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 el* *liptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology* * theory on the compactification of an S1-representation is given by the sheaf cohomology* * of a suitable line bundle on the curve. The construction is easy: by considering functions o* *n the elliptic curve with specified poles one may write down the representing S1-spectrum in t* *he first author's algebraic model of rational S1-spectra [6]. Contents 1. Introduction. * * 1 2. Formal groups from complex oriented theories. * * 3 3. The model for rational T-spectra. * * 5 4. The affine case: T-equivariant cohomology theories from additive and multiplicative groups. * * 9 5. Elliptic curves. * * 13 6. Coordinate data * *14 7. Local cohomology sheaves on elliptic curves. * * 15 8. A cohomology theory associated to an elliptic curve. * * 17 9. Multiplicative properties. * * 20 References * *21 1.Introduction. Two of the most important cohomology theories are associated to one dimension* *al group schemes in a way which is clearest in the equivariant context. Ordinary cohomol* *ogy of the Borel construction is associated to the additive group and equivariant K theory* * is associated to the multiplicative group. It is therefore natural to hope for an equivariant* * cohomology theory associated to an elliptic curve A, and it is the purpose of the present * *note to construct such a theory over the rationals which is equivariant for the circle group. A * *programme to extend this work to higher dimensional abelian varieties and higher dimensio* *nal tori is underway [7, 8, 9]. Let T denote the circle group, and z denote its natural representation on the* * complex numbers. The main purpose of this paper is to construct a rational T-equivarian* *t cohomology ___________ JPCG and MJH are grateful to the Mathematisches Forschungsinstitut Oberwolfac* *h for the opportunity to talk at the 1998 Homotopietheorie meeting, and to JPCG's audience for tolera* *ting the resulting delay. JPCG is grateful to M.Ando for useful conversations, and to H.R.Miller and the * *referee for stimulating comments 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 the points of order dividing n in A. The properties of the cohomology theory when w* *e work over a 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 c* *oefficient ring in degrees 0 and 1 is related to cohomology of the structure sheaf O by EA*T~=H*(A; O); and,Pmore generally, the value on the one point compactification SW of the rep* *resentation W = n anznPgives the sheaf cohomology of an associated line bundle O(-D(W )),* * where D(W ) = nanA[n]: gEA*T(SW ) ~=H*(A; O(-D(W )) and in homology we have gEAT*(SW ) ~=H*(A; O(D(W )): The construction and the isomorphisms in the statement are natural: for this it* * is necessary to specify suitable coordinate data on the elliptic curve. The first version of T-equivariant elliptic cohomology was constructed by Gro* *jnowksi in 1994 [10]. He was interested in implications for the representation theory of c* *ertain elliptic algebras: these implications are the subject of the work of Ginzburg-Kapranov-V* *asserot [5] and the context is explained further in [4]. For this purpose it was sufficient* * to construct a theory on finite complexes taking values in sheaves over the elliptic curve. La* *ter Rosu [13] used this sheaf-valued theory to give a proof of Witten's rigidity theorem for * *the equivariant elliptic genus of a spin manifold with non-trivial T-action. Ando [1] has rela* *ted the sheaf valued theory to the representation theory of loop groups. However, to exploit the theory fully, it is essential to have a theory define* *d on general T-spaces and T-spectra, and to have a conventional group-valued theory represen* *ted by a T-spectrum. This allows one to use the full apparatus of equivariant stable hom* *otopy theory. For example, twisted pushforward maps are immediate consequences of Atiyah dual* *ity; in more concrete terms, it allows one to calculate the theory on free loop spaces,* * and to describe algebras of operations. It is also likely to be useful in constructing an inte* *gral version of the theory, and we hope it may also prove useful in the continuing search for a* * geometric definition of elliptic cohomology. The theory we construct has these desirable properties, whilst retaining a ve* *ry close con- nection with the geometry of the underlying elliptic curve. Our construction di* *rectly models the representing spectrum EA in the first author's algebraic model As of ration* *al T-spectra [6]. Any object (such as that modelling T-equivariant elliptic cohomology) in t* *he algebraic model 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 coh* *omology theory gives a precise means by which the sheaf-valued cohomology can be recove* *red from a conventional theory with values in graded vector spaces. The construction of th* *e Grojnowski sheaf on the elliptic curve from the sheaf on the space of closed subgroups of * *T helps put the earlier construction in a topological context. It is intended