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RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY.

J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU

Abstract. We give a functorial construction of a rational S1-equivariant cohomology the-ory from an elliptic curve equipped with suitable coordinate data. The elliptic curve may berecovered from the cohomology theory; indeed, the value of the cohomology theory on thecompactification of an S1-representation is given by the sheaf cohomology of a suitable linebundle on the curve. The construction is easy: by considering functions on the elliptic curvewith specified poles one may write down the representing S1-spectrum in the first author’salgebraic model of rational S1-spectra [6].

Contents

1. Introduction. 12. Formal groups from complex oriented theories. 33. The model for rational T-spectra. 54. The affine case: T-equivariant cohomology theories from additive and

multiplicative groups. 95. Elliptic curves. 136. Coordinate data 147. Local cohomology sheaves on elliptic curves. 158. A cohomology theory associated to an elliptic curve. 179. Multiplicative properties. 20References 21

1. Introduction.

Two of the most important cohomology theories are associated to one dimensional groupschemes in a way which is clearest in the equivariant context. Ordinary cohomology of theBorel construction is associated to the additive group and equivariant K theory is associatedto the multiplicative group. It is therefore natural to hope for an equivariant cohomologytheory associated to an elliptic curve A, and it is the purpose of the present note to constructsuch a theory over the rationals which is equivariant for the circle group. A programmeto extend this work to higher dimensional abelian varieties and higher dimensional tori isunderway [7, 8, 9].

Let T denote the circle group, and z denote its natural representation on the complexnumbers. The main purpose of this paper is to construct a rational T-equivariant cohomology

JPCG and MJH are grateful to the Mathematisches Forschungsinstitut Oberwolfach for the opportunityto talk at the 1998 Homotopietheorie meeting, and to JPCG’s audience for tolerating the resulting delay.JPCG is grateful to M.Ando for useful conversations, and to H.R.Miller and the referee for stimulatingcomments on earlier versions of this paper.

1

2 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU

theory EA∗T(·) associated to any elliptic curve A over a Q-algebra. We write A[n] for thepoints of order dividing n in A. The properties of the cohomology theory when we work overa field may be summarized as follows; we give full details in Section 8 below.

Theorem 1.1. For any elliptic curve A over a field k of characteristic 0, there is a 2-periodic multiplicative rational T-equivariant cohomology theory EA∗T(·). The coefficient ringin degrees 0 and 1 is related to cohomology of the structure sheaf O by

EA∗T∼= H∗(A;O),

and, more generally, the value on the one point compactification SW of the representationW =

∑n anz

n gives the sheaf cohomology of an associated line bundle O(−D(W )), whereD(W ) =

∑n anA[n]:

ẼA∗T(S

W ) ∼= H∗(A;O(−D(W ))and in homology we have

ẼAT∗ (S

W ) ∼= H∗(A;O(D(W )).The construction and the isomorphisms in the statement are natural: for this it is necessaryto specify suitable coordinate data on the elliptic curve.

The first version of T-equivariant elliptic cohomology was constructed by Grojnowksi in1994 [10]. He was interested in implications for the representation theory of certain ellipticalgebras: these implications are the subject of the work of Ginzburg-Kapranov-Vasserot [5]and the context is explained further in [4]. For this purpose it was sufficient to construct atheory on finite complexes taking values in sheaves over the elliptic curve. Later Rosu [13]used this sheaf-valued theory to give a proof of Witten’s rigidity theorem for the equivariantelliptic genus of a spin manifold with non-trivial T-action. Ando [1] has related the sheafvalued theory to the representation theory of loop groups.

However, to exploit the theory fully, it is essential to have a theory defined on generalT-spaces and T-spectra, and to have a conventional group-valued theory represented by aT-spectrum. This allows one to use the full apparatus of equivariant stable homotopy theory.For example, twisted pushforward maps are immediate consequences of Atiyah duality; inmore concrete terms, it allows one to calculate the theory on free loop spaces, and to describealgebras of operations. It is also likely to be useful in constructing an integral version ofthe theory, and we hope it may also prove useful in the continuing search for a geometricdefinition of elliptic cohomology.

The theory we construct has these desirable properties, whilst retaining a very close con-nection with the geometry of the underlying elliptic curve. Our construction directly modelsthe representing spectrum EA in the first author’s algebraic model As of rational T-spectra[6]. Any object (such as that modelling T-equivariant elliptic cohomology) in the algebraicmodel As of [6] can be viewed as a sheaf over the space of closed subgroups of T [7]. More-over, the way a sheaf over the closed subgroups of T models a T-equivariant cohomologytheory gives a precise means by which the sheaf-valued cohomology can be recovered from aconventional theory with values in graded vector spaces. The construction of the Grojnowskisheaf on the elliptic curve from the sheaf on the space of closed subgroups of T helps put theearlier construction in a topological context. It is intended to give a full treatment elsewhere,giving an equivalence between a category of modules over the structure sheaf of A and acategory of modules over the representing spectrum for the cohomology theory.

RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. 3

Returning to the geometry, a very appealing feature is that although our theory is groupvalued, the original curve can still be recovered from the cohomology theory. It is alsonotable that the earlier sheaf theoretic constructions work over larger rings and certainlyrequire the coefficients to contain roots of unity: the loss of information can be illustratedby comparing the rationalized representation ring R(Cn) = Q[z]/(zn − 1) (with componentscorresponding to subgroups of Cn) to the complexified representation ring, isomorphic to thecharacter ring map(Cn,C) (with components corresponding to the elements of Cn).

Finally, the ingredients of the model are very natural invariants of the curve given bysheaves of functions with specified poles at points of finite order: Definition 8.4 simply writesdown the representing object in terms of these,1 and readers already familiar with ellipticcurves and the model of [6] need read nothing else. In fact the algebraic model of [6] gives ageneric de Rham model for all T-equivariant theories, and the models of elliptic cohomologytheories highlight this geometric structure. These higher de Rham models should allowapplications in the same spirit as those made for de Rham models of ordinary cohomologyand K-theory [11].

By way of motivation, we will discuss the way that a T-equivariant cohomology theoryis associated to several other geometric objects. Perhaps most familiar is the complete casediscussed in Section 2, where the Borel theory for a complex oriented cohomology theory isassociated to a formal group. Amongst global groups, the additive and multiplicative onesare the simplest, and in Section 4 we describe how they give rise to ordinary Borel cohomol-ogy and equivariant K-theory. This construction is notable in that it gives a construction ofequivariant cohomology theories from oriented 1-dimensional group schemes which is func-torial for isomorphisms. It is also functorial for certain isogenies as explained in 4.3.

2. Formal groups from complex oriented theories.

The purpose of this section is to recall that any complex orientable cohomology theory

E∗(·) determines a one dimensional, commutative formal group Ĝ and to explain how thecohomology of various spaces can be described in terms of the geometry of Ĝ. This is wellknown but it introduces the geometric language, and motivates our main construction, whichuses this geometric data to construct the cohomology theory. Indeed, we will show that themachinery of [6] permits a functorial construction of a 2-periodic rational T-equivariantcohomology theory EG∗T(·) from a one dimensional group scheme G over a Q-algebra. Fur-thermore, the construction is reversible in the sense that G can be recovered from EG∗T(·).The most interesting case of this is when G is an elliptic curve.

