ALGEBRAIC GROUPS AND EQUIVARIANT COHOMOLOGY THEORIES.

J.P.C.GREENLEES

Abstract. Many important cohomology theories are associated to algebraic groups, and this is clearest when equivariant theories are considered. The purpose of this article is to describe some familiar examples and to speculate on the existence of further examples.

Contents

1. Introduction. 2 1.A. The context. 2 1.B. Some familiar examples. 2 1.C. Some putative examples. 3 1.D. Some motivation. 3 2. K-theory and the multiplicative group. 3 2.A. The multiplicative group and the multiplicative formal group. 3 2.B. 𝕋-equivariant K-theory and 𝕋-equivariant Borel K-theory. 4 2.C. Reconstructing 𝕋-equivariant K-theory from the multiplicative group. 4 2.D. A-equivariant formal groups and A × 𝕋-equivariant Borel K-theory. 4 3. The shape of a cohomology theory. 5 3.A. The definition of the shape. 5 3.B. Equivariant K-theory. 6 3.C. Complex orientable theories. 7 3.D. What is special about K-theory and the multiplicative group? 7 4. The non-split torus. 8 4.A. The non-split torus and its formal group. 8 4.B. Constructing the associated cohomology theory. 8 5. Elliptic cohomology and elliptic curves. 8 5.A. Elliptic curves and their formal completions. 8 5.B. Reflecting the group structure on functions. 9 5.C. Elliptic cohomology theories and elliptic formal groups. 9 5.D. 𝕋-equivariant elliptic cohomology theories and elliptic curves. 10 5.E. A-equivariant elliptic cohomology theories and elliptic formal groups. 10 6. 𝕋-equivariant elliptic cohomology. 10 6.A. Sheaves on a curve. 10 6.B. An associated cohomology theory. 11 7. Shapes from projective varieties. 11 7.A. Sheaves on a Jacobian. 12 7.B. Further constraints. 13 8. Rational equivariant cohomology theories. 13 8.A. Algebraic models for categories of rational cohomology theories. 14 8.B. Consequences of the conjecture. 15 8.C. Status of the conjecture. 15 9. The model for the circle group G = 𝕋. 15 9.A. The category 𝒜(𝕋). 15 9.B. Rigid even objects. 16 9.C. Construction of a cohomology theory from an algebraic group. 17 9.D. The shape of a rigid even spectrum. 17 10. Reflecting the group structure of the elliptic curve. 17 References 19

The author is grateful to M.Ando, M.J.Hopkins, H.R.Miller and N.P.Strickland for collaborations and discussions which have informed this work.

1. Introduction.

1.A. The context. This article discusses a connection between the world of algebraic geometry and the world of algebraic topology. In the world of algebraic geometry, an algebraic group 𝔾 is a functor associating a group 𝔾(k) to a commutative ring k. In the world of algebraic topology, a G-equivariant cohomology theory associates a graded abelian group E_{G}^{∗}(X) to a space X with G-action, where G is the ambient compact Lie group of equivariance. Sometimes an algebraic group 𝔾 corresponds to an equivariant cohomology theory E_{G}^{∗}(·) in a way we will describe.

Going from topology to geometry is the more elementary process, and we will describe several well-known examples: in fact when G is the circle group 𝕋, suitable cohomology theories E_{𝕋}^{∗}(·) give rise to one dimensional algebraic groups 𝔾 by evaluation on the appropriate spaces. More striking is the fact that we can sometimes reverse the process and move from geometry to topology. From the easy direction, we shall see that a one dimensional algebraic group 𝔾 suggests the values of an equivariant cohomology theory E_{𝕋}^{∗}(·) on equivariant spheres. Rather remarkably, the algebraic model for rational 𝕋-equivariant cohomology theories [11] shows that, for rational theories, the values on spheres are very close to determining the cohomology theory, and in favourable cases they determine it completely. Going one step further, one can often construct a cohomology theory from its putative values on spheres, and this is often the case for values arising from algebraic groups.

To put flesh on this philosophy we need some examples.

1.B. Some familiar examples. In the most familiar case, the multiplicative group 𝔾_{m} is associated to K-theory, with the association being clearest for K_{𝕋}^{∗}(·) where 𝕋 is the circle group. This is discussed in detail in Section 2, and if the introduction seems too abstract, the reader should read Section 2 first.

It is also familiar that the additive group 𝔾_{a} is associated to ordinary cohomology with the association being clearest for the 𝕋-equivariant Borel theory H^{∗}(E𝕋×_{𝕋}(·)). Similarly, there is at least a formal group associated to any complex oriented theory E^{∗}(·), with the association being clearest for the 𝕋-equivariant Borel theory E^{∗}(E𝕋 ×_{𝕋} (·)). When the formal group is the formal completion of an ellipic curve around the identity we obtain elliptic cohomology theories, but again the association between the cohomology theory and the ellipitc curve is clearest for the 𝕋-equivariant version of the theory [19, 15]. It is the new behaviour of this 𝕋-equivariant elliptic theory, and the possibility of adapting its construction that led to the author writing this article.

1.C. Some putative examples. We will speculate on higher dimensional examples, at least when both the algebraic group and the compact Lie group of equivariance are abelian and connected. In this case it seems that if an algebraic group 𝔾 of dimension d is associated to a cohomology theory E_{G}^{∗}(·) then the compact Lie group G is of dimension d, and in general one might expect their ranks to agree. However we will see that it is very rare for E_{G}^{∗}(·) to be complex orientable, so that these theories will be of an unfamiliar sort.

1.D. Some motivation. One reason for this investigation is to understand familiar objects better. Equivariant versions of elliptic cohomology behave in a way which is unfamiliar to topologists. To develop appropriate intuition, we need to put this into context. One might even hope that the new context would help us to find a truly geometric definition.

History has shown that cohomology theories arising from algebraic groups are significant, having applications to index theory [32, 31], rigidity [25, 2, 3] and to geometric representation theory [10, 9]. This alone makes it worth seeking new ones.

In any case, the strength of the connection with algebraic geometry was already the motivation for the introduction of equivariant elliptic cohomology: applying the functor to natural objects in representation theory gives interesting algebras. Similar constructions are straightforward for other cohomology theories based on algebraic groups.

2. K-theory and the multiplicative group.

We will describe properties of the multiplicative group, K-theory and the relationship between them in some detail because the underlying phenomena are well known, and several of the issues arise in this case.

2.A. The multiplicative group and the multiplicative formal group. The multiplicative group 𝔾_{m} is the functor on commutative rings l defined by

𝔾_{m}(l) = Units(l) = Rings(ℤ[z, z^{−1}], l).

The first equality is the definition, and the second shows that 𝔾_{m} is represented by the ring ℤ[z, z^{−1}]. We also say that ℤ[z, z^{−1}] is the ring of functions on 𝔾_{m} and write

𝒪_{𝔾_{m}} = ℤ[z, z^{−1}].

The group operation is given by multiplication of units

Units(l) × Units(l) = Rings(ℤ[z, z^{−1}] ⊗ ℤ[z, z^{−1}], l) ⟶ Rings(ℤ[z, z^{−1}], l) = Units(l),

and this induced by the ring homomorphism

Δ : ℤ[z, z^{−1}] ⟶ ℤ[z, z^{−1}] ⊗ ℤ[z, z^{−1}]
z ⟼ z ⊗ z.

The identity element of Units(l) is 1 ∈ l and therefore the identity e of 𝔾_{m} is represented by the augmentation ℤ[z, z^{−1}] ⟶ ℤ setting z = 1. Accordingly, the ideal of functions vanishing at e is the principal ideal (y) where y = 1 − z. We say that y is a coordinate at the identity. In terms of y, the coproduct then takes the more familiar form y ⟼ 1 ⊗ y + y ⊗ 1 − y ⊗ y.