to give a full tr* *eatment elsewhere, giving an equivalence between a category of modules over the structure sheaf of* * A and a category 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 theo* *ry is group valued, the original curve can still be recovered from the cohomology theory. * *It is also notable that the earlier sheaf theoretic constructions work over larger rings a* *nd certainly require the coefficients to contain roots of unity: the loss of information can* * be illustrated by comparing the rationalized representation ring R(Cn) = Q[z]=(zn - 1) (with c* *omponents corresponding to subgroups of Cn) to the complexified representation ring, isom* *orphic to the character ring map (Cn; C) (with components corresponding to the elements of Cn* *). Finally, the ingredients of the model are very natural invariants of the curv* *e given by sheaves of functions with specified poles at points of finite order: Definition* * 8.4 simply writes down the representing object in terms of these,1 and readers already familiar w* *ith elliptic curves and the model of [6] need read nothing else. In fact the algebraic model* * of [6] gives a generic de Rham model for all T-equivariant theories, and the models of ellipti* *c cohomology theories highlight this geometric structure. These higher de Rham models shoul* *d allow applications in the same spirit as those made for de Rham models of ordinary co* *homology and K-theory [11]. By way of motivation, we will discuss the way that a T-equivariant cohomology* * theory is associated to several other geometric objects. Perhaps most familiar is the * *complete case discussed in Section 2, where the Borel theory for a complex oriented cohomolog* *y theory is associated to a formal group. Amongst global groups, the additive and multiplic* *ative ones are the simplest, and in Section 4 we describe how they give rise to ordinary B* *orel cohomol- ogy and equivariant K-theory. This construction is notable in that it gives a c* *onstruction of equivariant cohomology theories from oriented 1-dimensional group schemes which* * is func- torial for isomorphisms. It is also functorial for certain isogenies as explain* *ed in 4.3. 2. Formal groups from complex oriented theories. The purpose of this section is to recall that any complex orientable cohomolo* *gy theory E*(.) determines a one dimensional, commutative formal group bGand to explain h* *ow the cohomology of various spaces can be described in terms of the geometry of bG. T* *his is well known but it introduces the geometric language, and motivates our main construc* *tion, which uses this geometric data to construct the cohomology theory. Indeed, we will sh* *ow that the machinery of [6] permits a functorial construction of a 2-periodic rational T-e* *quivariant cohomology theory EG*T(.) from a one dimensional group scheme G over a Q-algebr* *a. Fur- thermore, the construction is reversible in the sense that G can be recovered f* *rom 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 g* *eometric language. Whilst all schemes are affine, the geometric language is equivalent * *to the ring theoretic language, and all geometric statements can be given meaning by transl* *ating them to 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 ar* *e defined in terms of the algebra. The geometric language becomes essential later, since ell* *iptic curves are not affine. A one dimensional commutative formal group law over a ring k is a commutative* * and associative coproduct on the complete topological k-algebra k[[y]]. Equivalent* *ly, it is a complete topological Hopf k-algebra O together with an element y 2 O so that O * *= k[[y]]. ___________ 1This 3rd version of the paper is the first to make the model completely expl* *icit. 4 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU A topological Hopf k-algebra O for which such a y exists is the ring of functio* *ns on a one dimensional commutative formal group bG. The counit O -! k, is viewed as evalua* *tion of functions at the identity e 2 bG, and the augmentation ideal I consists of func* *tions vanishing at 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 bG, and in the present af* *fine context these are specified by the O-module M = F of global sections. In particular, li* *ne bundles L over bGcorrespond to modules M which are submodules of the ring of rational fun* *ctions and free of rank 1. Line bundles can also be described in terms of the zeros and po* *les of their generating section: we only need this in special cases made explicit below. The* * generator f of 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 bGwhere f vanishes (with multiplicities), and D- is the sub* *scheme of bG where 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 to O(-ae). Next we may consider the [n]-series map [n] : O -! O, which corresponds* * to the n-fold sum map n : bG-! bG. We write bG[n] for the kernel of n, and its ring of* * functions is O=([n](y)). Hence, since n*y = [n](y) by definition, M = ([n](y)) corresponds t* *o O(-bG[n]), and M = ( ([n](y))a ) corresponds to O(-abG[n]). Finally, if M corresponds to * *O(-D) and M0 corresponds to O(-D0) then M_ := Hom (M; O) corresponds to O(D) and M M0 corresponds to O(-D - D0). This gives sense to enough line bundles for our purp* *oses. Now suppose that E is a 2-periodic ring valued theory with coefficients E* co* *ncentrated in even degrees. The collapse of the Atiyah-Hirzebruch spectral sequence for CP 1 * *shows that E is complex orientable. We may define the T-equivariant Borel cohomology by E** *T(X) = E*(ET xT X). We work over the ring k = E0T(T) = E0, and view E0T= E0(CP 1) as t* *he ring of functions on a formal group bGover k. The tensor product and duality of* * line bundles makes CP 1 into a group object, so E0(CP 1) is a Hopf algebra and bGis a group.