THE COEFFICIENT RING OF EQUIVARIANT HOMOTOPICAL BORDISM CLASSIFIES EQUIVARIANT FORMAL GROUP LAWS OVER

NOETHERIAN RINGS.

J.P.C.GREENLEES

Contents

1. Background. 1 2. Statement of results. 2 3. Outline. 3 4. Algebraic recollections. 4 5. Preliminaries. 5 6. The Strategy. 6 7. The pullback square for MU_{A}^{∗}. 8 8. Implications of topological localization and completion theorems. 9 9. At the Euler classes. 11 10. Equivariant Homotopical bordism. 13 11. Away from Euler classes. 15 References 19

1. Background.

The best behaved cohomological functors E_{A}^{∗}(·) of A-spaces for an abelian compact Lie group A are the complex orientable cohomology theories. These are the cohomology theories which behave as well as possible for complex line bundles. The archetypal complex orientable theory is tom Dieck’s homotopical bordism MU_{A}^{∗}(·) [6], and it is shown in [3] that it is topologically universal for complex oriented theories in the sense that a complex orientations on E correspond to multiplicative maps MU ⟶ E.

The notion of an A-equivariant formal group law was introduced in [2] to codify the algebraic behaviour of complex orientable cohomology theories. Indeed, if E_{A}^{∗}(·) is complex oriented, the ring E_{A}^{∗}(ℂP^{∞}) naturally has the structure of an equivariant formal group law. It is the purpose of the present paper to provide evidence that MU_{A}^{∗} is algebraically universal for equivariant formal group laws. Once this is proved, we may begin to erect topological counterparts of the theory of equivariant formal group laws in the study of complex oriented cohomology theories, as has been done so effectively, following Morava’s lead, in the nonequivariant context as a result of Quillen’s theorem [15]; this is surveyed by Ravenel [17, 18].

On our way to these results, we prove a number of results about equivariant bordism and equivariant formal group laws that may be of independent interest.

The author is grateful to the CRM, Barcelona where the penultimate stage of work on this paper was begun.

2. Statement of results.

To be more precise, we begin by recalling the definition of an equivariant formal group law. First we need the ring theoretic counterpart of the discrete group A^{∗} = Hom(A, S^{1}). We let k^{A^{∗}} denote the ring of k-valued functions on A^{∗}. If A is finite, this is a Hopf algebra over k using the group multiplication of A^{∗} to give the coproduct, and the inclusion of the identity element in A^{∗} to give the counit. If A is not finite k^{A^{∗}} is topologized as a product of copies of k and becomes a complete topological Hopf algebra. Definition 2.1. ([2]) If A is a finite abelian group, an A-equivariant formal group law over a commutative ring k is

(Afgl1): a complete topological Hopf k-algebra R with

(Afgl2): a homomorphism θ : R ⟶ k^{A^{∗}} of Hopf k-algebras whose kernel defines the

topology together with (Afgl3): an element y(ϵ) ∈ R which is (i) regular and (ii) generates the kernel of the

ϵth component, θ_{ϵ} of θ. Remark 2.2. If A is a general abelian compact Lie group the definition is the same except that in (Afgl2), R is required to be complete with respect to the system of principal ideals generated by finite products of Euler classes, and θ is required to be continuous.

It is shown in [2] that given any complete flag 0 = V^{0} ⊂ V^{1} ⊂ V^{2} ⊂ V^{3} ⊂ ⋯ in a complete A-universe 𝒰, there is an additive topological basis 1 = y(V^{0}), y(V^{1}), y(V^{2}), … of R, and we therefore write R = k{{𝒰}}. We may therefore express the structure maps in terms of this basis, so that it is specified by structure constants. It then follows quite easily [2] that there is a universal ring L_{A} classifying A-equivariant formal group laws. We refer to the paper [2] repeatedly for definitions and terminology.

Equivariant complex cobordism has a natural complex orientation, and therefore MU_{A}^{∗}(ℂP (𝒰)) has the structure of an equivariant formal group law. This is therefore classified by a ring homomorphism

ν : L_{A} ⟶ MU_{A}^{∗}.

The purpose of this paper is to prove the following counterpart of Quillen’s Theorem [15]. Theorem 2.3. For any finite abelian group A, the ring MU_{A}^{∗} classifies A-equivariant formal group laws over Noetherian rings. More precisely, the map ν is surjective and the kernel is Euler torsion, Euler-divisible and ℤ-torsion.

This provides evidence for a conjecture. Conjecture 2.4. The map ν : L_{A} ⟶ MU_{A}^{∗} is an isomorphism for any abelian compact Lie group A.

The conjecture follows from the special case in which A is a torus, and I have a strategy for approaching that case. The method of the present paper is restricted to finite groups since it works by induction, assuming the result for proper quotients of A, with Quillen’s theorem giving the base of the induction. In order to use the present method to prove the conjecture, it would be necessary to gain control of the Euler-power torsion in L_{A}; the advantage of the torus case is that the Euler classes are regular (at least in MU_{A}^{∗}).

By contrast with the non-equivariant case, neither MU_{A}^{∗} nor L_{A} is well understood. However we will show (Theorem 6.1) that MU_{A}^{∗} is determined by the information arising at various fixed point sets, and that the corresponding statment for L_{A} is close to being true. The comparison map at each fixed point set can be seen to be an isomorphism by induction, Quillen’s theorem and an analysis of normal data. This approach has also been used for studying MU_{A}^{∗} when A is of prime order by tom Dieck [6] and Kriz [12].

3. Outline.

Before proceeding, we outline the structure of the argument. The case of the group of prime order is particularly accessible because the subgroup lattice is so simple, and it is sufficient to work with principal ideals. Accordingly, we use this case to explain the idea of proof.

Given any ring k and principal ideal I = (e) we may form the Hasse square

k ⟶ k[1/e]
↓  ↓
k_{(e)}^{∧} ⟶ k_{(e)}^{∧}[1/e]

If k has bounded e-torsion (for instance if k is Noetherian) this is a pullback square. It is also a pullback for k = MU_{A}^{∗} when e is the Euler class of a non-trivial one dimensional representation, by the completion theorem for A-equivariant bordism [14, 5] and the fact MU_{A}^{∗} is concentrated in even degrees. From [7] it is immediate that if H_{1}^{(e)}(k) = 0, then there is a pullback square

k ⟶ k[1/e]
↓  ↓
H_{0}^{(e)}(k) ⟶ H_{0}^{(e)}(k)[1/e]

where H_{∗}^{(e)}(k) is the local homology group [10] (which calculates the left derived functors of completion in favourable cases). In general k is a square zero extension of the pullback ring k′.

Evidently ν induces a map of pullback squares, and to check ν gives an isomorphism L′_{A} ≅ MU_{A}^{∗} of pullbacks it suffices to show that it induces an isomorphism at the top right and the bottom two positions in the square. Since inverting e is exact, it therefore suffices to show (1)

H_{0}^{(e)}(ν) : H_{0}^{(e)}(L_{A}) ⟶^{≅} H_{0}^{(e)}(MU_{A}^{∗})

is an isomorphism and (2)

ν[1/e] : L_{A}[1/e] ⟶^{≅} MU_{A}^{∗}[1/e]

is an isomorphism.

The strategy of proof is the same for (1) and (2). Step1: The target is known for geometric reasons (the localization theorem in the first case, and the completion theorem in the second), and this provides a candidate for the domain. Step 2: The domain is characterized by a universal property. Step 3: The candidate can be seen to enjoy the universal property. This shows the map is between isomorphic rings. Step 4: Understand enough about the map to see it is an isomorphism.