We may then complete around the identity to form the multiplicative formal group (𝔾_{m})_{e}^{∧} with ring of functions

(ℤ[z, z^{−1}])_{(y)}^{∧} = ℤ[[y]].

2.B. 𝕋-equivariant K-theory and 𝕋-equivariant Borel K-theory. We can easily recover the multiplicative group from 𝕋-equivariant K-theory and the multiplicative formal group from 𝕋-equivariant Borel K-theory, where 𝕋 is the circle group. Just as the multiplicative formal group is only an approximation to the multiplicative group itself, so Borel K-theory K^{∗}(E𝕋 ×_{𝕋} X) is only an approximation to the true equivariant theory K_{𝕋}^{∗}(X) for a 𝕋-space X. Just as the multiplicative formal group is a completion of the multiplicative group, so the Atiyah-Segal completion theorem says Borel K-theory is a completion of equivariant K-theory.

The cohomology theory associated to the group 𝔾_{m} is 𝕋-equivariant K-theory for the circle group 𝕋. Indeed, the ring of functions on 𝔾_{m} is realized by K_{𝕋}^{0} = R(𝕋) = ℤ[z, z^{−1}]. The coproduct is realized by the multiplication map μ : 𝕋 × 𝕋 ⟶ 𝕋 in the sense that the coproduct is

K_{𝕋}^{0} ⟶^{μ^{∗}} K_{𝕋×𝕋}^{0} = K_{𝕋}^{0} ⊗ K_{𝕋}^{0}.

The cohomology theory associated to the formal group (𝔾_{m})_{e}^{∧} is Borel K-theory for the circle group 𝕋. Indeed, the ring of functions on the formal group (𝔾_{m})_{(e)}^{∧} is realized by its value on a point

K^{0}(B𝕋) = K^{0}(E𝕋 ×_{𝕋} pt) = (K_{𝕋}^{0})_{J}^{∧} = R(𝕋)_{(e(z))}^{∧} = (ℤ[z, z^{−1}])_{(y)}^{∧} = ℤ[[y]],

where the first equality is the Atiyah-Segal completion theorem [5] with J = ker(R(𝕋) ⟶ ℤ) the augmentation ideal, and the other equalities hold since J = (e(z)) and the Euler class e(z) = 1 − z = y. The coproduct is realized by the multiplication map Bμ : B𝕋 × B𝕋 = B(𝕋 × 𝕋) ⟶ 𝕋 since

K^{0}(B𝕋 × B𝕋) = K^{0}(B𝕋) ˆ⊗K^{0}(B𝕋).

2.C. Reconstructing 𝕋-equivariant K-theory from the multiplicative group. To fully justify saying that K-theory is associated to 𝔾_{m} we would need to reverse the process of making the multiplicative group from K-theory. In general it is not known how to make such a construction, but it can be done rationally using the algebraic model for rational 𝕋-spectra [11]. We return to this in Section 9.C below.

2.D. A-equivariant formal groups and A × 𝕋-equivariant Borel K-theory. There is an intermediate stage between the multiplicative group and its completion at the identity: we may complete 𝔾_{m} along a subgroup. This is known as an equivariant formal group [6, 14, 29]. More precisely, we choose a homomorphism ζ : A^{∗} ⟶ 𝔾_{m} of algebraic groups, where A is a finite abelian group and A^{∗} = Hom(A, S^{1}) is the dual group. The equivariant formal group is the formal neighbourhood (𝔾_{m})_{ζ}^{∧} of the image of ζ equipped with the map A^{∗} ⟶ (𝔾_{m})_{ζ}^{∧}.

To obtain an interesting construction, we need some points of finite order. Thus we want ζ to be specified by a map A^{∗} ⟶ Units(k) for a ring k with more units than ℤ. Accordingly, we change base to the version of the multiplicative group 𝔾_{m}|_{k} over k: this is the functor defined on k-algebras l and has ring of functions k[z, z^{−1}]; by the Yoneda lemma, ζ is now given by a homomorphism ζ : A^{∗} ⟶ Units(k) = 𝔾_{m}|_{k}(k) of ordinary groups.

The homomorphism ζ is represented by the Hopf algebra map θ : k[z, z^{−1}] ⟶ k^{A^{∗}} with αth component θ_{α}(z) = 1 − ζ(α) for α ∈ A^{∗}. Thus θ has kernel (∏_{α∈A^{∗}}(1 − ζ(α))) and the ring of functions on (𝔾_{m}|_{k})_{ζ}^{∧} is the completion

𝒪_{(𝔾_{m}|_{k})_{ζ}^{∧}} = k[z, z^{−1}]_{(∏_{α∈A^{∗}}(1−ζ(α)))}^{∧}.

The universal example of this is when k = R(A) = ℤ[A^{∗}] and ζ(α) = α.

This algebraic intermediate stage is also realized topologically, by using a suitable A × 𝕋equivariant Borel version of A-equivariant K-theory. To explain this, we start by observing that K_{A×𝕋}^{0} = R(A × 𝕋) = R(A)[z, z^{−1}] is the ring of functions on the multiplicative group 𝔾_{m}|_{k} over k = R(A), and the coproduct is still induced by μ : 𝕋 × 𝕋 ⟶ 𝕋. We can now construct the ring of functions on the A-equivariant formal group by using a Borel theory. This time we need the universal 𝕋-free A-space E_{A}𝕋, which is the total space of the universal bundle over the classifying space B_{A}𝕋 for A-equivariant line bundles. We may then take the Borel theory K_{A}^{0}(E_{A}𝕋×_{𝕋}X) where X is an A×𝕋-space. Again the Atiyah-Segal completion theorem [1] gives a connection with rings of functions on the equivariant formal group: the ring of functions on (𝔾_{m}|_{k})_{ζ}^{∧} when k = R(A) is

K_{A}^{0}(B_{A}𝕋) = K_{A×𝕋}^{0}(E_{A}𝕋) = (K_{A×𝕋}^{0})_{(e(ℂA⊗z))}^{∧} = (K_{A×𝕋}^{0})_{(∏_{α}(1−αz))}^{∧}.

The components of the fixed point set (B_{A}𝕋)^{A} correspond to the A-equivariant line bundles over a point (i.e., to points of A^{∗}), and the resulting A-map A^{∗} ⟶ B_{A}𝕋 induces the structure map A^{∗} ⟶ (𝔾_{m}|_{k})_{ζ}^{∧} of the equivariant formal group.

3. The shape of a cohomology theory.

Before we turn to other examples we must be more precise about what we mean by a cohomology theory being associated to an algebraic group. In this section we consider the example of K-theory in some detail. The description is phrased so that it can be easily transposed to give general principles in the following sections.

3.A. The definition of the shape. Non-equivariantly, the spheres S^{n} play a dual role. They are the building blocks for spaces and are also invertible as spectra. In G-equivariant homotopy theory, nice spaces are built from the suspended homogeneous spaces S^{n} ∧ G/K_{+}. These are not invertible unless K = G, and the group theory means that the cohomology groups Ẽ_{G}^{∗}(S^{n} ∧ G/K_{+}) = Ẽ_{K}^{∗}(S^{n}) can be complicated as modules over the coefficient ring E_{G}^{∗}. On the other hand, the one point compactification S^{V} of a representation V is invertible as a spectrum, and Ẽ_{G}^{∗}(S^{V}) is often rather simple. For this reason, we may view the values on representation spheres S^{V} as a rather basic invariant of the homology or cohomology theory. Partly because of variance, it is simpler to concentrate on homology, but the analogue for cohomology is also useful. Definition 3.1. The shape of an equivariant homology theory Ẽ_{∗}^{G}(·) is the composite functor

Rep_{ℂ}(G) ⟶ G-spectra ⟶^{Ẽ_{∗}^{G}(·)} AbGp

where Rep_{ℂ}(G) is the category of unitary representations of G and inclusions.