* * From this point of view, the augmentation ideal I = ker(E0T-! E0) consists of functions v* *anishing at the identity e 2 G. Now, if V is a complex representation of the circle group T, we also let V * *denote the associated bundle over CP 1 and the Thom isomorphism shows "E0((CP 1)V) = "E0T(* *SV ) is a rank 1 free module over E0T, and hence corresponds to a line bundle L(V ) ove* *r bG, whose global sections are naturally isomorphic to the module L(V ) = "E0T(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(-bG[n]) is the sheaf of functions vanishing on bG[n], and its * *module of sections is generated by the multiple [n](y) of the coordinate y. (4) L(azn) = O(-abG[n]) is the sheaf of functions vanishing on bG[n] with mu* *ltiplicity a, and its module of sections is generated ([n](y))a. Proof: The first statement is clear since "E0T(S0) = E0T. For the second we use* * the equivalence (CP 1)z ' (CP 1)0=(CP 0)0. The third statement follows from the Gysin sequence * *since zk RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. 5 is the pullback of z along the kth power map CP 1 -! CP 1. The final statement * *follows from the tensor product property. This gives the fundamental connection between the equivariant cohomology of a* * sphere and sections of a line bundle. Corollary 2.2. For any a 2 Z, n 6= 0 we have E"0T(Sazn) = O(-abG[n]): We want to finish this section by pointing out that if the formal group bG(wh* *ich is affine) is replaced by a group G with higher cohomology, we cannot expect a cohomology * *theory entirely in even degrees. Whenever the group is not affine, we write O for the * *structure sheaf of G. This is reconciled by the above usage since in the affine case the struc* *ture sheaf is determined 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 of sheaves O(-ae)=O(-(a + 1)e) - O(-ae) - O(-(a + 1)e): Any satisfactory cohomology theory will be functorial, and applying "E0T(.) wil* *l give sections of the associated sheaves. However the global sections functor on sheaves is no* *t usually right exact, 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 se* *quence of spaces should be the long exact cohomology sequence induced by the sequence of * *sheaves. This gives a natural candidate for the odd cohomology: "EiT(Saz) = Hi(G; O(-ae)) fori = 0; 1: This explains why it is possible for complex orientable cohomology theories to * *have coefficient rings in even degrees (formal groups are affine), and indeed how their values o* *n all complex spheres can be the same. It also explains why we cannot expect either property * *for a theory associated to an elliptic curve. 3.The model for rational T-spectra. For most of the paper we work with the representing objects of these cohomolo* *gy theories, namely T-spectra [3]. Thus we prove results about the representing spectra, an* *d deduce consequences about the cohomology theories. More precisely, any suitable T-equ* *ivariant cohomology theory E*T(.) is represented by a T-spectrum E in the sense that "E*T(X) = [X; E]*T: This enables us to define the associated homology theory EeT*(X) = [S0; E ^ X]T* in the usual way. We shall make use of the elementary fact that the Spanier-Whi* *tehead dual of the sphere SV is S-V , as one sees by embedding SV as the equator of SV 1. H* *ence, for example "E0T(SV ) = [SV ; E]T = [S0; S-V ^ E]T = ssT0(S-V ^ E) = "ET0(S-V ): 6 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU 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 s* *uffices to check the values on the homogeneous spaces T=H for closed subgroups T, since all spac* *es are built from these. Convention 3.1. Henceforth all spaces, groups and spectra are rationalized whet* *her or not this is indicated in the notation. Our results are made possible because there is a complete algebraic model of * *the category of rational T-spectra, and hence of rational T-equivariant cohomology theories * *[6]. There are two models for rational T-spectra, as derived categories of abelian categor* *ies: T-Spectra' D(As) ' D(At): The standard abelian category As has injective dimension 1, and the torsion abe* *lian category At is of injective dimension 2. It is usually easiest to identify the model for* * a T-spectrum in D(At), at least providing its model has homology of injective dimension 1. T* *his is then transported to the