Before introducing the cohomology theory into the picture, we introduce the geometriclanguage. Whilst all schemes are affine, the geometric language is equivalent to the ringtheoretic language, and all geometric statements can be given meaning by translating themto algebraic ones. This excuses us from setting up the geometric foundations of formal groups,and for the present the geometric language is purely suggestive: all notions are defined interms of the algebra. The geometric language becomes essential later, since elliptic curvesare not affine.

A one dimensional commutative formal group law over a ring k is a commutative andassociative coproduct on the complete topological k-algebra k[[y]]. Equivalently, it is acomplete topological Hopf k-algebra O together with an element y ∈ O so that O = k[[y]].

1This 3rd version of the paper is the first to make the model completely explicit.


A topological Hopf k-algebra O for which such a y exists is the ring of functions on a one

dimensional commutative formal group Ĝ. The counit O −→ k, is viewed as evaluation offunctions at the identity e ∈ Ĝ, and the augmentation ideal I consists of functions vanishingat e. The element y generates the ideal I, and is known as a coordinate (at e).

We also need to discuss locally free sheaves F over Ĝ, and in the present affine contextthese are specified by the O-module M = ΓF of global sections. In particular, line bundles L

over Ĝ correspond to modules M which are submodules of the ring of rational functions andfree of rank 1. Line bundles can also be described in terms of the zeros and poles of theirgenerating section: we only need this in special cases made explicit below. The generator fof the O-module M is a section of L, and as such it defines a divisor D = D+−D−, where D+is the subscheme of Ĝ where f vanishes (with multiplicities), and D− is the subscheme of Ĝwhere f has poles (with multiplicites). This divisor determines L, and we write L = O(−D).For example, M = I = (y) corresponds to O(−e), and M = Ia = (ya) corresponds toO(−ae). Next we may consider the [n]-series map [n] : O −→ O, which corresponds to then-fold sum map n : Ĝ −→ Ĝ. We write Ĝ[n] for the kernel of n, and its ring of functions isO/([n](y)). Hence, since n∗y = [n](y) by definition, M = ([n](y)) corresponds to O(−Ĝ[n]),and M = ( ([n](y))a ) corresponds to O(−aĜ[n]). Finally, if M corresponds to O(−D)and M ′ corresponds to O(−D′) then M∨ := Hom(M,O) corresponds to O(D) and M ⊗M ′corresponds to O(−D −D′). This gives sense to enough line bundles for our purposes.

Now suppose that E is a 2-periodic ring valued theory with coefficients E∗ concentrated ineven degrees. The collapse of the Atiyah-Hirzebruch spectral sequence for CP∞ shows thatE is complex orientable. We may define the T-equivariant Borel cohomology by E∗T(X) =E∗(ET ×T X). We work over the ring k = E0T(T) = E0, and view E0T = E0(CP∞) as thering of functions on a formal group Ĝ over k. The tensor product and duality of line bundlesmakes CP∞ into a group object, so E0(CP∞) is a Hopf algebra and Ĝ is a group. From thispoint of view, the augmentation ideal I = ker(E0T −→ E0) consists of functions vanishing atthe identity e ∈ G.

Now, if V is a complex representation of the circle group T, we also let V denote theassociated bundle over CP∞ and the Thom isomorphism shows Ẽ0((CP∞)V ) = Ẽ0T(SV ) isa rank 1 free module over E0T, and hence corresponds to a line bundle L(V ) over Ĝ, whoseglobal sections are naturally isomorphic to the module

ΓL(V ) = Ẽ0T(SV ).

From the fact that Thom isomorphisms are transitive we see that L(V ⊕W ) = L(V )⊗L(W ).The values of all these line bundles can be deduced from those of powers of z.

Lemma 2.1. (1) L(0) = O is the trivial bundle.(2) L(z) = O(−e) is the sheaf of functions vanishing at e, and its module of sections I

is generated by the coordinate y.

(3) L(zn) = O(−Ĝ[n]) is the sheaf of functions vanishing on Ĝ[n], and its module ofsections is generated by the multiple [n](y) of the coordinate y.

(4) L(azn) = O(−aĜ[n]) is the sheaf of functions vanishing on Ĝ[n] with multiplicity a,and its module of sections is generated ([n](y))a.

Proof: The first statement is clear since Ẽ0T(S0) = E0T. For the second we use the equivalence

(CP∞)z ' (CP∞)0/(CP 0)0. The third statement follows from the Gysin sequence since zk


is the pullback of z along the kth power map CP∞ −→ CP∞. The final statement followsfrom the tensor product property. �

This gives the fundamental connection between the equivariant cohomology of a sphereand sections of a line bundle.

Corollary 2.2. For any a ∈ Z, n 6= 0 we haveẼ0T(S

azn) = O(−aĜ[n]). �

We want to finish this section by pointing out that if the formal group Ĝ (which is affine)is replaced by a group G with higher cohomology, we cannot expect a cohomology theoryentirely in even degrees. Whenever the group is not affine, we write O for the structure sheafof G. This is reconciled by the above usage since in the affine case the structure sheaf isdetermined by its ring of global sections. In the non-affine case, the cofibre sequence

Saz ∧ T+ −→ Saz −→ S(a+1)z

forces there to be odd cohomology. Indeed, there is a corresponding short exact sequence ofsheaves

O(−ae)/O(−(a + 1)e)←− O(−ae)←− O(−(a + 1)e).Any satisfactory cohomology theory will be functorial, and applying Ẽ0T(·) will give sectionsof the associated sheaves. However the global sections functor on sheaves is not usually rightexact, and the sequence of sections continues with the sheaf cohomology groups H1(G; ·).It is natural to hope that the long exact cohomology sequence induced by the sequence ofspaces should be the long exact cohomology sequence induced by the sequence of sheaves.This gives a natural candidate for the odd cohomology:

ẼiT(Saz) = H i(G;O(−ae)) for i = 0, 1.

This explains why it is possible for complex orientable cohomology theories to have coefficientrings in even degrees (formal groups are affine), and indeed how their values on all complexspheres can be the same. It also explains why we cannot expect either property for a theoryassociated to an elliptic curve.

3. The model for rational T-spectra.

For most of the paper we work with the representing objects of these cohomology theories,namely T-spectra [3]. Thus we prove results about the representing spectra, and deduceconsequences about the cohomology theories. More precisely, any suitable T-equivariantcohomology theory E∗T(·) is represented by a T-spectrum E in the sense that

Ẽ∗T(X) = [X,E]∗T.

This enables us to define the associated homology theory

ẼT∗ (X) = [S0, E ∧X]T∗

in the usual way. We shall make use of the elementary fact that the Spanier-Whitehead dualof the sphere SV is S−V , as one sees by embedding SV as the equator of SV⊕1. Hence, forexample

Ẽ0T(SV ) = [SV , E]T = [S0, S−V ∧E]T = πT0 (S−V ∧ E) = ẼT0 (S−V ).