The main work here for both cases is in Step 3. We will return to give the proofs of (1) (in Section 9) and (2) (in Section 11) when we have established the formal framework for treating arbitrary abelian groups.

4. Algebraic recollections.

In this section we recall some standard constructions in commutative algebra from [7]. We suppose given a graded ring k and a finitely generated ideal I = (x_{1}, x_{2}, …, x_{n}). We shall only refer to homogeneous elements of graded objects.

We may define the flat stable Koszul complex

K(I) = (k ⟶ k[1/x_{1}]) ⊗ (k ⟶ k[1/x_{2}]) ⊗ ⋯ ⊗ (k ⟶ k[1/x_{n}])

and its projective replacement P K(I). Both K(I) and P K(I) are graded as a cochain complex non-zero in cohomological degrees 0 to n. Any cochain complex C^{•} can be viewed as a chain complex by C_{i} = C^{−i} or vice versa in the usual way.

It is easy to check that up to quasi-isomorphism K(I) is independent of the chosen generators, and only depends on the radical of I. There is a natural augmentation K(I) ⟶ k, and a fibre sequence

K(I) ⟶ k ⟶ Č(I),

which defines the Čech complex Č(I). Local homology and cohomology are defined by

H_{∗}^{I}(M) = H_{∗}(Hom(P K(I), M)) and H_{I}^{∗}(M) = H_{∗}^{I}(K(I) ⊗ M)).

The local Tate homology is defined by

Ĥ_{∗}^{I}(M) = H_{∗}(Hom(P K(I), M)) ⊗ Č(I)).

Note that it is immediate that local homology, local cohomology and local Tate cohomology only depend on the radical of I. In well behaved cases (for example if k is Noetherian), local cohomology calculates right derived functors of the I-power torsion functor M ⟼ Γ_{I}(M) = H_{I}^{0}(M) and local homology calculates the left derived functors of the (not necessarily right exact) I-adic completion functor M ⟼ M_{I}^{∧}. However, we emphasize that for us these statements are theorems and not definitions.

The definitions immediately give certain useful spectral sequences: Lemma 4.1. For any finitely generated ideals I, J there is a spectral sequence

H_{∗}^{I}(H_{∗}^{J}(M)) ⇒ H_{∗}^{I+J}(M)

for the composite of local homology. □

Taking I = J this shows that H_{0}^{I} is an idempotent functor, together with a completeness statement. Corollary 4.2. For any n ≥ 0 and any module M, the local homology group H_{n}^{I}(M) is HI-complete in the sense that the natural map H_{n}^{I}(M) ⟶^{≅} H_{0}^{I}(H_{n}^{I}(M)) is an isomorphism. Proof: The natural map together with the edge homomorphism of the spectral sequence show that the natural map is a split epimorphism. Hence H_{0}^{I}(H_{n}^{I}(·)) gives an exact sequence of δ-functors. Since we have equality H_{0}^{I}(H_{0}^{I}(M)) ≅ H_{0}^{I}(M) when n = 0, the equality in general follows. □

We shall also need a method for calculating the local Tate homology. Lemma 4.3. For any finitely generated ideal ideal I there is a spectral sequence

Ȟ_{I}^{∗}H_{∗}^{I}(M) ⇒ Ĥ_{∗}^{I}(M). □

We shall also need the following easy consequence of the definition. Lemma 4.4. (i) There is an exact sequence

0 ⟶ Ȟ_{I}^{0}(M) ⟶ ∏_{i} M[1/x_{i}] ⟶ ∏_{i<j} M[1/(x_{i}x_{j})].

(ii) Local cohomology and Čech cohomology are related by the exact sequence

0 ⟶ H_{I}^{0}(M) ⟶ M ⟶ Ȟ_{I}^{0}(M) ⟶ H_{I}^{1}(M) ⟶ 0

and the isomorphism

Ȟ_{I}^{i}(M) ≅ H_{I}^{i+1}(M) for i ≥ 1. □

We also recall the algebraic Tate diagram [7]

M ⊗ K(I)  ⟶  M  ⟶  M ⊗ Č(I)
≃↓  ↓  ↓
Hom(P K(I), M) ⊗ K(I) ⟶ Hom(P K(I), M) ⟶ Hom(P K(I), M) ⊗ Č(I).

The important fact for us is the left hand equivalence, which arises from the fact that K(I) ⟶ k, and hence E = Hom(k, M) ⟶ Hom(P K(I), M), are equivalences over V (I). We thus have a homotopy pullback square of chain complexes

\begin{matrix}M & ⟶ & M ⊗ Č(I) \\ ↓ & ^{•}□ & ↓ \\ Hom(P K(I), M) & ⟶ & Hom(P K(I), M) ⊗ Č(I). \end{matrix}

This gives a long exact Mayer-Vietoris sequence, including the sequence

0 ⟶ H_{1}^{I}(M) ⟶ Ĥ_{1}^{I}(M) ⟶ M ⟶ Ȟ_{I}^{0}(M) ⊕ H_{0}^{I}(M) ⟶ Ĥ_{0}^{I}(M) ⟶ 0.

5. Preliminaries.

We begin by choosing some notation. Let us suppose that A is of rank r, and that

A^{∗} = ⟨α_{1}, α_{2}, …, α_{r}|α_{1}^{n_{1}} = α_{2}^{n_{2}} = ⋯ = α_{r}^{n_{r}} = 1⟩.

The case with A infinite of dimension d is included by taking n_{1} = n_{2} = ⋯ n_{d} = ∞, so that α_{1}, α_{2}, …, α_{d} generate a free abelian summand of rank d. Note that for any m the group A acts freely on the product S(mα_{1}) × S(mα_{2}) × ⋯ × S(mα_{r}) of unit spheres.

Next, we need to introduce some ideals. Firstly, for both MU_{A}^{∗} and L_{A} we have the augmentation ideals J(MU_{A}^{∗}) = ker(MU_{A}^{∗} ⟶ MU) or J(L_{A}) = ker(L_{A} ⟶ L). However it can be hard to get control over the augmentation ideal, so we use ideals generated by Euler classes. Since Euler classes in L_{A} and MU_{A}^{∗} correspond under the map ν by definition, it is enough to consider the Euler class ideal in L_{A}. We shall work with I = (e(α_{1}), e(α_{2}), …, e(α_{r})) and I′ = (e(α) | α ∈ A^{∗}) thus

I ⊆ I′ ⊆ J(L_{A}).

Note that I is finitely generated, even when A is infinite.

Lemma 5.1. We have equality: I = I′. Proof: For an A-equivariant formal group law, the structure map θ is a map of Hopf algebras. More precisely, we evaluate the equality Δ(θ(y(ϵ))) = (θ ⊗ θ)(Δ(y(ϵ))) on α_{1}^{−1}α_{2}^{−1} to deduce that e(α_{1}α_{2}) lies in the ideal generated by e(α_{1}) and e(α_{2}). □

Remark 5.2. (i) When A is cyclic, the Gysin sequence shows that for MU_{A}^{∗} the ideal I is also equal to J(MU_{A}^{∗}). (ii) Since A acts freely on a product of r spheres, the results of [11] show that when treating MU_{A}^{∗}, any ideal I′′ in J(MU_{A}^{∗}) containing I gives the same local homology and cohomology. (iii) The results of [8] give similar conclusions for any Noetherian cohomology theory.