For example, if Ẽ_{∗}^{G}(·) is complex oriented then Ẽ_{∗}^{G}(S^{V}) is a shifted copy of the coefficient group E_{∗}^{G}. However, even in this case the shape includes useful information.

3.B. Equivariant K-theory. We concentrate on the circle group G = 𝕋. In this case, any complex representation is a sum of the one dimensional representations z^{n} for integers n, where z is the natural representaton of 𝕋 on ℂ. Applying K-theory to the inclusion 0 ⟶ V gives a diagram

0 ⟼ K̃_{0}^{𝕋}(S^{0}) = R(𝕋) =  R(T)
↓ ↓ ↓ χ(V) ↓ i_{V}
V ⟼ K̃_{0}^{𝕋}(S^{V}) ≅ R(𝕋) ≅ \frac{1}{χ(V)} · R(T).

The isomorphism K̃_{0}^{𝕋}(S^{V}) ≅ R(𝕋), and the fact that the inclusion S^{0} ⟶ S^{V} induces multiplication by the Euler class χ(V) come from equivariant Bott periodicity. The last isomorphism has replaced multiplication by the Euler class χ(V) by an inclusion, at the price of making 1/χ(V) the generator of the free R(T)-module. For a 1-dimensional representation α we have χ(α) = 1 − α, and χ(V ⊕ W) = χ(V)χ(W), so that we may easily calculate χ(V) in general. In particular for V = z^{n} then we find χ(z^{n}) = 1 − z^{n}, which is familiar as the function on the multiplicative group 𝔾_{m} whose vanishing defines the subgroup 𝔾_{m}[n] := ker(𝔾_{m} ⟶^{n} 𝔾_{m}) of points of order dividing n. Writing 𝒪_{𝔾_{m}}(𝔾_{m}[n]) for the ideal of functions permitted a simple pole on 𝔾_{m}[n], we may summarize the discussion by the calculation

K̃_{0}^{𝕋}(S^{z^{n}}) = \frac{1}{1 − z^{n}} · R(T) = 𝒪_{𝔾_{m}}(𝔾_{m}[n]).

Thus the shape of K-theory is specified by functions on the multiplicative group; more precisely, if V^{𝕋} = 0 we have

K̃_{0}^{𝕋}(S^{V}) = 𝒪_{𝔾_{m}}(D(V))

for a suitable divisor D(V); indeed if V = ∑_{n≠0} a_{n}z^{n} then D(V) = ∑_{n≠0} a_{n}𝔾_{m}[n].

Suspension by the trivial complex representation ϵ (i.e., double real suspension) corresponds to the tangent bundle (Ω_{𝔾_{m}}^{1})^{∨}. It is convenient to prove the dual statement that desuspension by ϵ corresponds to Ω_{𝔾_{m}}^{1}. Since we have K̃_{𝕋}^{0}(S^{W}) = K̃_{0}^{𝕋}(S^{−W}), we work in cohomology to prove K̃_{𝕋}^{0}(S^{2}) ≅ Ω_{𝔾_{m}}^{1}. First note that S^{2z} is obtained from S^{z} by attaching a single 𝕋-free 3-cell, so that we have a cofibre sequence

Σ^{2}𝕋_{+} ⟶ S^{z} ⟶ S^{2z}.

Now apply K̃_{𝕋}^{0}. Remembering that we are now working in cohomology, K̃_{𝕋}^{0}(S^{z}) = J and K̃_{𝕋}^{0}(S^{z}) = J^{2}, where J = 𝒪_{𝔾_{m}}(−(e)) is the ideal of functions vanishing at the identity, and we obtain the a short exact sequence

0 ⟵ K̃_{𝕋}^{0}(Σ^{2}𝕋_{+}) ⟵ J ⟵ J^{2} ⟵ 0.

Hence the value of the non-equivariant theory on S^{2} is the conormal bundle at the identity (i.e., the cotangent space at the identity since we are dealing with a point):

K̃^{0}(S^{2}) = K̃^{0}(Σ^{2}𝕋_{+}) ≅ J/J^{2} = ω_{e}^{1}.

Accordingly, double suspension in cohomology is given by tensoring with ω_{e}^{1}, and in particular the equivariant theory is the module of Kähler differentials

K̃_{𝕋}^{0}(S^{2}) = ω_{e}^{1} ⊗_{K^{0}} K_{𝕋}^{0} ≅ Ω_{𝔾_{m}}^{1}

as claimed, where the final isomorphism is given by using the group structure on 𝔾_{m} to trivialize the module, or in more concrete terms, by using the function z. The fact that Ω_{𝔾_{m}}^{1} is trivializable corresponds to the fact that K_{𝕋}^{∗} is 2-periodic.

Since the K-theory of all complex spheres is in even dimensions, this describes all of K-theory.

3.C. Complex orientable theories. The above discussion also applies to complex orientable cohomology theories E^{∗}(·) and their associated formal groups 𝔾̂. Since different notation may be more familiar to some readers, we provide a brief comparison. The relevant equivariant cohomology is the Borel theory E_{𝕋}^{∗}(X) = E^{∗}(E𝕋 ×_{𝕋} X), so we have

E_{𝕋}^{∗} = E^{∗}(B𝕋)

and

Ẽ_{𝕋}^{∗}(S^{V}) = Ẽ^{∗}(B𝕋^{V}),

where B𝕋^{V} denotes the Thom space of the bundle associated to the representation V. The ring E^{∗}(B𝕋) is complete for the skeletal topology, and is therefore the ring of functions on a formal scheme 𝔾̂, and since we have a completed Künneth theorem for E^{∗}(B𝕋 × B𝕋), multiplication on 𝕋 gives 𝔾̂ the structure of a formal group.

Now consider the ideal

J = ker(E_{𝕋}^{∗} = E_{𝕋}^{∗}(pt) ⟶ E_{𝕋}^{∗}(𝕋) = E^{∗})

of functions vanishing at the identity, which is principal by complex orientability. A generator y of J is a coordinate on 𝔾̂, and the coefficient ring is the power series ring E_{𝕋}^{∗} = E^{∗}[[y]]. The cofibre sequence 𝕋_{+} ⟶ S^{0} ⟶ S^{z} shows that y is the restriction of a Thom class in Ẽ_{𝕋}^{∗}(S^{z}) = Ẽ_{𝕋}^{∗}(BT^{z}). Since z^{n} = n^{∗}z we see that a Thom class for BT^{z^{n}} is given by [n](y), the pull back of y along the nth power map Bn : B𝕋 ⟶ B𝕋.

In these terms, and remembering we are working in cohomology, we recover the usual formula

Ẽ_{𝕋}^{∗}(S^{z^{n}}) = Ẽ^{∗}(BT^{z^{n}}) = ([n](y)) = 𝒪𝔾̂(−𝔾̂[n]).

Specializing, this gives

Ẽ_{𝕋}^{∗}(S^{z}) = Ẽ^{∗}(BT^{z}) = (y) = J = 𝒪𝔾̂(−(e)),

and hence the behaviour under untwisted suspensions

Ẽ^{∗}(S^{2}) ≅ J/J^{2} ≅ ω_{e}^{1}

and

Ẽ_{𝕋}^{∗}(S^{2}) ≅ Ẽ^{∗}(B𝕋_{+} ∧ S^{2}) ≅ Ω1 ; 𝔾̂.