We say that a cohomology theory is rational if its values are graded rational vector spaces.A spectrum is rational if the cohomology theory it represents is rational. It suffices to checkthe values on the homogeneous spaces T/H for closed subgroups T, since all spaces are builtfrom these.

Convention 3.1. Henceforth all spaces, groups and spectra are rationalized whether or notthis is indicated in the notation.

Our results are made possible because there is a complete algebraic model of the categoryof rational T-spectra, and hence of rational T-equivariant cohomology theories [6]. Thereare two models for rational T-spectra, as derived categories of abelian categories:

T-Spectra ' D(As) ' D(At).The standard abelian category As has injective dimension 1, and the torsion abelian categoryAt is of injective dimension 2. It is usually easiest to identify the model for a T-spectrumin D(At), at least providing its model has homology of injective dimension 1. This is thentransported to the standard category, where calculations are sometimes easier. To describethe categories, we need to use the discrete set F of finite subgroups of T. On this weconsider the constant sheaf R of rings with stalks Q[c] where c has degree −2. We need toconsider the ring R = map(F,Q[c]) of global sections. For each subgroup H , we let eH ∈ Rdenote the idempotent with support H . If w : F −→ Z≥0 is a function, we write cw for theelement of R with cw(H) = cw(H). Now consider the multiplicative set E generated by theuniversal Euler classes e(V ) for the representations V of T with V T = 0. These are definedby e(V ) = cv, where v(H) = dimC(V

H). In particular for V = zn we have e(zn) = csub(n)

where sub(n)(H) = 1 if H ⊆ T[n] and 0 otherwise. Equivalently,E = {cw | w : F −→ Z≥0 of finite support}.

We let tF∗ = E−1R: as a graded vector space this is

⊕H Q in positive degrees and

∏H Q in

degrees zero and below.The objects of the standard model As are triples (N, β, V ) where N is an R-module (called

the nub), V is a graded rational vector space (called the vertex) and β : N −→ tF∗ ⊗ V isa morphism of R-modules (called the basing map) which becomes an isomorphism when Eis inverted. When no confusion is possible we simply say that N −→ tF∗ ⊗ V is an objectof the standard abelian category. An object of As should be viewed as the module N withthe additional structure of a trivialization of E−1N . A morphism (N, β, V ) −→ (N ′, β ′, V ′)of objects is given by an R-map θ : N −→ N ′ and a Q-map φ : V −→ V ′ compatible underthe basing maps.

Since the standard abelian category has injective dimension 1, homotopy types of objectsof the derived category D(As) are classified by their homology in As, so that homotopy typescorrespond to isomorphism classes of objects of the abelian category As. In the sheaf theo-retic approach, N is the space of global sections of a sheaf on the space of closed subgroupsT, the vertex V is the value of the sheaf at the subgroup T and the fact that the basingmap β : N −→ tF∗ ⊗ V is an isomorphism away from E is the manifestation of the patchingcondition for sheaves.

The objects of the torsion abelian category At are triples (V, q, T ) where V is a gradedrational vector space T is an E-torsion R-module and q : tF∗ ⊗ V −→ T is a morphism of R-modules. The condition on T is equivalent to requiring (i) that T is the sum of its idempotentfactors T (H) = eHT in the sense that T =

⊕H T (H) and (ii) that each T (H) is a torsion


Q[c]-module. When no confusion is possible we simply say that tF∗ ⊗ V −→ T is an objectof the torsion abelian category. In the sheaf theoretic approach, the module T (H) is thecohomology of the structure sheaf with support at H . By contrast with the standard abeliancategory, the torsion abelian category has injective dimension 2. Thus not every object Xof the derived category D(At) is determined up to equivalence by its homology H∗(X) inthe abelian category At. We say that X is (intrinsically) formal if it is determined up toisomorphism by its homology. Evidently, X is formal if its homology has injective dimension0 or 1 in At. In general, if H∗(X) = (t

F∗ ⊗ V −→ T ), the object X is equivalent to the fibre

of a map (tF∗ ⊗ V −→ 0) −→ (tF∗ ⊗ 0 −→ ΣT ) (in the derived category) between objects inAt of injective dimension 1. This map is classified by an element of Ext(t

F∗ ⊗V,ΣT ), so that

X is formal if the Ext group is zero in even degrees. Thus X is formal if both V and T arein even degrees or if T is injective in the sense that each T (H) is an injective Q[c]-module.Definition 3.2. [6, 5.8.2] Suppose given a function w : F −→ Z with finite support. Thealgebraic w-sphere is the object of As defined by

Sw = (R(c−w) −→ tF∗ )where R(c−w) is the R-submodule of tF∗ generated by the Euler class c

−w.

Now for an object X of As there is an exact sequence

0 −→ ExtAs(S1+w,M) −→ [Sw,M ] −→ HomAs(Sw,M) −→ 0,so we shall need to calculate these Hom and Ext groups. For the present we restrict ourselvesto the Hom groups.

Lemma 3.3. For an object M = (Nβ−→ tF∗ ⊗ V ) of the abelian category As we have

HomAs(Sw, (N −→ tF∗ ⊗ V )) = N(c−w) := {n ∈ N | β(n) ∈ c−w ⊗ V }. �

We may now describe how to construct the counterparts Ms(E) = (N −→ tF∗ ⊗V ) (in thestandard abelian category As) and Mt(E) = (t

F∗ ⊗V −→ T ) (in the torsion abelian category

At) of a rational T-spectrum E. From the above discussion, the model Ms(E) determines Eitself, but Mt(E) only determines E if Mt(E) is formal. First, we may define the vertex V ,the nub N and torsion module T by formulae and then turn to practical computations interms of data easily accessible to us. To describe the answer, we need the universal F-spaceEF, and the basic cofibre sequence

EF+ −→ S0 −→ ẼFwhere ẼF is the join S0 ∗ EF+. We also use functional duality on T-spectra defined byDX = F (X,S0). The nub vertex and torsion modules associated to a T-spectrum E aregiven by

• N = πT∗ (E ∧DEF+)• V = πT∗ (E ∧ ẼF)• T = πT∗ (E ∧ ΣEF+)

The vertex is straightforward to calculate in terms of available data:

V = πT∗ (E ∧ ẼF) = lim→ V T=0πT∗ (E ∧ SV ).

An approach to the nub via limits is possible but not very illuminating.


The associated torsion sheaf T may be described by saying that its sections over the set[⊆ H ] of subgroups of H is πT∗ (ΣE[⊆ H ]+ ∧ E). Using idempotents from the Burnside ringof H this may be split up into stalks πT∗ (ΣE〈K〉 ∧ E) one for each subgroup K ⊆ H (itturns out that these are in independent of H , as is required for consistency). Now if H hasorder n, the infinite sphere S(∞zn) is a model for E[⊆ H ], and hence there is a long exactsequence

· · · −→ πT∗ (E) −→ πT∗ (S∞zn ∧E) −→ πT∗ (E[⊆ H ]+ ∧E) −→ · · · .