We may now proceed to describe a number of basic facts about the rings MU_{A}^{∗} and L_{A}. For MU_{A}^{∗} these are already known, and we just supply the appropriate references. On a historical note, the papers of Löffler are announcements; Comezana and May have filled this gap in the literature by publishing complete proofs. Theorem 5.3. (Löffler [14], Comezana [4]) The coefficient ring MU_{A}^{∗} is in even degrees. □

Combining the Completion theorem of Löffler and Comezana-May with that of Greenlees-May, we obtain a regularity condition. Theorem 5.4. (Löffler [14], Comezana-May [5], Greenlees-May [11]) The ring MU_{A}^{∗} is tame in the sense that the natural map H_{0}^{I}(MU_{A}^{∗}) ⟶^{≅} (MU_{A}^{∗})_{I}^{∧} is an isomorphism and H_{i}^{I}(MU_{A}^{∗}) = 0 for i ≥ 1. The completion theorem holds in the sense that

MU_{A}^{∗}(EA) = MU^{∗}(BA) = H_{0}^{I}(MU_{A}^{∗}) ⟶^{≅} (MU_{A}^{∗})_{I}^{∧}.

□

A useful corollary is that the local Tate cohomology is a functor of H_{0}^{I}(MU_{A}^{∗}). Corollary 5.5. The spectral sequence 4.3 for the local Tate cohomology collapses to give an isomorphism

Ĥ_{i}^{I}(MU_{A}^{∗}) ≅ Ȟ_{I}^{−i}(H_{0}^{I}(MU_{A}^{∗}))

for all i. □

6. The Strategy.

We are now equipped to explain the strategy in detail. The first step is to prove the existence of appropriate pullback squares. Theorem 6.1. For any compact abelian group A, the Tate diagram leads to a pullback of rings

\begin{matrix}MU_{A}^{∗} & ⟶ & Ȟ_{I}^{0}(MU_{A}^{∗}) \\ ↓ & ^{•}□ & ↓ \\ H_{0}^{I}(MU_{A}^{∗}) & ⟶ & Ĥ_{0}^{I}(MU_{A}^{∗}) \end{matrix}

This is proved in Section 7, and is of interest independent of equivariant formal group laws. It is the analogue of tom Dieck’s pullback square for groups of prime order [6]. The analogue for L_{A} is curiously hard to prove. It would follow easily from the tameness of L_{A}. We must make do with a formality. Proposition 6.2. For any compact abelian group A, the Tate diagram leads to a pullback of rings

\begin{matrix}L′_{A} & ⟶ & Ȟ_{I}^{0}(L_{A}) \\ ↓ & ^{•}□ & ↓ \\ H_{0}^{I}(L_{A}) & ⟶ & Ĥ_{0}^{I}(L_{A}), \end{matrix}

and there is a surjective map

L_{A} ⟶ L′_{A}

with kernel Euler-torsion, Euler divisible and ℤ-torsion.

We now proceed to describe how the pullback squares can be used to reduce the proof to two cases corresponding to the completion theorem and the localization theorem.

Certainly the natural map ν : L_{A} ⟶ MU_{A}^{∗} induces maps • H_{0}^{I}(ν) : H_{0}^{I}(L_{A}) ⟶ H_{0}^{I}(MU_{A}^{∗})

• Ȟ_{I}^{0}(ν) : Ȟ_{I}^{0}(L_{A}) ⟶ Ȟ_{I}^{0}(MU_{A}^{∗})

• Ĥ_{0}^{I}(ν) : Ĥ_{0}^{I}(L_{A}) ⟶ Ĥ_{0}^{I}(MU_{A}^{∗}) and it suffices to show that these three maps are isomorphisms for any finite abelian group A.

We shall show that H_{0}^{I}(ν) is an isomorphism in Section 9. Since the two maps to Ĥ_{0}^{I} factor through Ȟ_{I}^{0}H_{0}^{I} in both cases, the local Tate map may be replaced by Ȟ_{I}^{0}H_{0}^{I}(ν) : Ȟ_{I}^{0}H_{0}^{I}(L_{A}) ⟶ Ȟ_{I}^{0}H_{0}^{I}(MU_{A}^{∗}), which is an isomorphism since it is a functor of H_{0}^{I}(ν).

It remains to discuss Ȟ_{I}^{0}(ν). From the exact sequence 4.4 (i), we see that to show a map M ⟶ M′ induces an isomorphism of Ȟ_{I}^{0}, it is sufficient to show that M[1/x_{i}] ⟶ M′[1/x_{i}] is an isomorphism for a collection of radical generators x_{1}, x_{2}, …, x_{n} of I.

To find convenient radical generators we consider the representations V (B) = ⨁_{α^{B}=0} α. When A is finite we simply use their Euler classes e(V (B)) = ∏_{α^{B}=0} e(α). We will return to the case when A is infinite below. The reason for using these is their relevance to the geometric B-fixed point localization theorem, discussed in more detail in Section 11 below. Lemma 6.3. If A is finite, the ideal I = I′ = (e(α) | α ∈ A^{∗}) has the same radical as the ideal I′′ = (e(V (B)) | B ≠ 1 minimal ). Proof: Since I′′ ⊆ I′, it remains to show that for any non-trivial α, some power of e(α) lies in I′′. Suppose then that I′′ ⊆ ℘ for some prime ℘; it suffices to show that e(α) ∈ ℘ for all α ∈ A^{∗}. Indeed, by 5.1, it suffices to find a set of generators α_{1}, α_{2}, …, α_{N} of A^{∗} with e(α_{i}) ∈ ℘. Consider then the subgroup A′ ⊆ A^{∗} generated by those α ∈ A^{∗} with e(α) ∈ ℘.

The assumption A′ ≠ A^{∗} leads to a contradiction. Indeed, in this case we may choose a maximal subgroup (A/B)^{∗} of A^{∗} with A′ ⊆ (A/B)^{∗} ⊆ A^{∗}. However, e(V (B)) ∈ I′′ ⊆ ℘, so it follows that e(α) ∈ ℘ for some α with α^{B} = 0, or equivalently, for some α ∉ (A/B)^{∗}. □

To show Ȟ_{I}^{0}(ν) is an isomorphism it therefore suffices to show that ν[1/e(V (B))] is an isomorphism for B ≠ 1. This will be done in Sections 10 and 11.

Remark 6.4. When A is infinite we use the direct system of ideals I′′ = (e(V′(B)) | B ≠ 1 minimal) in which V′(B) runs through the finite dimensional subrepresentations of V (B). The statement of the lemma is correct for each such I′′. (If A is infinite M[1/e(V (B))] means M with the multiplicative set of all e(V′(B)) inverted).

7. The pullback square for MU_{A}^{∗}.

In this section we prove Theorem 6.1. The analysis is based on the Tate diagram [9]

L ∧ EA_{+}  ⟶  L  ⟶  L ∧ ẼA
≃↓  ↓  ↓
F (EA_{+}, L) ∧ EA_{+} ⟶ F (EA_{+}, L) ⟶ F (EA_{+}, L) ∧ ẼA.