3.D. What is special about K-theory and the multiplicative group? In algebra and topology there are several very special features of the examples described in this section. They are rather obvious in these cases, but the analogues become more interesting as we change the algebraic group and the cohomology theory: we will argue that the algebraic and topological special features correspond precisely. We concentrate on K-theory for simplicity, and we have in mind G = A × 𝕋 for a finite abelian group A.

• The ideal of functions on 𝔾_{m} vanishing at the identity is principal, and K_{G}^{∗}(·) is complex orientable.

• 𝔾_{m} is affine and K_{G}^{∗} is in even degrees.

• The group 𝔾_{m} is one dimensional and K_{G}^{∗} is 2-periodic. We will see in Section 4 that the first property fails for the non-split torus, in Section 5

that the second property fails for the elliptic case, and in Section 7 that the third fails for abelian varieties of dimension ≥ 2.

4. The non-split torus.

We describe a one dimensional algebraic group whose ideal of functions vanishing at the origin is not principal. There is a corresponding cohomology theory which is therefore not complex orientable.

4.A. The non-split torus and its formal group. Let C be a 1-dimensional torus over a ℚ-algebra k, with ring of functions 𝒪. The standard torus is the multiplicative group C = 𝔾_{m} with 𝒪 = k[z, z^{−1}], but there is also a non-split (non-deployé) form 𝔾_{nd} with 𝒪 = k[a, b]/(a^{2} + b^{2} = 1), which we consider here. This is related to 𝔾_{m} since they become isomorphic when we adjoin a square root i of −1. We then have z = a + ib, and the usual formula (a_{1}+ib_{1})(a_{1}+ib_{1}) = (a_{1}a_{2}−b_{1}b_{2})+i(a_{1}b_{2}+a_{2}b_{1}) gives the coproduct a ⟼ a⊗a−b⊗b b ⟼ a ⊗ b + b ⊗ a. Now the counit sets a = 1 and b = 0 so that the ideal of functions vanishing at the identity is (1 − a, b). This is not a principal ideal. Furthermore, even if we complete at the identity, the ideal (1 − a, b) is still not principal. Thus even the Borel theory of a theory associated to the non-split torus will not be complex orientable.

One way to obtain 𝔾_{nd} is as a quotient of 𝔾_{m}|_{k(i)}. Indeed, k[a, b]/(a^{2} + b^{2} = 1) is the fixed ring of k(i)[z, z^{−1}] under the action of the group of order 2 acting by Galois automorphisms on k(i) and exchanging z with z^{−1}.

4.B. Constructing the associated cohomology theory. Rationally, one may construct a 𝕋-equivariant cohomology theory associated to 𝔾_{nd} by the method of Section 9. It would be interesting to have a bundle theoretic construction.

Complex orientable theories E_{𝕋}^{∗}(·) have the property that for any complex representation V the E_{𝕋}^{∗}-module Ẽ_{𝕋}^{∗}(S^{V}) is free on one generator. The cohomology theory associated to 𝔾_{nd} is therefore not complex orientable since its value on S^{z} is the non-principal ideal of functions vanishing at the identity.

5. Elliptic cohomology and elliptic curves.

We now replace the multiplicative group by an elliptic curve C over k. This makes both the algebra and the topology more complicated, but probably the biggest change is that C is no longer affine. For general background on elliptic curves and algebraic geometry see [28, 20].

5.A. Elliptic curves and their formal completions. Because C is not affine we must replace the ring of functions by the sheaf of functions 𝒪 = 𝒪_{C}. Since all regular functions on C are constant, if we want to recover C from a knowledge of functions, we need to permit some poles. These are measured by considering formal sums of points of C, known as divisors. Given a divisor D = ∑_{P} n(P)P on C we may form the line bundle 𝒪(D), of functions which have poles of order ≤ n(P) at P. The Riemann-Roch theorem states that if D has degree ≥ 1 then

dim_{k}H^{0}(C; 𝒪(D)) = deg(D),

where deg(D) = ∑_{P} n(P). Thus in particular if we concentrate on the identity e and consider a multiple D = ne, the sheaf 𝒪(ne) allows functions with a pole of order n ≥ 1 at the identity e, and we obtain n independent functions since dim_{k}H^{0}(C; 𝒪(ne)) = n. Thus if n = 1 we only obtain the constants, for n = 2 we obtain a new function, call it x, and for n = 3 we obtain a new function, call it y. For n = 4 we may use x^{2} as the new function, and for n = 5 we may use xy. However for n = 6 we have two potential new functions, y^{2} and x^{3}. There is therefore a linear relation amongst the 7 functions so far, and this may be put into the Weierstrass form

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

for suitable a_{1}, a_{2}, a_{3}, a_{4}, a_{6} in k. It turns out that this implies all relations, and defines C. Indeed, we may consider the graded ring

k(C, e) := {H^{0}(C; 𝒪(ne))}_{n≥0},

and the above discussion my be restated briefly in geometric language as

C = Proj(k(C, e)).

In the Weierstrass form, we find there is one point at infinity, and we may choose this as the identity of the group C.

5.B. Reflecting the group structure on functions. For an elliptic curve C the graded ring k(C, e) is not a Hopf algebra. This is because the pullback of a line bundle ℒ, such as 𝒪(ne), along the multiplication map C × C ⟶ C does not decompose as a tensor product. However, for any line bundle ℒ we may form ℳ = ℳ(ℒ) = p_{1}^{∗}ℒ ⊗ p_{2}^{∗}ℒ, and then if we define ξ : C × C ⟶ C × C by ξ(x, y) = (x + y, x − y), the theorem of the square shows that ξ^{∗}ℳ ≅ ℳ^{⊗2}. This gives

ξ^{∗} : H^{0}(C; ℒ) ⊗ H^{0}(C; ℒ) ⟶ H^{0}(C; ℒ^{⊗2}) ⊗ H^{0}(C; ℒ^{⊗2}),

and hence the multiplication on C is reflected in the functions. The properties satisfied by the function ξ are fairly complicated [24].

On the other hand, we may construct a formal group by completing C at the identity, and since C is a smooth curve the resulting representing ring is isomorphic k[[t]]. Since C × C is a smooth surface, if we complete it at its identity we obtain a ring of functions

k[[1 ⊗ t, t ⊗ 1]].

Thus, the group operation on C gives a coproduct on k[[t]], and when we choose a suitable coordinate, this takes the familiar form of an elliptic formal group law.

5.C. Elliptic cohomology theories and elliptic formal groups. An elliptic spectrum in the sense of Ando-Hopkins-Strickland [4] is a triple (E, C, ϕ) where E is a 2-periodic even spectrum, necessarily complex orientable, an elliptic curve C over E^{0} and an isomorphism ϕ between the formal group E^{0}(B𝕋) of E and C_{e}^{∧}. In the present terms, the 𝕋-equivariant Borel theory of E is associated to the formal group C_{e}^{∧}.

5.D. 𝕋-equivariant elliptic cohomology theories and elliptic curves. Since C is not determined by a ring of functions, it is now rather less clear what it means for a 𝕋-equivariant cohomology theory to be associated to an elliptic curve. However the way the graded ring k(C, e) was used in reconstructing C suggests that the sections of various line bundles should occur as values of the cohomology theory. The way the multiplication on C was reflected in ξ : C × C ⟶ C × C suggests that the 𝕋 × 𝕋 equivariant theory for C × C should play a role in describing this structure. All of this structure should reduce to elliptic formal groups in the Borel theory.

In Section 10 we make precise the properties it is natural to hope for. Such a cohomology theory has been constructed [15] by the methods described in Section 8 below.

5.E. A-equivariant elliptic cohomology theories and elliptic formal groups. Just as we constructed A-equivariant formal groups from 𝔾_{m} for a finite abelian group A, we may construct them by completing around the image of ζ : A^{∗} ⟶ C. Strickland has recently shown [30] that provided 2 is inverted, these correspond to complex oriented A-equivariant cohomology theories. Devoto earlier constructed a version for the Jacobi quartic when the group order is inverted [7] and investigated its role in index theory.