Since πT∗ (S∞zn ∧E) = lim

→ aπT∗ (S

azn ∧ E) we may conclude there is a short exact sequence

0 −→ ΣE∗V (H)T /e(zn)∞ −→ πT∗ (E[⊆ H ]+ ∧E) −→ e(zn)-power torsion(E∗V (H)T ) −→ 0

where E∗V (H)T is the ring graded by multiples of z

n with azn-th component πT0 (Sazn ∧E) and

e(zn) is the degree −zn Euler class.In this account we have described the calculation of V and T in terms of available data. If

this is to determine E we must show in addition that Mt(E) is formal. In our case this willhold because V and T are in even degrees. It is convenient for calculation to deduce Ms(E).

Lemma 3.4. If Mt(E) = (tF∗ ⊗V

q−→ T ) has surjective structure map, then Mt(E) is formaland

Ms(E) = (N −→ tF∗ ⊗ V )where

N = ker(tF∗ ⊗ V −→ T ),and the basing map is the inclusion. Furthermore we have the explicit injective resolution

0 −→Ms(E) =

N↓tF∗ ⊗ V

−→ tF∗ ⊗ V↓

tF∗ ⊗ V

−→ T↓

0

−→ 0in As.

Proof: To see that Mt(E) is formal, it is only necessary to remark that T is the quotient ofan E-divisible group and therefore injective [6, 5.3.1]. �

Finally, we should record that spheres and suspensions in the algebraic and topologicalcontexts correspond.

Lemma 3.5. [6, 5.8.3] Suppose W is a virtual representation with W T = 0 and let w =dimC(W ). The object modelling the sphere S

W with V T = 0 in As is the algebraic sphereSw:

Ms(SW ) = Sw = (R(c−w) −→ tF∗ )

where R(c−w) is the R-submodule of tF∗ generated by c−w = e(−W ).

Convention 3.6. In the present paper we are interested in cohomology theories with aperiodicity element u of degree 2. We may therefore shift even degree elements into degreezero. For example uc is the degree 0 counterpart of c. For the rest of the paper we use c todenote the degree 0 version.


4. The affine case: T-equivariant cohomology theories from additive andmultiplicative groups.

The algebraic models of equivariant K-theory and Borel cohomology are easily described[6]. In this section we show they are special cases of a general functorial construction ofa cohomology theory EG∗T(·) associated to a one dimensional affine group scheme G. Thiswill serve to illustrate the algebraic categories described in Section 3 and also complete themotivation of our construction for elliptic curves.

The additive group scheme Ga and the multiplicative group scheme Gm are affine, andtherefore the construction of associated cohomology theories is considerably simpler thanthat for elliptic curves. Nonetheless the general features are the same, and it is useful tohave seen the phenomena first in a familiar setting. It turns out that the associated 2-periodicT-equivariant theories are concentrated in even degrees and

(EGa)0T(X) = H∗(ET×T X)and

(EGm)0T(X) = K0T(X),and models for these theories were given in [6]. We will repeat the answer here in our presentlanguage.

We start by summarizing the properties we want of such a construction, and then observethat the algebraic categories of Section 3 immediately gives a unique construction.

• The subgroup T[n] of order n corresponds to the subgroup G[n] of elements of orderdividing n• The family F of finite subgroups corresponds to the set G[tors] of elements of torsion

points.• The suspension Sazn ∧ EG corresponds to the sheaf O(aG[n]) and more generally,

suspension by zn corresponds to tensoring with O(G[n]).• The inclusion S0 −→ Szn which induces multiplication by the Euler class (in the

presence of a Thom isomorphism) corresponds to O −→ O(G[n]).• We extend the notation to allow

S∞zn

:= lim→ a

Sazn

to correspond to the sheaf

O(∞G[n]) := lim→ aO(aG[n])

andẼF := lim

→ a,nSaz

n

to correspond toO(∞G[tors]) := lim

→ a,nO(aG[n]).

(This description of ẼF requires us to be working rationally; more generally one onlyhas ẼF = lim

→ V T=0SV .)

We need to say more about Euler classes. Consider the subgroup T[n] of order n. Thenatural geometric construction is the Euler class induced by S0 −→ Szn . Pulling back aThom class for Sz

ngives the function e(zn) in R, which vanishes at all subgroups of T[n].


Evidently, if we take cd to be the function vanishing to the first order on the group of orderd and taking the value 1 elsewhere, we have

e(zn) =∏d|n

cd,

so that we may view cd as a universal cyclotomic function.We have already motivated the idea that S0 −→ Szn should correspond to O −→ O(G[n]).

The Thom class for Szn

corresponds to a generating section of O(G[n]) and hence e(zn)should correspond to a function χ(zn) defining G[n] in G.

Now choose a coordinate y =: χ(z) at e ∈ G. We may then takeχ(zn) := [n](y) := n∗(y).

so that χ(zn) is a function vanishing to first order on G[n].Next, we may a decompose the divisor G[n]:

G[n] =∑d|n

G〈d〉

where G〈d〉 is the divisor of points of exact order d. Now we define a function φ〈d〉 := φG〈d〉vanishing to the first order on G〈d〉 recursively by the condition

χ(zn) =∏d|n

φ〈d〉 :

the formula defines φ〈n〉 directly for n = 1, and for larger values of n, it is defined by dividingχ(zn) by the previously defined φ〈d〉.Definition 4.1. Given a virtual complex representation V =

∑n anz

n, we define a divisorby D(V ) =

∑n anG[n]. We say that a 2-periodic T-equivariant cohomology theory E∗T(·) is

of type G ifẼiT(S

V ) ∼= H i(G;O(−D(V ))whenever V or −V is a complex representation.

We also make a naturality requirement. For this it will be clearer if we insist V is anactual representation, and reformulate the other case as the isomorphism

ẼTi (SV ) ∼= H i(G;O(D(V )).

Now we require these isomorphisms to be natural for inclusions j : V −→ V ′ of represen-tations. First note that such a map induces a map SV −→ SV ′ of T-spaces and hencemaps

j∗ : ẼiT(SV ′) −→ ẼiT(SV )

andj∗ : Ẽ

Ti (S

V ) −→ ẼiT(SV′).

On the other hand we have inclusion of divisors D(V ) −→ D(V ′), inducing mapsO(−D(V ′)) −→ O(−D(V ))

andO(D(V )) −→ O(D(V ′)).

The induced maps in sheaf cohomology are required to j∗ and j∗.


Theorem 4.2. Given a commutative 1-dimensional affine group scheme G over a ring con-taining Q, and a coordinate y at e ∈ G there is a 2-periodic cohomology theory EG∗T(·) oftype G. Furthermore, EG∗T is in even degrees and G = spec(EG0T). The construction isnatural for isomorphisms.

Remark 4.3. The construction is also natural for quotient maps p : G −→ G/G[n] in thesense that there is a map p∗ : E(G/G[n]) −→ inflTT/T[n]EG of T-spectra, where EG is viewedas a T/T[n]-spectrum and inflated to a T-spectrum.