The important fact for us is the left hand equivalence, which arises from the fact that EA_{+} ⟶ S^{0}, and hence E = F (S^{0}, L) ⟶ F (EA_{+}, L), are non-equivariant equivalences. We thus have a pullback square of spectra

\begin{matrix}L & ⟶ & L ∧ ẼA \\ ↓ & ^{•}□ & ↓ \\ F (EA_{+}, L) & ⟶ & F (EA_{+}, L) ∧ ẼA. \end{matrix}

The Tate pullback square gives a long exact Mayer-Vietoris sequence. We prove in 7.3 below that the map MU ∧ ẼA ⟶ F (EA_{+}, MU) ∧ ẼA is surjective in odd degrees. Since MU_{A}^{∗} is in even degrees by 5.3, we obtain a pullback square

\begin{matrix}MU_{A}^{∗} & ⟶ & MU_{even}^{A}(ẼA)) \\ ↓ & ^{•}□ & ↓ \\ MU_{A}^{even}(EA_{+}) & ⟶ & t(MU)_{even}^{A}. \end{matrix}

Furthermore, by the completion theorem [14, 5, 11], MU_{A}^{∗}(EA_{+}) = H_{0}(MU_{A}^{∗}) = (MU_{A}^{∗})_{I}^{∧}. Therefore the left hand vertical of this pullback square is of the stated form.

However the right hand vertical is not yet of the stated form. Nonetheless, for any complex stable Y, there is a spectral sequence

E_{2}^{∗,∗} = Ȟ_{I}^{∗}(Y_{∗}^{A}) ⇒ Y_{∗}^{A}(ẼA)

which we can use to proceed. First we consider the effect on E_{2} terms. Lemma 7.1. If H_{j}^{I}(M) = 0 for i ≥ 0 then M ⟶ H_{0}^{I}(M) induces an isomorphism in local cohomology H_{I}^{∗}(·). It therefore also induces an isomorphism in Ȟ_{I}^{i}(·) for i ≥ 1. Proof: It suffices to observe that M ⟶ Hom(P K(I), M) becomes an isomorphism after tensoring with K(I) as in [7, 4.6 (v)]. □

It follows that the map MU ⟶ F (EA_{+}, MU) induces an isomorphism of E_{2}^{s,∗} for s ≥ 1. This is enough to show we may replace the map MU_{A}^{even} ⟶ t(MU)_{A}^{even} by Ȟ_{I}^{0}(MU_{A}^{∗}) ⟶ Ĥ_{0}^{I}(MU_{A}^{∗}) in the above pullback square. Indeed, we may argue as follows. The edge homomorphism allows us to define the submodule K(Y) by the exact sequence

0 ⟶ K(Y) ⟶ Y_{∗}^{A}(ẼA) ⟶ Ȟ_{I}^{0}(Y_{∗}^{A}).

Lemma 7.2. The map K(c) : K(MU) ⟶ K(F (EA_{+}, MU)) is an isomorphism.

Proof: It follows from 7.1, and the fact that MU_{A}^{∗} is tame by 5.4, that MU_{A}^{∗} ⟶ MU_{A}^{∗}(EA_{+}) induces an isomorphism in Ȟ_{I}^{i} for i ≥ 1. We thus have a map of finite spectral sequences which is an isomorphism except on the 0th column.

From the naturality of the spectral sequence it now follows that the map K(c) is an epimorphism. Since MU_{A}^{∗} and MU_{A}^{∗}(EA_{+}) are in even degrees by 5.3 and 5.4, the 0th columns are also both in even degrees. Therefore all differentials originating on the 0th column end in odd degrees: the map K(c) is therefore a monomorphism in even degrees.

Finally, note that K(Y) = Y_{∗}^{A}(ẼA) in odd degrees for Y = MU or F (EA_{+}, MU). The fact that K(c) is a monomorphism in odd degrees follows from the pullback square since MU_{A}^{∗} is in even degrees. □

Corollary 7.3. The map c : MU ∧ ẼA ⟶ F (EA_{+}, MU) ∧ ẼA is an isomorphism in odd degrees. □

We next notice by a Snake Lemma argument that if

\begin{matrix}A & ⟶ & B \\ ↓ & ^{•}□ & ↓ \\ C & ⟶ & D \end{matrix}

is a pullback square we may replace B ⟶ D by B′ ⟶ D′ if we have a diagram

0 ⟶ K ⟶ B ⟶ B′ ⟶ 0
≅↓ ↓ ↓
0 ⟶ L ⟶ D ⟶ D′ ⟶ 0

or a diagram

0 ⟶ B′ ⟶ B ⟶ E
↓ ↓ ≅↓
0 ⟶ D′ ⟶ D ⟶ F

The first diagram is applied in conjunction with 7.2 to replace MU_{∗}^{A}(ẼA) = (MU ∧ ẼA)_{∗}^{A} ⟶ (F (EA_{+}, MU) ∧ ẼA)_{∗}^{A} = t(MU)_{∗}^{A} by E_{∞}^{0,∗}(MU) ⟶ E_{∞}^{0,∗}(F (EA_{+}, MU)). The spectral sequence collapses at a finite stage, so the E_{∞} term is the E_{N} term for some sufficiently large N. Now, the second diagram is used to replace E_{k+1}^{0,∗}(MU) ⟶ E_{k+1}^{0,∗}(F (EA_{+}, MU). by E_{k}^{0,∗}(MU) ⟶ E_{k}^{0,∗}(F (EA_{+}, MU)), and applying this as often as necessary we eventually reach k = 2 as claimed in the statement. □

8. Implications of topological localization and completion theorems.

In this section we examine how the topological formal group laws behave under the constructions in the Tate diagram. On the one hand this motivates certain algebraic results below, and on the other it is necessary to establishing the parallelism between the algebraic and topological worlds. Lemma 8.1. The natural map

[ℂP (𝒰), F (EA_{+}, E)]_{∗}^{A} ⟶^{≅} [ℂP (𝒰_{ϵ}), F (EA_{+}, E)]_{∗}^{A}

is an isomorphism.

Proof: The inclusion j : ℂP (𝒰_{ϵ}) ⟶ ℂP (𝒰) is a non-equivariant equivalence. □

The first step is to observe that because A is abelian, Ẽ⟨⊉ B⟩ is of a particularly simple form. This also simplifies the algebra.

Lemma 8.2. Ẽ⟨⊉ B⟩ = S^{∞V (B)} where V (B) = ⨁_{α^{B}=0} α.

Proof: It is clear that if C ⊇ B, then V (B)^{C} = 0. On the other hand, if C ⊉ B, then there is a one dimensional representation α of A/C in which B acts non-trivially, so that V (B)^{C} ≠ 0. □

We now want to apply a localization theorem, and since we wish to apply it to the infinite complex ℂP (𝒰) it is essential to use pro-group valued theories: the value on an A-space K is the inverse system E_{A}^{∗}(K_{α}) where K_{α} runs through finite subcomplexes of K. For any space K we have [K, E ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} = [K^{B}, E ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} by obstruction theory. However, to obtain a useful result we need this to be algebraically related to E_{A}^{∗}(K). If K is finite then by definition of Euler classes we have [K, E ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} ≅ E_{A}^{∗}(K)[e(V (B))^{−1}]. The same statement holds for infinite K provided both sides are interpreted as pro-groups. We use the notation

E_{A}^{∗}(K)([e(V (B))^{−1}]) = {E_{A}^{∗}(K_{α})[e(V (B))]^{−1}}

to emphasize that pro-groups have been used.

Lemma 8.3. We have a natural isomorphism

[ℂP (𝒰), E ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} ≅ E_{A}^{∗}(B^{∗} × ℂP (𝒰^{B}))([e(V (B))^{−1}]).

Proof: This is immediate from the preceding discussion since ℂP (𝒰)^{B} = B^{∗} × ℂP (𝒰^{B}). □

Finally, the bottom right corner may be deduced from the bottom left with special care about the order of operations.

Lemma 8.4. We have an isomorphism

[ℂP (𝒰), F (EA_{+}, E) ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} ≅ E_{A}^{∗}(B^{∗} × ℂP (𝒰^{B}))_{ℱ}^{∧}([e(V (B))^{−1}]),

where the topology is defined by the skeleta of ℂP (𝒰^{B}).