6. 𝕋-equivariant elliptic cohomology.

If we are looking for other interesting shapes in the sense of Definition 3.1, it is natural to consider other ways of getting a graded abelian group from a representation, and we might consider replacing the triangulated category of spectra by a different one, such as the category of sheaves on an algebraic variety.

6.A. Sheaves on a curve. As with K-theory, we expect to associate to an invertible (i.e., one dimensional) representation α, an invertible sheaf (i.e., a line bundle) ℒ_{α} = 𝒪_{C}(D(α)) on the curve C. Sums of representations should correspond to sums of divisors or tensor products of bundles:

\begin{aligned}α & ⟼ & ℒ_{α} & = & 𝒪_{C}(D(α)) \\ V ⊕ W & ⟼ & ℒ_{V} ⊗ ℒ_{W} & = & 𝒪_{C}(D(V) + D(W)). \end{aligned}

One also hopes to make the tensor product of representations correspond to some structure in the category of sheaves, but this seems less straightforward. One can then obtain a potential shape of a homology theory Ẽ_{∗}^{𝕋}(·) by taking sheaf cohomology of the associated line bundle

Ẽ_{−i}^{𝕋}(S^{V}) = H^{i}(C; 𝒪_{C}(D(V)))

for suitable values of i. Since C is a curve, the sheaf cohomology on the right will only be non-zero for i = 0, 1, so we expect to deal with other values of i by twisting the sheaf with the line bundle ℒ_{ϵ} = 𝒪_{C}(D(ϵ)) associated to the trivial representation ϵ of 𝕋 on ℂ; just as in the case of the multiplicative group, we expect ℒ_{ϵ} to have a rather different character to the line bundles associated to representations with no fixed points.

When C is an elliptic curve, the identity element e provides a preferred divisor, and the example of the multiplicative group suggests we should choose D(z) = e. We then observe that z^{n} = n^{∗}z and hence choose

D(z^{n}) = n^{∗}e = C[n] = ker(C ⟶^{n} C),

and this too is analogous to the case of the multiplicative group. Again the example of the multiplicative group suggests the trivial representation ϵ should be associated to the tangent sheaf ℒ_{ϵ} = (Ω_{C}^{1})^{∨}. However, since C is a group, this tangent sheaf is trivializable. Thus we have the association

\begin{aligned}0 & ⟼ & ℒ_{0} & = & 𝒪_{C} \\ ϵ & ⟼ & ℒ_{ϵ} & = & (Ω_{C}^{1})^{∨} \\ z & ⟼ & ℒ_{z} & = & 𝒪_{C}(e) \\ z^{n} & ⟼ & ℒ_{z^{n}} & = & 𝒪_{C}(C[n]) \\ V & ⟼ & ℒ_{V} & = & 𝒪_{C}(D(V)); \end{aligned}

as with K-theory and 𝔾_{m}, if V = ∑_{n≠0} a_{n}z^{n}, we take D(V) = ∑_{n} a_{n}C[n].

6.B. An associated cohomology theory. In fact one may associate a 𝕋-equivariant cohomology theory to an elliptic curve. Theorem 6.1. [15] If C is an elliptic curve over a field of characteristic 0 then there is a 𝕋-equivariant cohomology of the shape described in Subsection 6.A. In particular, it is 2-periodic and for i = 0, 1 we have

Ẽ_{−i}^{𝕋}(S^{V}) = H^{i}(C; 𝒪(D(V)))

and

Ẽ_{𝕋}^{i}(S^{V}) = H^{i}(C; 𝒪(−D(V))).

For example Ẽ_{0}^{𝕋}(S^{nz}) = k^{n} for n ≥ 1, so that the theory is not complex orientable, and Ẽ_{i}^{𝕋}(S^{0}) = k for all i, so that the theory is not concentrated in even degrees.

The construction depends on an auxiliary piece of structure: a coordinate divisor, which is the divisor of a function t_{e} vanishing to the first order at e with all zeros and poles at points of finite order. For instance if the points of order 2 are P, P′, P′′ then e + P − P′ − P′′ is a coordinate divisor. From the function t_{e} one can construct functions t_{s} vanishing to the first order at points of exact order s for each s ≥ 2. Indeed, the space of functions vanishing at the n_{s} points of exact order s and with a pole of order ≤ n_{s} at e is one dimensional, and we may normalize by requiring that t_{s}/t_{e}^{n_{s}} has the value 1 at e.

In fact, the connection between elliptic curves and 𝕋-equivariant cohomology goes a bit further. The representing spectrum EC is a commutative ring spectrum, and we may consider modules over it. Inverting equivariant equivalences (which are equivalences in 𝕋-fixed points and in 𝕋[n]-fixed points for all n ≥ 1), we obtain its homotopy category Ho(𝕋-EC-mod). On the other hand we may consider sheaves of modules over 𝒪_{C}; here we only consider sheaves as taking values on open sets which are the complements of complete sets of points of specified finite order. Inverting strong equivalences (which are equivalences in all H^{∗}(C; 𝒪 ⊗ (·)) and in H^{∗}(C; 𝒪(C[n]) ⊗ (·)) for all n ≥ 1), we obtain its derived category D(𝒪_{C}-mod). These algebraic and topological derived categories are equivalent. Theorem 6.2. [15] There is an equivalence of triangulated categories

D(𝒪_{C}-mod) ≃ D(𝕋-EC-mod).

7. Shapes from projective varieties.

Encouraged by the success described in the previous section, we seek suitable shapes associated to curves of higher genus, and discuss the properties of a cohomology theory with such a shape. At present the author does not know if cohomology theories with this shape exist, but for suitable curves, one can construct what would be 𝕋-equivariant restrictions of them. The geometry is too intricate to be described here in detail, but we give an outline.

7.A. Sheaves on a Jacobian. If C is a curve of genus g, we may consider shapes associated to the Jacobian J = Jac(C) (for background see [23, 22, 21]). If the genus is 1 then C = J, and we are reduced to the case of an elliptic curve, as discussed in Subsection 6.A above.

Suppose C is a curve over the complex numbers. The genus g of C may be calculated either topologically via the singular homology H_{1}(C; ℤ) ≅ ℤ^{2g} or geometrically, via the space H^{0}(C; Ω_{C}^{1}) ≅ ℂ^{g} of holomorphic 1-forms. We may construct the Jacobian Jac(C) as follows. A topological 1-chain Z on C defines a linear form on the holomorphic 1-forms by integration ω ⟼ ∫_{Z} ω; this defines an embedding

H_{1}(C; ℤ) ⟶ H^{0}(C; Ω_{C}^{1})^{∨},

and Jac(C) is the quotient.

The most important features of J are (1) that it is a group, (2) that it has a preferred divisor Θ (the theta divisor), uniquely specified up to translation and (3) that it has the structure of an algebraic variety. Any other principally polarized abelian variety of dimension g would do as well for the general discussion, although special properties of the abelian variety will probably required to actually construct such a theory.

Since J is of dimension g, one might expect the associated group of equivariance to be a torus 𝕋^{g} of dimension g. The group of torsion points of J is J_{tors} ≅ (ℚ/ℤ)^{2g} as an abelian group, whereas (𝕋^{g})^{∗} ≅ ℤ^{g}, so we may choose a complete level structure

(𝕋^{g})^{∗} ⊗ (ℚ/ℤ × ℚ/ℤ) ≅ J_{tors}.

