1}f_{1,2n−1}^{1} + f_{1,5}^{1}f_{1,2n−3}^{1} + ⋯ + f_{1,2n−1}^{1}f_{1,3}^{1}] = 0.

We really want to examine the consequences of these relations e-adically and with e inverted. As a first step, note that mod e^{2} the first states 2e = 0, and the rest are all consequences. Mod 2 we the first reads e^{2}f_{1,1}^{1} = 0, and the second follows from it. The remaining even terms read

e^{2} [f_{1,3}^{1}f_{1,2n−2}^{1} + f_{1,5}^{1}f_{1,2n−4}^{1} + ⋯ + f_{1,2n+1}^{1}f_{1,0}^{1}] = 0

and the remaining odd terms are trivial if 2n + 1 = 3 mod 4 and gives

e^{2}(f_{1,2m+1}^{1})^{2} = 0

in the remaining case (where 2n + 1 = 4m + 1). If we instead invert e, the first relation states that ef_{1,1}^{1} = −2, the second is a consequence. The relation from y(V^{2}) gives f_{1,3}^{1} = 0, and the successive relations from y(V^{4}), y(V^{6}), … allow us to deduce f_{1,2n+1}^{1} = 0 for n ≥ 1. The relation from y(V^{2n+1}) is automatically satisfied.

To obtain the recursive formula for f^{n+1} we use t^{∗}y(V^{n+1}) = t^{∗}y(V^{n})t^{∗}y(α_{n+1}), and there are then two cases, since α_{n+1} is ϵ or η depending on whether n + 1 is odd or even. The first case is straightforward, but the second uses t^{∗}(y(η)) = (1 ⊗ l_{η})(t^{∗}(y(ϵ)); together these give the formulae

f_{s,t}^{n+1} = \{ \begin{matrix}∑_{i,j,k,l} f_{i,j}^{n}f_{k,l}^{1}d(η^{ik})_{s−i−k+1}d(η^{jl})_{t−j−l+1} & if n + 1 is odd \\ ∑_{i,j,k,l,m} f_{i,j}^{n}f_{k,l}^{1}d(η^{ik})_{s−i−k+1}d_{m}^{l}d(η^{jm})_{t−j−m+1} & if n + 1 is even \end{matrix}

Combining this with our expressions for the d’s we obtain

f_{s,t}^{n+1} = ∑_{i,j,k,l,m} f_{i,j}^{n}f_{k,l}^{1} [δ_{s}^{i+k} + e(η^{ik})f_{1,s−i−k+1}^{1}] [δ_{m}^{l} + e(η^{n})f_{1,m−l+1}^{1}] [δ_{t}^{j+m} + e(η^{jm})f_{1,t−j−m+1}^{1}] .

It is perhaps worth making explicit the structure of the additive and multiplicative laws in this case. For an additive law, the action is given by

l_{η}y(V^{n}) = y(V^{n}) + e(η^{n})y(V^{n−1})

and the product is given by

y(V^{i})y(V^{j}) = y(V^{i+j}) + e(η^{ij})y(V^{i+j−1}).

Let x = y(V^{2}) = y(ϵ)y(η), and y = y(ϵ). The ring k{{F }} is the x-adic completion of the ring k[x, y]/(y^{2} = x + ey) = k[y]. The action by A^{∗} is given by y ⟼ e + y, which 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

l_{η}y(V^{n}) = [1 − e(η^{n})] y(V^{n}) + e(η^{n})y(V^{n−1})

and the product is given by

y(V^{i})y(V^{j}) = [1 − e(η^{ij})] y(V^{i+j}) + e(η^{ij})y(V^{i+j−1}).

Let x = y(V^{2}) = y(ϵ)y(η), and y = y(ϵ). The ring k{{F }} is the x-adic completion of k[x, y]/(y^{2} = (1 − e)x + ey). The action by A^{∗} is given by y ⟼ 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. The completion is with respect to x = y(e + (1 − e)y). We may let z = 1 − y, and it is reasonable to view 1 − e as the image of η in k, under the classifying map R(A) ⟶ k so that we obtain the completed ring k[z]_{(1−z)(1−ηz)}^{∧} in the form familiar from K-theory.

References

[1] J.F.Adams “Stable homotopy and generalised homology.” Chicago University Press (1974)

[2] G.E.Bredon “Equivariant cohomology theories” Lecture notes in maths. 34 (1972) Springer-Verlag.

[3] M.Cole “Complex oriented RO(G)-graded equivariant cohomology theories and their formal group laws.” Thesis, University of Chicago (1996)

[4] M.Cole “Complex oriented equivariant cohomology theories and equivariant complex projective spaces” (In preparation)

[5] T.tom Dieck “Bordism of G-manifolds and integrality theorems.” Topology 9 (1970) 345-358

[6] T.tom Dieck “Characteristic numbers of G-manifolds II” J. Pure and Applied Algebra 4 (1974) 31-39 [7] J.P.C.Greenlees “Equivariant formal groups: the geometric approach.” (In preparation)

[8] J.P.C.Greenlees “The coefficient ring of equivariant bordism is the universal ring for equivariant formal group laws.” (In preparation)

[9] J.P.C.Greenlees and N.P.Strickland “Varieties and local cohomology for chromatic group cohomology rings”. (Submitted for publication)

[10] M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel. Generalised group characters and complex oriented cohomology theories. Preprint, (various editions, since late 1980’s).

[11] G.Lewis, J.P.May and J.E.McClure “Ordinary RO(G)-graded cohomology” Bull American Math. Soc. 4 (1981) 208-212 [12] G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) “Equivariant stable homo-

topy theory” Lecture notes in maths. 1213 (1986) Springer-Verlag

[13] C.Okonek “Der conner-Floyd-Isomorphismus für Abelsche Gruppen” Math. Z. 179 (1982) 201-212.

[14] C.Okonek “Die äquivariante unitäre Kobordimentheorie und ihre formale Gruppe.” Arch. Math. 40 (1983) 208-220.

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