Proof: From the identification of B-fixed points we have

[ℂP (𝒰), F (EA_{+}, E) ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} ≅ [B^{∗} × ℂP (𝒰^{B}), F (EA_{+}, E) ∧ Ẽ⟨⊉ B⟩]_{∗}^{A}.

With care about the order of operations we see

[T, F (EA_{+}, E) ∧ Ẽ⟨⊉ B⟩]_{∗}^{A} = E_{A}^{∗}(T × Eℱ)([e(V (B))^{−1}])

where the topology comes from the filtration of T by finite subcomplexes, and the result follows. □

9. At the Euler classes.

We implement the strategy for H_{0}^{I}(ν) outlined in Section 6. Theorem 9.1. The map

H_{0}^{I}(ν) : H_{0}^{I}(L_{A}) ⟶^{≅} H_{0}^{I}(MU_{A}^{∗})

is an isomorphism.

Step 1: The completion theorem 5.4 states H_{0}^{I}(MU_{A}^{∗}) = MU^{∗}(BA_{+}) and the usual calculation with the Gysin sequence [13] gives

H_{0}^{I}(MU_{A}^{∗}) = MU^{∗}(BA_{+}) = MU^{∗}[[x_{1}, x_{2}, …, x_{r}]]/([n_{1}](x), [n_{2}](x), …, [n_{r}](x)).

Step 2: Since H_{0}^{I} is an idempotent functor, H_{0}^{I}(L_{A}) classifies A-equivariant formal group laws which are HI-complete in the sense that k ⟶ H_{0}^{I}(k) is an isomorphism.

Step 3: Suppose given an equivariant formal group law k{{𝒰}} so that k ⟶ H_{0}^{I}(k) is an isomorphism. The main ingredient in this section (proved below as Theorem 9.2) is that the natural map

k{{𝒰}} ⟶^{≅} k{{𝒰_{ϵ}}}

induced by the inclusion of universes is an isomorphism.

The formal group law is thus given by a classical non-equivariant formal group law F together with Euler classes, determined by a homomorphism

e : A^{∗} ⟶ nil(k)

where the group operation on the set nil(k) of nilpotent elements is defined by F in the usual way. This specifies a map

MU^{∗}[[x_{1}, x_{2}, …, x_{r}]] ⟶ k

with x_{i} mapping to e(α_{i}), and [n_{i}](e(α_{i})) = 0, so the map factors through

MU^{∗}[[x_{1}, x_{2}, …, x_{n}]]/([n_{1}](x), [n_{2}](x), …, [n_{r}](x)).

On the other hand it does not factor any further in general since

MU_{A}^{∗}(EA_{+}) = MU^{∗}BA = MU^{∗}[[x_{1}, x_{2}, …, x_{n}]]/([n_{1}](x), [n_{2}](x), …, [n_{r}](x))

is itself an example.

Step 4: Any A-equivariant formal group law k{{𝒰}} defines a canonical non-equivariant formal group law k{{𝒰_{ϵ}}}, using flags with successively larger numbers of copies of the trivial representation at the start. This defines a ring homomorphism L ⟶ L_{A}.

Since the composite of ν with L ⟶ L_{A} does classify the classical non-equivariant formal group law on k{{𝒰_{ϵ}}} by 8.1, the nonequivariant Quillen theorem shows that H_{0}^{I}(ν) is an isomorphism on the subalgebra MU_{∗}. Furthermore the Euler classes in L_{A} are named so that they correspond to the Euler class in k so the H_{0}^{I}(ν) is an isomorphism as required.

It remains to establish the main input. Theorem 9.2. If k is complete in the sense that k ≅ H_{0}^{I}(k) then the natural map

k{{𝒰}} ⟶^{≅} k{{𝒰_{ϵ}}}

is an isomorphism.

Proof: This time we need to do a little more work, and we are guided by the topology. In fact the proof consists of translating the Adams-Haeberly-Jackowski-May [1] proof of the Atiyah-Segal completion theorem into our context. For the usual reason, it is convenient to use progroups. We think of 𝒰 equipped with its finite dimensional subspaces, and hence consider the pro-group k{{𝒰}} = {k{{V }}}_{V ⊆𝒰}. Note that any exhaustive flag gives a countable cofinal sub-pro-group. Note too that the terms in k{{𝒰}} are finitely generated free k-modules. Now any isometry V ⟶ W induces an epimorphism k{{W }} ⟶ k{{V }}, and we let k{{W, V }} denote the kernel. It should be thought of as analogous to E_{A}^{∗}(ℂP (W), ℂP (V)). We extend this to allow V and W to be universes, using pro-groups. In particular we have the inclusion i : 𝒰_{ϵ} ⟶ 𝒰, and we want to prove that the induced map

i^{∗} : k{{𝒰}} ⟶ k{{𝒰_{ϵ}}}

is an isomorphism. It is equivalent to show

k{{𝒰, 𝒰_{ϵ}}} = 0.

Next we need an analogue of the fact that if k ≅ k_{I}^{∧} we can recover vanishing from that of the I-adic pro-completion pro-group k_{I}^{∧}{{𝒰}} = {k{{V }}/I^{s}}_{s,V}. Indeed this suffices in case k has bounded torsion. However in our present context we only know k ≅ H_{0}^{I}(k), so we consider instead the inverse system of chain complexes of progroups

k_{I}^{h∧}{{𝒰}} = {K_{s}^{•}(I) ⊗ k{{V }}}_{s,V} .

Here the complexes K_{s}^{•}(I) depend on the generators e(α) of the ideal I and are defined by

K_{s}^{•}(I) = ⨂_{α∈A^{∗}}(k ⟶^{e(α)^{s}} k).

However, the abuse of notation is justified by the fact that the homotopy type of the direct system is independent of generators in a suitable sense. These complexes are relevant since local homology is defined by

H_{∗}^{I}(M) = H_{∗}(Hom(holim_{→}_{s} K_{s}^{•}(I), M)),

and this does not depend on the chosen generators. The important features of the definition of k_{I}^{h∧}{{𝒰}} are (i) it gives a system of chain complexes of finitely generated free modules, so that the obvious extension of the functor to other universes is therefore exact and (ii) H_{0}^{I}(·) may be recovered as the zeroth homology of the homotopy inverse limit of this system of chain complexes. In view of the Milnor short exact sequence

0 ⟶ lim_{←}_{s}^{1} H_{n+1}(M_{s}) ⟶ H_{n}(holim_{←}_{s} M_{s}) ⟶ lim_{←}_{s} H_{n}(M_{s}) ⟶ 0

we may prove k{{𝒰, 𝒰_{ϵ}}} = H_{0}^{I}(k{{𝒰, 𝒰_{ϵ}}}) is zero by establishing that the inverse system k_{I}^{h∧}{{𝒰, 𝒰_{ϵ}}} of chain complexes is pro-zero in the homotopy category.