More precisely, if y is a coordinate on G then its norm Πa∈G[n]Tay is a coordinate onG/G[n] (where Ta denotes translation by a). Using these coordinates, we obtain equivariantspectra EG/G[n] and EG. As a first step to maps between them, note that we have mapsp∗V : V (G/G[n]) −→ VG and p∗T : T (G/G[n]) −→ TG corresponding to pullback of functions.However p∗V and p

∗T do not give a map of T-spectra E(G/G[n]) −→ EG; for example the

non-equivariant part of E(G/G[n]) corresponds to functions on G/G[n] with support at theidentity, and these pull back to functions on G supported on G[n], which correspond to thepart of EG with isotropy contained in T[n]. The answer is to view the circle of equivarianceof EG as T/T[n], and then to use the inflation functor studied in Chapters 10 and 24 of [6]to obtain a T-spectrum.Proof: The construction was motivated in Section 2. We take

VG = O(∞tors),

TG = O(∞tors)/O,and use the map

qG : tF∗ ⊗ O(∞tors) −→ O(∞tors)/Ogiven by

s/e(W )⊗ f 7−→ s · f/χ(W ).We must explain how TG is a module over R, and why β is a map of R-modules. We

make TG into a module over R by letting cd act as φd. Since any function only has finitelymany poles, all but finitely many cd act as the identity on any element of TG, and sincepoles are of finite order, TG is a E-torsion module. The definition of the map qG shows it isan R-map.

Finally, we must show that the homotopy groups of the resulting object are as required in4.1. By 3.4 we have Ms(EG) = (βG : NG −→ tF∗ ⊗VG), where NG = ker(tF∗ ⊗VG −→ TG),and we need to calculate

[SW , EG]T∗ = [Sw,Ms(EG)]∗.Since qG is epimorphic, βG is monomorphic, and TG is injective. Thus by 3.4 we have theexplicit injective resolution

0 −→ Ms(EG) =

NG↓tF∗ ⊗ VG

−→ tF∗ ⊗ VG↓

tF∗ ⊗ VG

−→ TG↓

0

−→ 0.Now, applying 3.3 we see Ext(Sw,Ms(EG)) = 0 since any torsion element t ∈ TG lifts tof ∈ VG and hence to 1/e(W )⊗ χ(W )f . It is immediate from the definition that

Hom(Sw,Ms(EG)) = {c−w ⊗ f | f/χ(W ) regular }.


By construction the divisor associated to the function χ(V ) is D(V ), so f/χ(V ) is regular ifand only if f ∈ O(−D(V )) as required. �

Remark 4.4. In the above proof we made use of the fact that the Euler class χ(W ) exists asa function in VG. The point of this comment will become apparent when we treat the ellipticcase which behaves rather differently: there the Euler class is given by different functionsat different points, corresponding to the fact that the cohomology theory is not complexorientable, so that the bundle specified by W is not trivializable.

We make the construction explicit in a few cases.The ring of functions on Ga is Q[x], and the group structure is defined by the coproduct

x 7−→ 1⊗ x+ x⊗ 1. We choose x as a coordinate about the identity, zero. The group Ga[n]of points of order dividing n is defined by the vanishing of χ(zn) = nx, so the identity is theonly element of finite order over Q-algebras. This case becomes rather degenerate in that itonly detects isotropy 1 and T.Proposition 4.5. The model of 2-periodic Borel cohomology in the torsion model is formal,concentrated in even degrees and in each even degree is the map

tF∗ ⊗ O(∞tors) = tF∗ ⊗Q[x, x−1] −→ Q[x, x−1]/Q[x] = O(∞tors)/O

s/e(V )⊗ f 7−→ s · f/χ(V ).Here O = Q[x] and χ(zn) = nx. The ring O(∞tors) = Q[x, x−1] of functions with poles onlyat points of finite order is obtained by inverting the Euler class of z. Accordingly, 2-periodicBorel cohomology is the theory associated to the additive group in the sense of 4.2. �

The ring of functions on Gm is O = R(T) = Q[z, z−1], and the group structure is definedby the coproduct z 7−→ z ⊗ z. We choose y = 1 − z as a coordinate about the identityelement, 1. The coproduct then takes the more familiar form y 7−→ 1 ⊗ y + y ⊗ 1 − y ⊗ y.The group Gm[n] of points of order dividing n is defined by the vanishing of χ(zn) = 1− zn.Proposition 4.6. [6, 13.4.4] The model of equivariant K-theory in the torsion model isformal, concentrated in even degrees and in each even degree is the map

tF∗ ⊗ O(∞tors) −→ O(∞tors)/O

s/e(V )⊗ f 7−→ s · f/χ(V ).Here O = Q[z, z−1] and χ(zn) = 1− zn. The ring O(∞tors) of functions with poles only atpoints of finite order is obtained by inverting all Euler classes. Accordingly, equivariant Ktheory is the theory associated to the multiplicative group in the sense of 4.2. �

By way of completeness we also record the analogue for formal groups. This completesthe circle by establishing the universality of the motivation described in Section 2. However,since we must work over Q, there is little difference from the additive group above. Supposegiven a commutative one dimensional formal group Ĝ over a ring k containing Q, with acoordinate y. We may identify the ring of functions on Ĝ with k[[x]], and the group structureis the coproduct x 7−→ F (x⊗1, 1⊗x). The group Ĝ[n] of points of order dividing n is definedby the vanishing of χ(zn) = [n](x) so the identity is the only element of finite order over


Q-algebras. We may now make the direct analogue of the construction in 4.2. This casebecomes rather degenerate in that it only detects isotropy 1 and T.Proposition 4.7. The model of the 2-periodic Borel cohomology associated to a complexorientable theory E∗(·) in the torsion model is formal, concentrated in even degrees and ineach even degree is the map

tF∗ ⊗ O(∞tors) = tF∗ ⊗ E0((x)) −→ E0((x))/E0[[x]] = O(∞tors)/Os/e(V )⊗ f 7−→ s · f/χ(V ).

Here O = E0[[x]] and χ(zn) = [n](x). The ring O(∞tors) = E0[[x]][1/x] = E0((x)) offunctions with poles only at points of finite order is obtained by inverting the Euler class ofz. Accordingly, 2-periodic E-Borel cohomology is the theory associated to the formal groupof E in the sense of 4.2. �

5. Elliptic curves.

In this section we record the well known facts about elliptic curves that will play a partin our construction. We use [15] as a basic reference for facts about elliptic curves, and [12]as background from algebraic geometry.

Let A be an elliptic curve (i.e. a smooth projective curve of genus 1 with a specified pointe) over an algebraically closed field k of characteristic 0 and let O = OA be its sheaf ofregular functions. Note that ΓO = k, so the sheaf contains a great deal more informationthan its ring of global sections. A divisor on A is a finite Z-linear combination of points,and associated to any rational function f on A we have the divisor div(f) = ΣPordP (f)(P ),where ordP (f) ∈ Z is the order of vanishing of f at P . In the usual way, if D is a divisor onA, we write O(D) for the associated invertible sheaf. Its global sections are given by

ΓO(D) = {f | div(f) ≥ −D} ∪ {0},so that for a point P , the global sections of O(−P ) are the functions vanishing at P .

We also have O(D1)⊗ O(D2) = O(D1 +D2).Since the global sections functor is not right exact, we are led to consider cohomology, but

since A is one-dimensional this only involves H0(A; ·) = Γ(·) and H1(A; ·), which are relatedby Serre duality. This takes a particularly simple form since the canonical divisor is zero onan elliptic curve:

H0(A;O(D)) = H1(A;O(−D))∨,where (·)∨ = Homk(·, k) denotes vector space duality.