For the Jacobian we have a natural isomorphism J_{tors} = H_{1}(C; ℚ/ℤ), so it is natural to choose the factors of 𝕋^{g} to correspond to a decomposition of H_{1}(C; ℤ) into a sum of 2dimensional hyperbolic forms; in more concrete terms we choose copies of ℚ/ℤ × ℚ/ℤ to correspond to pairs of curves on C with a single point of intersection. For some purposes, it may be more appropropriate to use the torus T = Hom(NS(J), 𝕋) dual to the Neron-Severi group as the group of equivariance. Here NS(J) := im(H^{1}(J; 𝒪^{×}) ⟶ H^{2}(J; ℤ)) is a free abelian group. Its rank is called the Picard number, and this is ≤ g^{2}.

With this structure available, we may attempt to define a suitable shape. It turns out that the divisors n^{∗}Θ and n^{2}Θ define isomorphic line bundles. However, the divisor n^{∗}Θ is usually not decomposable. It is therefore better to replace it by a sum of translates of Θ. Indeed, if z is a generating one dimensional representation, our level structure provides an embedding z̃ : ℚ/ℤ × ℚ/ℤ ⟶ J_{tors}, and we may consider Θ_{z,n} := ∑_{na=0} Θ_{z̃(a)}, where Θ_{x} denotes the translation of Θ by x. The divisor Θ_{z,n} is of degree n^{2}, and linearly equivalent to n^{2}Θ since {z̃(a) | na = 0} is a subgroup of J_{tors} with the property that the sum of its elements is the identity. This leads to the following association.

\begin{aligned}0 & ⟼ & ℒ_{0} & = & 𝒪_{J} \\ ϵ & ⟼ & ℒ_{ϵ} & = & ? \\ z & ⟼ & ℒ_{z} & = & 𝒪_{J}(Θ) \\ z^{n} & ⟼ & ℒ_{z^{n}} & = & 𝒪_{J}(Θ_{z,n}) \\ V & ⟼ & ℒ_{V} & = & 𝒪(D(V)) \end{aligned}

The constraints on the choice of ℒ_{ϵ} are suggested by the discussion in Subsection 7.B. Note in particular that (Ω_{C}^{1})^{∨} is not a line bundle for g ≥ 2, so we need another candidate.

Since J is of dimension g, the sheaf cohomology of these line bundles will be in degrees between 0 and g, so we now only expect a spectral sequence

E_{2}^{i,2j} = H^{i}(J; 𝒪_{J}(D(V)) ⊗ ℒ_{ϵ}^{j}) ⇒ Ẽ_{∗}^{𝕋^{g}}(S^{V}),

where odd rows are zero. The cohomology of all these sheaves is well-known, and for g = 2, a spectral sequence of this form necessarily collapses for some natural choices of ℒ_{ϵ}. This leads to predictions for the values of the cohomology theory on spheres.

7.B. Further constraints. We argue here that if there is a cohomology theory associated to an abelian surface, it cannot be 2-periodic.

Suppose then that C is a curve of genus 2, and that E_{𝕋^{2}}^{∗}(·) is a cohomology theory associated to it. Since g = 2, we may identify the theta divisor Θ with the curve C embedded in J by the Abel-Jacobi map (once we have picked a basepoint on C).

We choose a circle subgroup 𝕋 and restrict attention to the 𝕋-equivariant cohomology theory. Now we consider the two cofibre sequences

S^{0} ⟶ S^{z} ⟶ Σ𝕋_{+} and S^{z} ⟶ S^{2z} ⟶ Σ^{3}𝕋_{+},

where the second is obtained by smashing the first with S^{z}. The point to note is the untwisting 𝕋_{+} ∧ S^{z} ≃ 𝕋_{+} ∧ S^{2} that has been used to identify the last term in the second cofibre sequence. These cofibre sequences should correspond to the exact sequences

𝒪 ⟶ 𝒪(C) ⟶ Q_{1} and 𝒪(C) ⟶ 𝒪(2C) ⟶ Q_{2}

of sheaves over J, where the second is obtained by tensoring the first with 𝒪(C). Here we note that Q_{1} and Q_{2} are supported on C. Indeed, we have Q_{1} ≅ i_{∗}q_{1} and Q_{2} ≅ i_{∗}q_{2} for suitable sheaves q_{1} and q_{2} on C. We even expect q_{1} and q_{2} to be line bundles on C. Since the second exact sequence is obtained from the first by tensoring with 𝒪(C), we find

Q_{2} ≅ Q_{1} ⊗ 𝒪(C) ≅ i_{∗}(q_{1} ⊗ 𝒪(C)|_{C}).

On the other hand, by the untwisting we have

Q_{2} ≅ Q_{1} ⊗ ℒ_{ϵ} ≅ i_{∗}(q_{1} ⊗ (ℒ_{ϵ})|_{C}).

This gives

ℒ_{ϵ}|_{C} ≅ 𝒪_{J}(C)|_{C} ≅ Ω_{C}^{1},

where the final equality is the adjunction formula. An alternative argument adapts the discussion in Subsection 3.B to show ℒ_{ϵ}|_{C} is the normal bundle of C in J; we see that this is Ω_{C}^{1}, since the canonical bundle on J is trivial.

Since C is of genus 2, the line bundle Ω_{C}^{1} is of degree 2, and H^{0}(C; (Ω_{C}^{1})^{⊗n}) depends on n (it is of dimension 2 for n = 1 and of dimension 2n − 1 for n ≥ 2). This shows that even the non-equivariant form E^{∗}(·) = E_{𝕋}^{∗}(𝕋 × (·)) of the theory will not be 2-periodic.

8. Rational equivariant cohomology theories.

So far we have only presented evidence for a one-way connection between algebraic groups and equivariant cohomology theories. In the one dimensional case, suitable cohomology theories E_{𝕋}^{∗}(·) give rise to one dimensional algebraic groups 𝔾 by evaluation on suitable spaces. On the other hand, we have argued that a one dimensional algebraic group 𝔾 suggests the values of an equivariant cohomology theory E_{𝕋}^{∗}(·) on spheres (i.e., the shape of the theory). In this section we show that the shape of a cohomology theory is very close to determining a cohomology theory over the rationals, and that it does determine a cohomology theory in favourable cases.

8.A. Algebraic models for categories of rational cohomology theories. The idea is that for any compact Lie group G there is an abelian category 𝒜(G) modelling rational G-equivariant cohomology theories. As usual, it is worth working with the category of objects representing the cohomology theories, namely rational G-spectra. This category admits the structure of a Quillen model category, and its homotopy category gives the cohomology theories. Thus homotopy equivalence classes of G-spectra correspond to Gequivariant rational cohomology theories, and all cohomology theories are of the form

E_{G}^{∗}(X) = [X, E]_{G}^{∗}.

On a practical level, we want to be able to calculate in this homotopy category, but if we understand the category completely we can also construct interesting new cohomology theories. The idea is that objects of 𝒜(G) should be a rather small, and based on information easily accessible from the cohomology theories they represent.

One natural way to construct simpler cohomology theories from E_{G}^{∗}(·) is to take Borel cohomology of fixed points. Indeed, if X is a G-space we may form the H-fixed point space X^{H}; the group W_{G}(H) = N_{G}(H)/H acts on this, as does the maximal torus T W_{G}(H) of its identity component. We may therefore form the Borel theory Ẽ_{T W_{G}(H)}^{∗}(ET W_{G}(H)_{+} ∧ X^{H}). At least when G is abelian, this is closely related to the shape of the cohomology theory. Indeed, each of the two steps can be performed (up to extension) using the shape. For the first step, we use the formal localization theorem, which states that in suitable circumstances X^{H} may be obtained by inverting formal Euler classes. To explain this, we suppose H is normal, and consider the infinite sphere S^{∞V (H)} = lim_{→}_{V^{H}=0} S^{V}. We say that G has enough

representations to detect H if (S^{∞V (H)})^{K} is contractible when K does not contain H. When H is normal in G and G has enough representations to detect H, obstruction theory gives

[T^{H}, X^{H}]^{G/H} = [T, X ∧ S^{∞V (H)}]^{G}

as required. For the second step, we note that for any torus T we may form the universal free T -space ET from spheres. More precisely, if we choose a direct product decomposition T = 𝕋 × ⋯ × 𝕋 and z_{1}, z_{2}, …, z_{r} are the natural representations of the factors, we have

ET ≃ S(∞z_{1}) × S(∞z_{2}) × ⋯ × S(∞z_{r}).