Since

k_{I}^{h∧}{{𝒰, 𝒰_{ϵ}}} = lim_{←}_{V^{A}=0} k_{I}^{h∧}{{𝒰_{ϵ} ⊕ V, 𝒰_{ϵ}}},

it suffices to show that k_{I}^{h∧}{{𝒰_{ϵ} ⊕ V, 𝒰_{ϵ}}} = 0. Using the long exact sequence associated to

(𝒰_{ϵ} ⊕ V, 𝒰_{ϵ}) ⟶ (𝒰_{ϵ} ⊕ V ⊕ α, 𝒰_{ϵ}) ⟶ (𝒰_{ϵ} ⊕ V ⊕ α, 𝒰_{ϵ} ⊕ V)

we see that it suffices to show k_{I}^{h∧}{{𝒰_{ϵ} ⊕ V ⊕ α, 𝒰_{ϵ} ⊕ V }} = 0 when α^{A} = 0. Recalling that ℂP (V ⊕ α)/ℂP (V) ≅ S^{V ⊗α^{−1}}, the exact analogue of the fact that

S^{(kϵ⊕V)⊗α^{−1}} ⟶ S^{((k+l)ϵ⊕V)⊗α^{−1}}

induces multiplication by the Euler class e(lα) is the following. Lemma 9.3. The inclusion

((kϵ ⊕ V) ⊕ α, kϵ ⊕ V) ⟶ (((k + l)ϵ ⊕ V) ⊕ α, (k + l)ϵ ⊕ V)

induces multiplication by the Euler class e(lα^{−1}). Proof: The k-module k{{kϵ ⊕ V ⊕ α, kϵ ⊕ V }} is generated by y(kϵ ⊕ V) mod y(kϵ ⊕ V)y(α). Thus we simply need to identify y((k + l)ϵ ⊕ V) = y(kϵ ⊕ V)y(lϵ) as a multiple of this generator. Evidently, the multiple is the constant term when y(lϵ) is expressed in terms of a basis associated to a flag beginning with α: this is e(lα^{−1}) as required. □

We may now complete the proof of the theorem. Indeed, since e(α) is a named generator of I, the e(α)^{2s} is chain null-homotopic as a self-map of K_{s}^{•}(I). Therefore the pro-group

k_{I}^{h∧}{{𝒰_{ϵ} ⊕ V ⊕ α, 𝒰_{ϵ} ⊕ V }} = {K_{s}^{•}(I) ⊗ k{{kϵ ⊕ V ⊕ α, kϵ ⊕ V }}}_{s,k}

is pro-zero, since for chosen (s, k) the map from the (s, k +2s)th term is chain nullhomotopic by 9.3. □

10. Equivariant Homotopical bordism.

We identify the geometric fixed point spectra of homotopical bordism, following tom Dieck [6], and record their homology following [3]. The idea is that the geometric B-fixed point A/B-spectrum Φ^{B}MU encodes the normal data to the B-fixed point manifold.

Before we give the description we need some notation. We shall need to choose a splitting s : B^{∗} ⟶ A^{∗} to the natural surjection A^{∗} ⟶ B^{∗}. We write \overline{B^{∗}} for the non-trivial elements of B^{∗}. Next, we consider the additive group J(B) = ker(R(B) ⟶ ℤ), and we grade it by by deg(j) = j^{B} (i.e., the dimension of j^{B} as a virtual real vector space). Thus it is a free abelian group on the generators β − ϵ of degree −2 for non-zero β ∈ B^{∗}. When we consider the group ring k[J(B)], as is usual with a free abelian group, it is convenient to view this as a ring of Laurent series on variables z_{s(β)} for non-zero β ∈ B^{∗}, with z_{s(β)} corresponding to the basis element β − ϵ ∈ J(B).

In terms of this we may state the result. Proposition 10.1. If B is finite, the geometric B-fixed point A/B-spectrum of MU is given by

Φ^{B}MU ≃ ⋁_{j∈J(B)} Σ^{j^{B}}MU ∧ (BU_{+})^{∧\overline{B^{∗}}}
= MU ∧ (BU_{+})^{∧\overline{B^{∗}}}[J(B)]
= MU ∧ (BU_{+})^{∧\overline{B^{∗}}}[z_{s(β)}, z_{s(β)}^{−1} | β ∈ \overline{B^{∗}}]

The ring structure combines the ring structure of MU, Whitney sum on each of the factors BU_{+} and the group ring of J(B). The splitting of Φ^{B}MU depends on the splitting s : B^{∗} ⟶ A^{∗}.

Proof: We present the proof when A is finite, because the description of the functor Φ^{B} is notationally simpler. It is convenient to use multiples of the regular representation as the indexing sequence for the spectra concerned. First note that if X is an A-spectrum, then Φ^{B}X is an A/B-spectrum, and the kℂA/B’th term in the prespectrum Φ^{B}X is X(kℂA)^{B}. For the prespectrum MU we may easily calculate the fixed points of the V th term of MU. First note that

(Gr_{m}(U))^{B} = ∐_{|W |=m} ∏_{β} Gr_{|W_{β}|}(U_{β}),

where W runs over isomorphism classes of m-dimensional representations of B, and where W_{β} is the β-isotypical summand of W. Now take Thom spaces to find

(Gr_{m}(U)^{γ})^{B} = ⋁_{|W |=m} Gr_{|W_{ϵ}|}(U_{ϵ})^{γ} ∧ ⋀_{β≠ϵ} Gr_{|W_{β}|}(U_{β})_{+}.

Now we apply this to MU(Y) = Gr_{|Y |}(𝒰 ⊕ Y)^{γ}, and to the structure map S^{X} ∧ MU(Y) ⟶ MU(Y ⊕X), which results from classifying γ ⊕X and taking Thom spaces. The W ’th factor (where |W | = |Y |), maps into the W ⊕ X’th factor:

S^{X_{ϵ}} ∧ Gr_{|W_{ϵ}|}(𝒰_{ϵ} ⊕ Y_{ϵ})^{γ} ∧ ⋀_{β≠ϵ} Gr_{|W_{β}|}(𝒰_{β} ⊕ Y_{β})_{+} ⟶

Gr_{|W_{ϵ}⊕X_{ϵ}|}(𝒰_{ϵ} ⊕ Y_{ϵ} ⊕ X_{ϵ})^{γ} ∧ ⋀_{β≠ϵ} Gr_{|W_{β}⊕X_{β}|}(𝒰_{β} ⊕ Y_{β} ⊕ X_{β})_{+}.

On the first smash factor the structure map is the usual structure map S^{X_{ϵ}} ∧ MU(Y_{ϵ}) ⟶ MU(Y_{ϵ} ⊕ X_{ϵ}), and on the βth factor with β ≠ ϵ, it is a stabilization map of Gr_{∞}(𝒰_{β}). Furthermore, tensor product with s(β) gives an A/B-equivariant equivalence

Gr_{∞}(𝒰_{β}) ≃ Gr_{∞}(𝒰^{B}) = BU.

Since tensor product distributes over direct sum, this is compatible with the Whitney sum operation. This shows that each term in the wedge contributes a suspension of the spectrum MU ∧ BU_{+}^{×\overline{B^{∗}}}. To identify the appropriate suspension, note that the W ’th summand in MU(Y)^{B} contributes to the W^{B}th term in a spectrum MU ∧ BU_{+}^{×\overline{B^{∗}}}. The summand is therefore Σ^{Y^{B}−W^{B}}MU ∧ BU_{+}^{×\overline{B^{∗}}}. □

Remark 10.2. (Invertibility of Euler classes)

There is a canonical complex orientation, and so we have canonical Euler classes. The Thom class of V in MU_{A}^{|V |} is defined by choosing a copy of V in 𝒰 to give a map ∗ ⟶ Gr_{|V |}(𝒰 ⊕ |V |), and then taking Thom spaces, to give a map S^{V} ⟶ MU(|V |). The Euler class results by composing with the inclusion S^{0} ⟶ S^{V}, giving a map S^{0} ⟶ MU(|V |), and hence χ(V) ∈ MU_{−|V |}^{A}.