From the Riemann-Roch theorem we deduce that the canonical divisor is 0 and the coho-mology of each line bundle:

dim(H0(A;O(D)) =

{degD if deg(D) ≥ 10 if deg(D) ≤ −1

and

dim(H1(A;O(D)) =

{| degD| if deg(D) ≤ −10 if deg(D) ≥ 1.

For the trivial divisor one has

dim(H0(A;O)) = dim(H1(A;O)) = 1.


Now if D = ΣPnP (P ) is a divisor of degree 0, we may form the sum P (D) = ΣPnPP in A,and D is linearly equivalent to (P (D))− (e). If P (D) = e then the sheaf O(D) has the samecohomology as O. Otherwise, since no function vanishes to order exactly 1 at P , we find

H0(A;O(D)) = H1(A;O(D)) = 0.

We may recover A from the graded ring Γ(O(∗e)) = {ΓO(ne)}n≥0. Indeed, this is thebasis of the proof in [15, III.3.1] that any elliptic curve is a subvariety of P2 defined by aWeierstrass equation. We choose a basis {1, x} of ΓO(2e) and a extend it to a basis {1, x, y}of ΓO(3e). Now observe that since ΓO(6e) is 6-dimensional, there is a relation betweenthe seven elements 1, x, x2, x3, y, xy and y2: this is the Weierstrass equation, and it may beverified that A is the closure in P2 of the plane curve it defines. The graded ring Γ(O(∗e))has generator Z of degree 1 corresponding to the constant function 1 in ΓO(e), X of degree2 corresponding to x, and Y of degree 3 corresponding to y. These three variables satisfythe homogeneous form of the Weierstrass equation. The statement that A is the projectiveclosure of the plane curve defined by the Weierstrass equation may be restated in terms ofProj:

A = Proj(Γ(O(∗e))).

6. Coordinate data

Our main theorem constructs a cohomology theory of type A from an elliptic curve togetherwith suitable coordinate data. In this section we describe the data, and the choices offunctions that they permit.

Definition 6.1. Coordinate data for an elliptic curve is a choice of two functions xe witha pole of order 2 at the identity and nowhere else, and ye with a pole of order 3 at theidentity and nowhere else. We also require that xe and ye only vanish at torsion points. Thiscoordinate data determines a local uniformizer te = xe/ye of Oe, and hence also a uniformizertP at P by translating te.

Remark 6.2. (i) Since te is a uniformizer, t2exe = ue is a unit in Oe.

However, we note that any global representative of te must have two poles Z,Z′ away from

e, so ue cannot be a constant.(iii) One popular choice of coordinate data involves choosing a point P of order 2. This

determines a choice of xe and ye up to a constant multiple by the conditions

div(xe) = −2(e) + 2(P ) and div(ye) = −3(e) + (P ) + (P ′) + (P ′′)where A[2] = {e, P, P ′, P ′′}. Thus

div(te) = (e) + (P )− (P ′)− (P ′′). �

The divisor A〈n〉 of points of exact order n will play a central role. Note that

A[n] =∑d|n

A〈d〉,

and

A[tors] =∑d≥1

A〈d〉.

The coordinate data allow us to specify a function defining the points of exact order d.


Lemma 6.3. Given a choice of coordinate data on the elliptic curve d, for each d ≥ 2, thereis a unique function td with the properties

(1) td vanishes exactly to the first order on A〈d〉,(2) td is regular except at the identity e ∈ A where it has a pole of order |A〈d〉|,(3) t

|A〈d〉|e td takes the value 1 at e

Proof: Consider the divisor A〈d〉 − |A〈d〉|(e). Note that the sum of the points of A〈d〉 inA is the identity: if d 6= 2 this is because points occur in inverse pairs, and if d = 2 it isbecause the A[2] is isomorphic to C2 ×C2. It thus follows from the Riemann-Roch theoremthat there is a function f with A〈d〉 − |A〈d〉|(e) as its divisor. This function (which satisfiesthe first two properties in the statement) is unique up to multiplication by a non-zero scalar.The third condition fixes the scalar. �

Remark 6.4. If we choose any finite collection π = {d1, . . . , ds} of orders ≥ 2, there is againa unique function φ〈π〉 with analogous properties. Indeed, the good multiplicative propertyof the normalization means we may take

φ〈π〉 =∏i

φ〈di〉.

This applies in particular to the set A[n] \ {e}. �

For some purposes, it is convenient to have a basis for functions with specified poles. Wealready have the basis 1, x, y, x2, xy, . . . if all the poles are at the identity. Multiplication bya function f induces an isomorphism

f · : ΓO(D) −→ ΓO(D + (f))so we can translate the basis we have.

Lemma 6.5. For the divisor D = Σd≥1n(d)A〈d〉 let t∗(D) :=∏

b≥2 tn(b)b . Multiplication by

t∗(D) gives an isomorphism

t∗(D)· : H0(A;O(deg(D)(e))∼=−→ H0(A;O(D)).

A basis of H0(A;O(D)) is given by t∗(D) if deg(D) = 0, and by the first deg(D) terms inthe sequence

t∗(D), t∗(D)x, t∗(D)y, t∗(D)x2, t∗(D)xy, . . .

otherwise. �

Remark 6.6. It is essential to be aware of the exceptional nature of the degree zero case.

7. Local cohomology sheaves on elliptic curves.

The basic ingredients of the torsion model of a the cohomology theory associated to anelliptic curve A are analogous to the affine case. The vertex

V A = ΓO(∞tors)


consists of rational functions whose poles are all at torsion points, however the torsion moduleis not simply the quotient of this by regular functions, but rather

TA = Γ(O(∞tors)/O).Before we work with this definition we need some basic tools.

Convention 7.1. Here and elsewhere, we only consider open sets obtained by deletingtorsion points. Thus localization only permits poles at torsion points: for example OP is thesubsheaf of O(∞tors) consisting of functions regular at P .

For any effective divisor D we may use the short exact sequence

0 −→ O −→ O(aD) −→ Q(aD) −→ 0of sheaves to define the quotient sheaf Q(aD) for 0 ≤ a ≤ ∞. The cohomology of Q(∞D)is the cohomology of A with support on D.

In fact we may reduce constructions to the case when the divisor D is a single point P .Evidently, Q(∞P ) is a skyscraper sheaf concentrated at P , so we may localize at P to obtain

0 −→ OP −→ O(∞P )P −→ Q(∞P ) −→ 0.Notice that O(∞P )P = O(∞tors).

Since A is a smooth curve, the local ring OP is a discrete valuation ring, and if we choosea local uniformizer tP any element of Γ(Q(∞P )) may be represented by an element of theform

a−1t−1P + a−3t

−3P + · · ·+ a−nt−nP

for suitable scalars a−i. Thus the sequence becomes

0 −→ OP −→ OP [1/tP ] −→ OP /t∞P −→ 0.This gives the basis of the Thom isomorphism.