The factors are related to spheres relevant to the shape by the cofibre sequences

S(∞z)_{+} ⟶ S^{0} ⟶ S^{∞z},

so that we obtain a model for ET_{+} like the stable Koszul complex. Conjecture 8.1. For a compact Lie group G there is an abelian category 𝒜(G) and a Quillen equivalence

G-spectra/ℚ ≃ dg𝒜(G)

such that (1) 𝒜(G) is abelian

(2) InjDim(𝒜(G)) = rank(G)

(3) The category consists of sheaves of modules over a space of closed subgroups of G; the object corresponding to a cohomology theory E_{G}^{∗}(·) has fibre over H built from the Borel theory E_{T W_{G}(H)}^{∗}(ET W_{G}(H)_{+} ∧ X^{H}). The additional structure is built from these Borel theories using their relation under localization and inflation.

(4) The model of E_{G}^{∗}(·) is built from its shape and a little extra structure.

8.B. Consequences of the conjecture. Note immediately that if the conjecture holds we have an equivalence of homotopy categories

Ho(G-spectra/ℚ) ≃ D(𝒜(G))

as triangulated categories. This reduces to algebra the problem of classifying rational equivariant cohomology theories and the process of calculation with them. Furthermore, it provides a universal homology theory

π_{∗}^{𝒜(G)} : G-spectra ⟶ 𝒜(G)

and an Adams spectral sequence

Ext_{𝒜(G)}^{∗,∗} (π_{∗}^{𝒜(G)} (X), π_{∗}^{𝒜(G)} (Y)) ⇒ [X, Y]_{∗}^{G}

for calculation. Finally, because of the injective dimension of 𝒜(G), the Adams spectral sequence is only non-zero on s-line for 0 ≤ s ≤ r, so the calculation is very accessible.

8.C. Status of the conjecture.

• G finite. The conjecture is true. Indeed, all rational cohomology theories are ordinary. This follows easily from three facts: (1) rational cohomology is rational homotopy, (2) the rational cohomology of a finite group is trivial, and (3) rational Mackey functors are all injective. The idempotents in the rationalized Burnside ring allow us to express this in terms of representation theory, so that

𝒜(G) = ∏_{(H)} ℚW_{G}(H)-mod.

• The circle group G = 𝕋. Again the theorem is true. Indeed, [11] constructs 𝒜(G) and shows that there is a triangulated equivalence of homotopy categories. Shipley [27] upgraded this to a Quillen equivalence. The model is briefly described in Section 9 below.

• The groups G = O(2), SO(3) and their double covers. In this case the equivalence of homotopy categories is proved in [12, 13].

• The tori G = 𝕋^{g}. The Adams spectral sequence exists [16], and in [17, 18] we work towards showing that the Quillen equivalence holds.

9. The model for the circle group G = 𝕋.

9.A. The category 𝒜(𝕋). The objects of 𝒜(𝕋) are sheaves over the discrete set ℱ ∪ {𝕋}, where ℱ is the set of finite subgroups. It is thus natural that the structure sheaf should be the constant sheaf ℚ[c], with global sections

R = map(ℱ, ℚ[c]) = ∏_{n} ℚ[c].

Next we need to discuss suspension. If w : ℱ ⟶ ℤ_{≥0} is a function zero almost everywhere, we may form

Σ^{w}R = ∏_{n} Σ^{2w(n)}ℚ[c].

For example, if W is a complex representation with W^{𝕋} = 0 we might take w(n) = dim_{ℂ}(W^{𝕋[n]}), and Σ^{w} would correpond to suspension by W; this explains the factor of 2. The inclusion R ⟶ Σ^{w}R corresponds to the inclusion S^{0} ⟶ S^{W}, and hence we think of it as division by a universal Euler class. Now form

t = ⋃_{w} Σ^{w}R = ℰ^{−1}R,

where the last equality refers to the idea that the map R ⟶ Σ^{w}R is multiplication by an Euler class, so that in forming the union we have inverted the multiplicative set of Euler classes.

The objects M of 𝒜(𝕋) are of the form

M = (N ⟶^{β} t ⊗ V),

where N is an R-module, and β is a map of R-modules which becomes an isomorphism when ℰ is inverted. The morphisms are diagrams

M = (N ⟶ t ⊗ V)
↓ θ ↓ ↓ 1 ⊗ ϕ
M′ = (N′ ⟶ t ⊗ V′)

It is not hard to show that 𝒜(𝕋) is abelian and of injective dimension 1. Theorem 9.1. [11, 27] There is a Quillen equivalence

𝕋-spectra/ℚ ≃ dg𝒜(𝕋).

9.B. Rigid even objects. Consider an object M = (N ⟶ t ⊗ V) of 𝒜(𝕋). If N is concentrated in even degrees and ℰ-torsion free we have

N = ker(t ⊗ V ⟶ T)

where T is the ℰ-torsion module (t ⊗ V)/N. Furthermore, it then turns out that T = ⨁_{n} T_{n} where T_{n} is a torsion module over ℚ[c]. We may thus define the object M by using the map

q : t ⊗ V ⟶ ⨁_{n} T_{n},

and this always defines an object M provided q is surjective. Now if M represents the homology theory E_{∗}^{𝕋}(·) it turns out that we can very nearly recover M from the shape of E_{∗}^{𝕋}(·):

• V = lim_{→}_{W^{𝕋}=0} Ẽ_{∗}^{𝕋}(S^{W})

• T = cok(Ẽ_{∗}^{𝕋}(S^{0}) ⟶^{i} lim_{→}_{W^{𝕋}=0} Ẽ_{∗}^{𝕋}(S^{W})).

The reason that the shape does not quite determine the object is that the map q also encodes information about division by Euler classes. Summary 9.2. If i is injective, we may almost write down the object ME of 𝒜(T) representing the homology theory E_{∗}^{𝕋}(·) purely in terms of the shape.

9.C. Construction of a cohomology theory from an algebraic group. When C is a 1-dimensional algebraic group we may now describe the construction of a 2-periodic rational 𝕋-equivariant cohomology theory E_{𝕋}^{∗}(·) associated to it.

Indeed, since the theory is 2-periodic we need only describe the model in degrees 0 and 1, and we take the odd part to be zero. In degree 0 we take

V_{0} = lim_{→}_{W^{𝕋}=0} Ẽ_{∗}^{𝕋}(S^{W}) = lim_{→}_{W^{𝕋}=0} H^{0}(C; 𝒪(D(W)) = 𝒦_{C},

where 𝒦_{C} is the ring of meromorphic functions with poles only at points of finite order. For the torsion we take

T_{0} = H^{0}(C; 𝒦_{C}/𝒪_{C}) = ⨁_{s} H_{C⟨s⟩}^{1}(C),

where C⟨s⟩ consists of the points of exact order s and H_{C⟨s⟩}^{1}(C) is the local cohomology, and therefore consists of the principal parts of functions at points of exact order s.