If V^{B} = 0 it is not hard to find an explicit inverse for χ(V) as a map S^{|V |} ⟶ Φ^{B}MU. Indeed, there is a variant on the above construction. Suppose that |V | = k|ℂA|. Then there is a corresponding point

V ∈ ∏_{β} Gr_{|V_{β}|}(𝒰_{β} ⊕ kℂA_{β}) ⊆ Gr_{|V |}(𝒰 ⊕ kℂA).

Taking Thom spaces, this gives a map

S^{V^{B}} ⟶ MU(kℂA)^{B},

and hence a homotopy element

t(V) ∈ (Φ^{B}MU)_{(V −kℂA)^{B}}^{A/B},

not necessarily in an integer degree. In fact if α^{B} = 0, then z_{α} is the special case of this with V = ℂA ⊕ α − ϵ giving t(ℂA ⊕ α − ϵ) = z_{α}. The invertibility of z_{β} follows from easy properties of t(·). Lemma 10.3. The construction t has the following properties.

(i) t(kℂA) = 1 (ii) t(V ⊕ W) = t(V)t(W). □

To find an inverse for z_{α} we thus take V = ℂA ⊕ ϵ − α.

In [3] it is shown that, using the canonical orientation, a complete flag in 𝒰 determines certain elements γ_{1}, γ_{2}, … in MU_{∗}^{A/B}(BU) as follows (the more usual letter β conflicts with other notation below). First we use the orientation to form a flag basis y(W^{0}), y(W^{1}), ⋯, y(W^{n−1}) of the cohomology rings MU_{A/B}^{∗}(ℂP (W^{n})), and then form the dual basis γ_{0}, γ_{1}, …, γ_{n−1} of the homology MU_{∗}^{A/B}(ℂP (W^{n})). These are compatible with the inclusions W^{n} ⊂ W^{n+1}, and hence give a basis γ_{0}, γ_{1}, γ_{2}, … of MU_{∗}^{A/B}(ℂP (𝒰^{B})); note that γ_{n} depends on the part of the flag up to W^{n+1}. The elements γ_{i} are the images of these basis elements under the inclusion ℂP (𝒰) ⟶ BU (γ_{0} becomes the unit in the course of the stabilization process). There is then an isomorphism

MU_{∗}^{A/B}(BU) = MU_{∗}^{A/B}[γ_{1}, γ_{2}, …],

of rings. This gives a calculation of the homotopy of the geometric fixed point spectrum. Corollary 10.4. The coefficient ring of the geometric B-fixed point spectrum is

\begin{aligned}(Φ^{B}MU)_{∗}^{A/B} & = & MU_{∗}^{A/B}[γ_{s(β),i}|i ≥ 1, β ∈ \overline{B^{∗}}][z_{s(β)}, z_{s(β)}^{−1}|β ∈ \overline{B^{∗}}] \\ & ≅ & MU_{∗}^{A/B}[γ_{1}, γ_{2}, γ_{3}, …][z, z^{−1}]^{⊗\overline{B^{∗}}} \end{aligned}

where the elements γ_{β,i} and z_{s(β)} depend on the choice of a complete flag in 𝒰 and the splitting s : B^{∗} ⟶ A^{∗}. □

11. Away from Euler classes.

We are now equipped to provide the second main step in the proof of the main theorem. We show that ν[1/e] is an isomorphism, for the Euler class e of V (B) = ⨁_{α^{B}=0} α, with the understanding that if A is infinite, [1/e] means the inversion of the multiplicative set of e(V′(B)) where V′(B) runs over the finite dimensional subrepresentations of V (B). Theorem 11.1. If A is a finite abelian group, the map

ν[1/e] : L_{A}[1/e] ⟶^{≅} MU_{A}^{∗}[1/e]

is an isomorphism.

Step 1: The localization theorem states MU_{A}^{∗}[1/e] = (Φ^{B}MU)_{∗}^{A/B}, and as described in 10.4, this captures the normal data to the fixed point set, in the sense that

MU_{A}^{∗}[1/e] = (Φ^{B}MU)_{∗}^{A/B} = MU_{∗}^{A/B}(BU)[z, z^{−1}]^{⊗\overline{B^{∗}}},

where z is of degree 2. As before \overline{B^{∗}} is the set of non-trivial simple representations of B, and the βth factor corresponds to the summand of the normal bundle for the associated β-isotypical representation.

Step 2: Since [1/e] is an idempotent functor, L_{A}[1/e] classifies A-equivariant formal group laws in which Euler classes are invertible.

Step 3: Suppose given an equivariant formal group law k{{𝒰}} so that e is invertible in k. The main input is the observation that if the Euler class e = e(V (B)) is invertible, an A-equivariant formal group law simplifies considerably. Theorem 11.2. If e = e(V (B)) is invertible in k, the natural map

k{{𝒰}} ⟶^{≅} ∏_{β} k{{𝒰_{β}}}

induced by the inclusion of universes is an isomorphism. Proof: This is essentially the Chinese Remainder Theorem. It is convenient to choose our complete flag to contain multiples of the regular representation. Thus the natural map is obtained from

q : k{{𝒰}}/(y(kℂA)) = k{{𝒰}}/(∏_{β} y(kℂA_{β})) ⟶ ∏_{β} k{{𝒰}}/(y(kℂA_{β}))

by passage to inverse limits. The kernel of this system of maps is the pro-ideal {⋂_{β}(y(kℂA_{β}))}_{k}, which is pro-zero integrally. To ensure q is surjective it suffices that y(ℂA_{β}) is coprime to ∏_{γ≠β} y(ℂA_{γ}). For this we simply express ∏_{γ≠β} y(ℂA_{γ}) with respect to the basis associated to a flag beginning with β. The constant term is the non-zero Euler class ∏_{γ≠β} e(ℂA_{γβ^{−1}}), and thus invertible by hypothesis. □

This allows us to express the condition that k{{𝒰}} is an equivariant formal group law in more elementary terms. Neither of the following two lemmas require invertibility of Euler classes. Lemma 11.3. If k{{𝒰}} is an A-equivariant formal group law then it determines the structure of an A/B-eqivariant formal group law on k{{𝒰^{B}}}. Proof: By definition, we have the structure of a topological ring on k{{𝒰^{B}}}. To see that the coproduct Δ on k\{\{𝒰\}\} determines a coproduct \overline{Δ} on k\{\{𝒰^{B}\}\}, we note that k\{\{𝒰^{B}\}\} is the inverse limit of the quotients k{{𝒰}}/(y(W)) = k{{W }} with W running through a complete flag in 𝒰^{B}. Now we need to know that for any pair of subspaces W′, W′′ of 𝒰^{B}, if W is sufficiently large the coproduct Δ(y(W)) is divisible by y(W′) ⊗ y(W′′). This is a special case of the vanishing condition (VT) in [2, 14.1].

The group homomorphism (A/B)^{∗} ⟶ A^{∗} induces a Hopf algebra map k^{A^{∗}} ⟶ k^{(A/B)^{∗}}, which by composition with θ gives the structure map \overline{θ} for k\{\{𝒰^{B}\}\}.