Lemma 7.2. A choice of local uniformizer at P gives isomorphisms

O((a + r)P )/O(rP ) = Q((a + r)P )/Q(rP ) ∼= Q(aP ),and hence

Q(∞P )⊗ O(rP ) ∼= Q(∞P ). �

Note that it is immediate from the Riemann-Roch formula that for 0 ≤ a ≤ ∞ thecohomology group H0(A;Q(aP )) is a dimensional, and H1(A;Q(aP )) = 0.

Now we may assemble these sheaves for each point. Indeed, we have a diagram

O −→ O(∞D) −→ Q(∞D)↓ ↓O −→ O(∞(D +D′)) −→ Q(∞(D +D′))

of sheaves, and hence a map Q(∞D) −→ Q(∞(D +D′)).Proposition 7.3. If P, P ′ are distinct points of A then the natural map

Q(∞P )⊕Q(∞P ′)∼=−→ Q(∞(P + P ′))

is an isomorphism.


Proof: We apply the Snake Lemma to the diagram

O⊕ O −→ O(∞P )⊕ O(∞Q) −→ Q(∞P )⊕Q(∞P ′)↓ ↓ ↓O −→ O(∞(P + P ′)) −→ Q(∞(P + P ′))

in the abelian category of sheaves on A. The first vertical is obviously surjective with kernelO. The kernel of the second vertical is also O, since if f and f ′ are local sections of O(∞P )and O(∞P ′) (ie f only has poles at P and f ′ only at P ′) then f + f ′ = 0 implies thatf and f ′ are regular. Finally we must show that O(∞(P + P ′)) is the sheaf quotient ofO −→ O(∞P )⊕ O(∞P ′). However, this may be verified stalkwise, where it is clear. �

Let us now collect what we need for the construction. To give a Thom isomorphism forQ(∞A〈d〉) we need to choose local uniformizers tP at each point P of exact order d. Forexample we explained in Section 6 how coordinate data determines a function td vanishingon A〈d〉 to the first order at all points of A〈d〉, and we could take tP = td for all points P ofexact order d.

Corollary 7.4. The natural map gives an isomorphism⊕d

Q(∞A〈d〉)∼=−→ Q(∞tors),

and a choice of coordinate tP at each P ∈ A〈d〉 gives a Thom isomorphism

Td : Q(∞A〈d〉)⊗ O(A〈d〉)∼=−→ Q(∞A〈d〉).

The sheaf Q(∞A〈d〉) has no higher cohomology and its global sections areΓQ(∞A〈d〉) = V A/{f | f is regular on A〈d〉}. �

Remark 7.5. This corresponds to the fact that there is a rational splitting

ΣEF+ '∨H

ΣE〈H〉

where E〈H〉 = cofibre(E[⊂ H ]+ −→ E[⊆ H ]+) [6, 2.2.3].

8. A cohomology theory associated to an elliptic curve.

We are now ready to state and prove the main theorem. Indeed, the paper so far hasconsisted entirely of motivation and repackaging of known results by way of preparation.

Theorem 8.1. Given an elliptic curve A over a field k of characteristic zero, and coordinatedata (xe, ye), there is an associated 2-periodic rational T-equivariant cohomology theory oftype A, so that for any representation W with W T = 0 we have

ẼAi

T(SW ) = H i(A;O(−D(W )))

and

ẼATi (S

W ) = H i(A;O(D(W )))

where the divisor D(W ) is defined by taking

D(W ) =∑n

anA[n] when W =∑n

anzn.


This association is functorial for isomorphisms of elliptic curves with coordinate data.

Remark 8.2. (i) The elliptic curve can be recovered from the cohomology theory. Indeed,we may form the graded ring

ẼA0

T(S−∗z) := {ẼA

0

T(S−az)}a≥0

from the products S−az ∧S−bz −→ S−(a+b)z, and the elliptic curve can be recovered from thecohomology theory via

A = Proj(ẼA0

T(S−∗z)),

as commented in Section 5.(ii) The coordinate data on A can therefore be recovered from suitable elements of homology:

xe ∈ ẼAT0 (S

2z) and ye ∈ ẼAT0 (S

3z).

Remark 8.3. We have not required the field to be algebraically closed. To see the advantageof this, note that even for the multiplicative group, the individual points of order n are onlydefined over k if k contains appropriate roots of unity. However Gm[n] (defined by 1−zn) andhence also Gm〈n〉 (defined by the cyclotomic polynomial φn(z)) are defined over Q. Henceequivariant K theory itself is defined over Q. For an elliptic curve A we require that thereis a basis for ΓO(aG〈d〉)) consisting of functions defined over k.Proof: We must describe a vector space V = V A, an R-module TA and an R-map

qA : tF∗ ⊗ V A −→ TA.It is easy to describe V A and TA; indeed, we take

V A = ΓO(∞tors)consisting of rational functions whose poles are all at torsion points, and torsion module

TA = Γ(Q(∞tors)).The splitting

Q(∞tors) ∼=⊕d

Q(∞A〈d〉)

of 7.4 gives

TA =⊕d

TA〈d〉

whereTA〈d〉 = V A/{f | f is regular on A〈d〉}.

It is not hard to describe the R-module structure on TA. The direct sum splitting of TAcorresponds to the splitting

R ∼=∏d

Q[c],

and TA〈d〉 is a Q[c]-module where c acts as multiplication by the function td defining A〈d〉.Since the order of any pole is finite, TA〈d〉 is a torsionQ[c]-module. Notice that the definitionof the Thom isomorphism is arranged so that the composite

φD : Q(∞D) = Q(∞D)⊗ O −→ Q(∞D)⊗ O(D) ∼= Q(∞D)is multiplication by td.


Definition 8.4. If u : F −→ Z is a function positive almost everywhere, we defineqA : cu ⊗ V −→ TA =

⊕d

TA〈d〉

by specifying its dth component

qA(cu ⊗ f)d = tu(d)d f.Lemma 8.5. The definition does determine an R-map qA : tF∗ ⊗ V −→ TA.Proof: Since any function is regular at all but finitely many points, the map qA maps intothe sum.

Now, R-maps q : tF∗ ⊗ V −→⊕

d Td are determined by the idempotent pieces qd :Q[c, c−1] ⊗ V −→ Td, and conversely, any set of Q[c]-maps qd so that qd(c0 ⊗ f) is non-zero for only finitely many d determines an R-map q. It is easy to see that the componentsof qA (ie qd(c

s ⊗ f) = qA(csδ(d) ⊗ f)d) have these properties, and that the function theydetermine agrees with qA(cu ⊗ f) wherever it is defined. �

Now we can check that the resulting homology and cohomology of spheres agrees with thecohomology of the corresponding divisors on the elliptic curve.

Consider the complex representation W with W T = 0 and the corresponding functionw(H) = dimC(W

H). We see from 3.3 and 3.5 (as in the proof of 4.2 that

ẼAT0 (S

W ) = ker(qA : cw ⊗ V A −→ TA)and

ẼAT1 (S

W ) = cok(qA : cw ⊗ V −→ TA)and similarly with W replaced by −W . Since the kernel and cokernel are vector spaces overk, it is no loss of generality to extend scalars to assume it is algebraically closed. This isconvenient because it is simpler to treat separate points of order n one at a time.