Finally, to define the map q : t ⊗ V_{0} ⟶ T_{0} we must choose some additional structure. For each s ≥ 1 we must choose a function t_{s} vanishing to the first order at points of C⟨s⟩ and with poles only at points of finite order. The geometry of C may suggest natural ways to make this choice but for now we assume the choice has been made. For instance if 𝔾 = 𝔾_{m} we may take t_{s} to be the sth cyclotomic polynomial ϕ_{s}(z), and for an elliptic curve we may use the functions constructed from the coordinate divisor in Section 6.

Having made this choice, we may define q(c^{w} ⊗ f) for a meromorphic function f to have as sth component

q_{s}(c^{w} ⊗ f) = \overline{t_{s}^{w(s)}f}.

In fact this is legitimate since t_{s}^{w(s)}f is regular on C⟨s⟩ for all but finitely many values of s, and in fact q is determined by the values on these elements.

9.D. The shape of a rigid even spectrum. The sphere S^{W} is modelled by (Σ^{w}R ⟶ t⊗ℚ. From this it is easy to calculate Hom_{𝒜(𝕋)} (S^{W} , ME) and Ext_{𝒜(𝕋)} (S^{W} , ME). This gives

Ẽ_{0}^{𝕋}(S^{W}) = Ẽ_{𝕋}^{0}(S^{−W}) = ker(q : c^{w} ⊗ 𝒦_{C} ⟶ T),

and

Ẽ_{−1}^{𝕋}(S^{W}) = Ẽ_{𝕋}^{1}(S^{−W}) = cok(q : c^{w} ⊗ 𝒦_{C} ⟶ T).

Since functions are regular if they are regular at all points, it is clear that

H^{0}(C; 𝒪(D(W))) = ker(q : c^{−w} ⊗ 𝒦_{C} ⟶ T) = Ẽ_{𝕋}^{0}(S^{W}),

as required. The argument for Ẽ_{−1}^{𝕋}(S^{W}) involves a little more geometry [15, Section 5].

10. Reflecting the group structure of the elliptic curve.

We explain here how the map ξ^{∗} of Subsection 5.B should be realized in topology. Since we do not yet have a complete algebraic model of rational 𝕋 × 𝕋-equivariant spectra, we can only show that the structures arising in algebra would be precisely mirrored in topology under the assumption that the model works as expected.

Suppose there exists a 𝕋×𝕋-equivariant cohomology theory with shape suggested by C×C. Constructing such a theory is significantly easier than constructing a 𝕋×𝕋-equivariant theory for an arbitrary abelian surface. To the representation w^{i} ⊗ z^{j} of 𝕋 × 𝕋 we associate the divisor C[i] × C[j] and extend this to arbitrary representations in the usual way. The theory E(C × C)_{∗}^{𝕋×𝕋}(·) should then come with a spectral sequence

H^{∗}(C × C; 𝒪_{C×C}(D(W))) ⇒ { \atop E(C × C)}_{∗}^{𝕋×𝕋}(S^{W}).

Since some line bundles have cohomology in degree 2, this does not determine { \atop E(C × C)}_{∗}^{𝕋×𝕋}(S^{W}) in general. However when 𝒪_{C×C}(D(W)) has no cohomology in dimension 2 we would find

{ \atop E(C × C)}_{0}^{𝕋×𝕋}(S^{W}) = H^{0}(C × C; 𝒪_{C×C}(D(W))).

Next, the map ξ : C × C ⟶ C × C is an isogeny with kernel

ΔA[2] = {(a, a) | a + a = e}.

We also consider the corresponding group homomorphism

ξ̂: 𝕋 × 𝕋 ⟶ 𝕋 × 𝕋,

defined by ξ̂(w, z) = (wz, w/z), which is surjective with kernel

Δ𝕋[2] = {(z, z) | z^{2} = 1}.

To minimize confusion, we identify the second 𝕋 × 𝕋 with \overline{𝕋 × 𝕋} = (𝕋 × 𝕋)/Δ𝕋[2]. The map ξ should correspond to a map

ξ_{i}^{∗} : inf𝕋×𝕋 ; \overline{𝕋×𝕋}E(C × C) ⟶ E(C × C)

(i for inflation) of 𝕋 × 𝕋-spectra or adjointly, to a map

ξ_{f}^{∗} : E(C × C) ⟶ E(C × C)^{Δ𝕋[2]}

(f for fixed point) of \overline{𝕋 × 𝕋}-spectra. Lemma 10.1. For any representation \overline{W} of \overline{𝕋 × 𝕋}, the map ξ_{f} induces

ξ_{f}^{∗} : E(C × C)\overline{𝕋×𝕋} ; 0(S\overline{W}) ⟶ E(C × C)_{0}^{𝕋×𝕋}(S\overline{W}).

Proof: The map ξ_{f}^{∗} induces

ξ_{f}^{∗} : [S^{0}, S\overline{W} ∧ E(C × C)]\overline{𝕋×𝕋} ; 0 ⟶ [S^{0}, S\overline{W} ∧ E(C × C)^{Δ𝕋[2]}]\overline{𝕋×𝕋} ; 0,

so it suffices to identify the domain and codomain. By definition { \atop E(C × C)}\overline{𝕋×𝕋} ; 0(S\overline{W}) = [S^{0}, S\overline{W} ∧ E(C × C)]\overline{𝕋×𝕋} ; 0 so we turn to the codomain and calculate

[S^{0}, S\overline{W} ∧ E(C × C)^{Δ𝕋[2]}]\overline{𝕋×𝕋} ; 0 = [S^{−\overline{W}} , E(C × C)^{Δ𝕋[2]}]\overline{𝕋×𝕋} ; 0
= [S^{−\overline{W}} , E(C × C)]_{0}^{𝕋×𝕋}
= [S^{0}, S\overline{W} ∧ E(C × C)]_{0}^{𝕋×𝕋}
= { \atop E(C × C)}_{0}^{𝕋×𝕋}(S\overline{W}).
□

To model M = p_{1}^{∗}L ⊗ p_{2}^{∗}L with L = 𝒪(D(W)) for a representation W of 𝕋 we take \overline{W} = (W ⊗ 1) ⊕ (1 ⊗ W). Direct sum of representations corresponds to tensor product of line bundles and to sums of divisors, so if

W corresponds to the line bundle L and the divisor D(W), then

\overline{W} corresponds to the line bundle p_{1}^{∗}L ⊗ p_{2}^{∗}L and the divisor [D(W) × C] + [C × D(W)].

Viewed as a representation of 𝕋 × 𝕋 by pullback along ξ̂ we find

ξ̂^{∗}(\overline{W}) = ξ̂_{1}^{∗}W ⊕ ξ̂_{2}^{∗}W.

In particular if W = z^{n} we find

ξ̂^{∗}(\overline{W}) = (w^{n} ⊗ z^{n}) ⊕ (w^{n} ⊗ z^{−n}).

Finally, we need to observe that for any n, the bundles associated to

(w^{n} ⊗ z^{n}) ⊕ (w^{n} ⊗ z^{−n}) and (w^{2n} ⊗ 1) ⊕ (1 ⊗ z^{2n})

are isomorphic: this is precisely the same argument as showed ξ^{∗}M ≅ M^{2} above. With L = 𝒪(D(W)), we thus expect a commutative diagram

H^{0}(C; L)^{⊗2} = H^{0}(C × C; p_{1}^{∗}L ⊗ p_{2}^{∗}L) ⟶^{ξ^{∗}} H^{0}(C × C; p_{1}^{∗}L^{2} ⊗ p_{2}^{∗}L^{2}) = H^{0}(C; L^{2})^{⊗2}
↓  ↓
{ \atop E(C × C)}\overline{𝕋×𝕋} ; 0(S\overline{W}) ⟶^{ξ_{f}^{∗}}  { \atop E(C × C)}_{0}^{𝕋×𝕋}(S\overline{W}).

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Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.

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