It is immediate that the conditions required of the coproduct and augmentation are satisfied. We will return to discussion of orientations below. □

We next claim that the structure of k{{𝒰^{B}}} determines most of the structure on k{{𝒰}}. Lemma 11.4. Suppose that an A-equivariant formal group law k{{𝒰}} has the property that the inclusion of universes induces an isomorphism

k{{𝒰}} ⟶^{≅} ∏_{β} k{{𝒰_{β}}}

The structure maps \overline{Δ} and \overline{θ} of k\{\{𝒰^{B}\}\}, together with the group extension 0 ⟶ (A/B)^{∗} ⟶ A^{∗} ⟶ B^{∗} ⟶ 0 determine the structure maps Δ and θ for k{{𝒰}}. Proof: First note that the ring structure of k{{𝒰}} is simply that of a B^{∗}-indexed product of rings k{{𝒰_{β}}}. Using s(β) ∈ A^{∗} to represent β ∈ B, tensor product with s(β) gives a ring isomorphism

s(β)^{∗} : k{{𝒰_{β}}} ⟶^{≅} k{{𝒰^{B}}}.

We now note that there is a diagram

0 ⟶ k^{B^{∗}} ⟶ k{{𝒰}} ⟶ k{{𝒰^{B}}} ⟶ 0
=↓ ↓ ↓
0 ⟶ k^{B^{∗}} ⟶ k^{A^{∗}} ⟶ k^{(A/B)^{∗}} ⟶ 0

of Hopf algebras in which both rows are exact. In the topological case this corresponds to the diagram

0 ⟵ B^{∗} ⟵ ℂP (𝒰)^{B} ⟵ ℂP (𝒰^{B}) ⟵ 0
=↑  ↑  ↑
0 ⟵ B^{∗} ⟵  A^{∗}  ⟵ (A/B)^{∗} ⟵ 0

of group objects in the homotopy category of A/B-spaces. In the category of groups it is well known that such a diagram determines the middle term of the top row and all maps involving it: the explicit form is the use of a factor set associated to a splitting to define the multiplication. The same proof works for the Hopf algebras. Our chosen splitting s : B^{∗} ⟶ A^{∗} determines A^{∗} from the factor set f : B^{∗} × B^{∗} ⟶ (A/B)^{∗} defined by f(β_{1}, β_{2}) = s(β_{1}β_{2})^{−1}s(β_{1})s(β_{2}). The co-factor set for k{{𝒰}} is the composite

k{{𝒰^{B}}} ⟶ k^{(A/B)^{∗}} ⟶^{f} k^{B^{∗}×B^{∗}} ⟶^{≅} k^{B^{∗}} ⊗ k^{B^{∗}}.

The construction ensures that the map θ constructed from \overline{θ} is a map of Hopf algebras. □

Let us consider orientations. It may help the reader if we first consider the special case B = A, so that an A/B-equivariant formal group law is a classical nonequivariant formal group law, and k{{𝒰_{α}}} = k[[y_{α}]] is a power series ring. An element (f_{α}(y_{α})) gives an element of k{{𝒰, ϵ}} if f_{ϵ}(y_{ϵ}) has zero constant term. To give a generator of k{{ϵ⊕ϵ, ϵ}} the coefficient of y_{ϵ} must be a unit. Finally, to give a generator of k{{ϵ ⊕ α, ϵ}} the constant term in f_{α}(y_{α}) must be a unit (it must necessarily be a unit multiple of χ(α), so this is just as well). Thus, if Euler classes are given, an orientation is specified by (i) a classical orientation of the classical formal group law k{{𝒰_{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}). This shows that MU_{∗}((BU[z, z^{−1}])^{×\overline{A^{∗}}}) is a representing ring for A-equivariant formal group laws with e invertible.

Now consider a general subgroup B. An element f = (f_{β}) with f_{β} ∈ k{{𝒰_{β}}} gives an element of k{{𝒰, ϵ}} if and only if f_{ϵ} defines an element of k{{𝒰^{B}, ϵ}}. Supposing this to be the case, the condition that f gives a generator of k{{ϵ ⊕ α, ϵ}} for all α fixed by B is equivalent to the condition that f_{ϵ} is an orientation of k{{𝒰^{B}}}. This completes the proof of 11.3. For a chosen non-trivial β ∈ B^{∗}, the conditions that f gives a generator k{{ϵ ⊕ α, ϵ}} for the various α restricting to β are equivalent to the condition that f_{β} has unit constant term in every flag basis.

Thus, if Euler classes are given, an orientation is freely specified by (i) an orientation of the A/B-equivariant formal group law k{{𝒰^{B}}} and (ii) for each each non-trivial β ∈ B^{∗}, a unit f_{β} ∈ k{{𝒰^{B}}}.

Now by 10.4,

(Φ^{B}MU)_{∗}^{A/B} = MU_{∗}^{A/B}[γ_{s(β),i}|i ≥ 1, β ∈ \overline{B^{∗}}][z_{s(β)}, z_{s(β)}^{−1}|β ∈ \overline{B^{∗}}].

A map from this to a ring k is thus freely determined by the map on MU_{∗}^{A/B} (defining an A/B-equivariant formal group law by induction, and hence the coproduct Δ and augmentation θ by 11.4) and the images of γ_{s(β),i} and z_{s(β)} subject to the requirement that the image of z_{s(β)} is invertible.

Remark 11.5. Since z_{s(β)}^{−1} is in the ring, it is clear that a classifying map (Φ^{B}MU)_{∗}^{A/B} ⟶ k forces the Euler class e(s(β)) to be invertible. We may deduce the invertibility of e(α) when α^{B} = 0 as follows. Indeed, by definition e(α) is the image of z_{α}, which we showed to be invertible for geometric reasons in Section 10. We have not made the formula for z_{α}^{−1} in terms of the named generators explicit, but its image necessarily gives a formula for the inverse of e(α).

Step 4: Any A-equivariant formal group law k{{𝒰}} defines a canonical A/B-equivariant formal group law k{{𝒰^{B}}} by 11.3. This defines a ring homomorphism L_{A/B} ⟶ L_{A}.

Since the composite of ν with the map L_{A/B} ⟶ L_{A} does classify the A/B-equivariant formal group law on k{{𝒰^{B}}} by 8.3, the A/B-equivariant version of the theorem (which we may suppose proved by Quillen or induction) shows that ν[1/e] is an isomorphism on the subalgebra MU_{∗}^{A/B}. Next z_{s(β)} is the Euler class of s(β), so is mapped correctly. By construction of the classifying map, the element γ_{s(β),i} is mapped to the coefficient of y(W^{i}) in the βth factor of the orientation, and it remains to identify this coefficient when k = (Φ^{B}MU)_{∗}^{A/B}.

By definition the canonical orientation is the based cohomology class

ℂP (𝒰) = Gr_{1}(𝒰 ⊕ ϵ) ≃ MU(ϵ) ⟶ Σ^{2}MU

Taking fixed points we find this is the inclusion of

∐_{β} ℂP (𝒰_{β}) ⟶ Gr_{1}(𝒰_{ϵ} ⊕ ϵ)^{γ} ∧ ⋀_{β≠ϵ} Gr_{1}(𝒰_{β})_{+}

into Σ^{2}Φ^{B}MU. In other words, the ϵth factor (which contains the basepoint) is included in Σ^{2}MU by the A/B-equivariant canonical orientation, and each of the other factors is the inclusion of BU(1) ⟶ BU_{+}. From the construction it is a tautology that the coefficient of y(W^{i}) in the βth factor is γ_{s(β),i}. [The classical procedure (which could be used if B = A) interposes an extra identification because of its familiarity. We compose with cohomology classes BU_{+} ⟶ Σ^{2k}H given by monomials in the Chern classes, it is immediate from the construction of the classes γ_{i} (usually known as β_{i}) that the coefficients are as claimed.]

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School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.

E-mail address: j.greenlees@sheffield.ac.uk