The following two lemmas complete the proof. �

Lemma 8.6. If W is a representation with W T = 0 then

ẼAT0 (S

W ) = H0(A;O(D(W ))),

and if W 6= 0,ẼA

T0 (S

−W ) = 0.

Proof: By definition

qA(cw ⊗ f)d = tw(d)d f.Since the function td vanishes to exactly the first order on A〈d〉, the condition that f liesin the kernel is that ordP (f) ≥ −w(d) for each point P of exact order d. Since D(W ) =ΣPw(dP )(P ) we have

ker(qA : cw ⊗W −→ TA) = {f ∈ V A | div(f) +D(W ) ≥ 0}as required.

Replacing W by −W , the second statement is immediate. �


Remark 8.7. The proof is local and therefore shows the kernel is actually the subsheafO(D(W )) of the constant sheaf V A.

The calculation of the odd cohomology is less elementary.

Proposition 8.8. If W is a representation with W T = 0 then

ẼAT1 (S

−W ) = H1(A;O(−D(W ))),and if W 6= 0,

ẼAT1 (S

W ) = 0.

Proof: We have to calculate cok(qA : c−w ⊗ V A −→ TA). The following proof that this isH1(A;O(−D(W ))) is that given in [14, Proposition II.3].

We have already considered the kernel, and we have an exact sequence of sheaves

0 −→ O(−D(W )) −→ V A −→ Q(−D(W )) −→ 0.The exact sequence in cohomology ends

V Aφ−→ H0(A;Q(−D(W ))) −→ H1(A;O(−D(W ))) −→ 0,

so it remains to observe that cok(φ) may be identified with cok(qA−w).However Q(−D(W )) is a skyscraper sheaf concentrated its space of sections is W/W (D),

where

W = {(xP )P | xP ∈ V A, and almost all xP ∈ k}is the space of adèles (for torsion points P ) and

W (D) = {(xP ) ∈W | ordP (xP ) + ordP (D) ≥ 0}.Thus cok(φ) = W/(W (−D(W )) + V A) = cok(qA−w) as required. �

Remark 8.9. It is possible to give a more explicit proof as follows. First, one checks anyelement (g1, g2, . . .) ∈

⊕d TA〈d〉 is congruent to one with g2 = g3 = · · · = 0. Now, identify a

subspace of the correct codimension in the image. Using divisors one sees the cokernel mustbe at least this big. Finally, the cokernel is naturally dual to H0(A;O(D(W )), and hencenaturally isomorphic to H1(A;O(−D(W ))) by Serre duality.

9. Multiplicative properties.

Theorem 9.1. If E is constructed from a 1-dimensional group scheme (ie if E = EG orEA) then E is a commutative ring spectrum.

For the rest of this section we suppose E = (N −→ tF∗ ⊗ V ), and that there is a shortexact sequence

0 −→ N β−→ tF∗ ⊗ Vq−→ Q −→ 0.

It is natural to use the geometric terminology, and talk of V as a space of sections (of animagined bundle), and N(c0) as the space of regular sections

First we note that E is flat.

Lemma 9.2. A spectrum E with monomorphic structure β map is flat.


Proof: Tensor product on As is defined termwise. First, note that tF∗ ⊗V is exact for tensor

product with objects P with E−1P ∼= tF∗ ⊗W for some W , so the tensor product is exact onthe vertex part.

For the nub, we use the fact that the category As is of flat dimension 1 by [6, 23.3.5],together with the fact that N is a submodule of tF∗ ⊗ V . �

It follows that tensor product with E models the smash product.

Lemma 9.3. Suppose that V has a commutative and associative product (so we may referto it as an algebra of sections).

If the product of two regular sections is regular then the associated object E admits acommutative and associative product.

Proof: By hypothesis the product on tF∗ ⊗V takes N ⊗RN to N , and therefore gives a mapE ⊗ E −→ E

in As. Associativity and commutativity are inherited from V . �

Corollary 9.4. (i) If V is an affine algebra of functions the product of two regular sectionsis regular.(i) If V is an elliptic algebra of functions then the product of two regular sections is regular.

Proof: Suppose s and t are sections. We must show that if q(s) = 0 and q(t) = 0 thenq(st) = 0. This is clear since regularity is detected one point at at time and ordx(fg) =ordx(f) + ordx(g). �

Now that we have a product structure we can tie up topological and geometric duality ina satisfactory way.

Lemma 9.5. Spanier-Whitehead duality for spheres corresponds to Serre duality in the sensethat the Serre duality pairing

H1(A;O(−D(W )))⊗H0(A;O(D(W ))) −→ H1(A;O)‖ ‖

[S0, S−W ∧ ΣEA]T ⊗ [S0, SW ∧ EA]T [S0,ΣEA]T

is induced by the algebraically obvious Spanier-Whithead pairing

S−W ∧ EA ∧ SW ∧ EA ' S−W ∧ SW ∧ EA ∧ EA −→ S0 ∧ EA ∧ EA −→ EA.Proof: Both maps are induced by multiplication of functions and a residue map (see [14,Chapter II]). �

References

[1] M. Ando “Power operations in elliptic cohomology and representations of loop groups” Trans. AmericanMath. Soc. (to appear)

[2] G. Carlsson and R.Cohen “The cyclic groups and the free loop space.’ Comm. Math. Helv. 62 (1987)423-449

[3] L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) “Equivariant stable homo-topy theory.” Lecture notes in maths. 1213, Springer-Verlag (1986)


[4] V.Ginzburg (notes by V. Baranovsky) “Geometric methods in the representation theory of Hecke alge-bras and quantum groups.” Preprint (1998) (AG/9802004)

[5] V.Ginzburg, M. Kapranov, E. Vasserot “Elliptic algebras and equivariant elliptic cohomology” Preprint(1995) (q-alg/9505012)

[6] J.P.C.Greenlees “Rational S1-equivariant stable homotopy theory.” Mem. American Math. Soc. 661(1999) xii + 289pp

[7] J.P.C.Greenlees “Rational torus equivariant cohomology theories I: the standard model.” (In prepara-tion)

[8] J.P.C.Greenlees “Rational torus equivariant cohomology theories II: Adams spectral sequences.” (Inpreparation)

[9] J.P.C.Greenlees and B.E. Shipley “Rational torus equivariant cohomology theories III: Quillen equiva-lence with the standard model.” (In preparation)

[10] I. Grojnowski “Delocalized elliptic cohomology.” Yale University Preprint (1994).[11] V. Guillemin “Riemann-Roch for toric orbifolds.” J. Diff. Geom. 45 (1997) 53-73[12] R. Hartshorne “Algebraic geometry” Springer-Verlag, 1977[13] I. Rosu “Equivariant elliptic cohomology and rigidity.” Thesis (1998), Preprint (1999).[14] J.-P. Serre “Algebraic groups and class fields” Springer-Verlag (1988)[15] J.H. Silverman “The arithmetic of elliptic curves.” Springer-Verlag (1986)

Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.E-mail address: [email protected]

Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.E-mail address: [email protected]

Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA.E-mail address: [email protected]

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