standard category, where calculations are sometimes easier. * *To describe the categories, we need to use the discrete set F of finite subgroups of T. On* * this we consider the constant sheaf R of rings with stalks Q[c] where c has degree -2. * *We need to consider the ring R = map (F; Q[c]) of global sections. For each subgroup H, we* * let eH 2 R denote the idempotent with support H. If w : F -! Z0 is a function, we write c* *w for the element of R with cw(H) = cw(H). Now consider the multiplicative set E generate* *d by the universal Euler classes e(V ) for the representations V of T with V T= 0. These* * are defined by 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 -! Z0 of finite support}: L * * Q We let tF*= E-1R: as a graded vector space this is H Q in positive degrees an* *d H Q in degrees zero and below. The objects of the standard model As are triples (N; fi; V ) where N is an R-* *module (called the nub), V is a graded rational vector space (called the vertex) and fi : N -!* * tF* V is a morphism of R-modules (called the basing map) which becomes an isomorphism wh* *en E is inverted. When no confusion is possible we simply say that N -! tF* V is an * *object of the standard abelian category. An object of As should be viewed as the modul* *e N with the additional structure of a trivialization of E-1N. A morphism (N; fi; V ) -!* * (N0; fi0; V 0) of objects is given by an R-map : N -! N0 and a Q-map OE : V - ! V 0compatible* * under the basing maps. Since the standard abelian category has injective dimension 1, homotopy types* * of objects of the derived category D(As) are classified by their homology in As, so that h* *omotopy types correspond 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 cl* *osed subgroups T, the vertex V is the value of the sheaf at the subgroup T and the fact that * *the basing map fi : N -! tF* V is an isomorphism away from E is the manifestation of the p* *atching condition for sheaves. The objects of the torsion abelian category At are triples (V; q; T ) where V* * is a graded rational vector space T is an E-torsion R-module and q : tF* V - ! T is a morph* *ism of R- modules. The condition on T is equivalent toLrequiring (i) that T is the sum of* * its idempotent factors T (H) = eH T in the sense that T = H T (H) and (ii) that each T (H) i* *s a torsion RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. 7 Q[c]-module. When no confusion is possible we simply say that tF* V - ! T is an* * object of the torsion abelian category. In the sheaf theoretic approach, the module T* * (H) is the cohomology of the structure sheaf with support at H. By contrast with the stand* *ard abelian category, the torsion abelian category has injective dimension 2. Thus not ever* *y object X of the derived category D(At) is determined up to equivalence by its homology H* **(X) in the abelian category At. We say that X is (intrinsically) formal if it is dete* *rmined up to isomorphism by its homology. Evidently, X is formal if its homology has injecti* *ve dimension 0 or 1 in At. In general, if H*(X) = (tF* V - ! T ), the object X is equivalent* * to the fibre of a map (tF* V - ! 0) -! (tF* 0 -! T ) (in the derived category) between objec* *ts in Atof injective dimension 1. This map is classified by an element of Ext(tF* V; * *T ), so that X is formal if the Ext group is zero in even degrees. Thus X is formal if both * *V and T are in even degrees or if T is injective in the sense that each T (H) is an injecti* *ve Q[c]-module. Definition 3.2. [6, 5.8.2] Suppose given a function w : F -! Z with finite supp* *ort. The algebraic 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] -! Hom As(Sw ; M) -! 0; so we shall need to calculate these Hom and Ext groups. For the present we rest* *rict ourselves to the Hom groups. fi F Lemma 3.3. For an object M = (N -! t* V ) of the abelian category As we have Hom As(Sw ; (N -! tF* V )) = N(c-w ) := {n 2 N | fi(n) 2 c-w V }: We may now describe how to construct the counterparts Ms(E) = (N -! tF* V ) (* *in the standard abelian category As) and Mt(E) = (tF* V - ! T ) (in the torsion abelia* *n category At) of a rational T-spectrum E. From the above discussion, the model Ms(E) dete* *rmines E itself, but Mt(E) only determines E if Mt(E) is formal. First, we may define th* *e vertex V , the nub N and torsion module T by formulae and then turn to practical computati* *ons in terms of data easily accessible to us. To describe the answer, we need the univ* *ersal F-space EF, and the basic cofibre sequence EF+ -! S0 -! "EF where E"F is the join S0 * EF+. We also use functional duality on T-spectra de* *fined by DX = F (X; S0). The nub vertex and torsion modules associated to a T-spectrum * *E are given by o N = ssT*(E ^ DEF+) o V = ssT*(E ^ "EF) o T = ssT*(E ^ EF+) The vertex is straightforward to calculate in terms of available data: V = ssT*(E ^ "EF) = lim ssT*(E ^ SV ): ! V T=0 An approach to the nub via limits is possible but not very illuminating. 8 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU The associated torsion sheaf T may be described by saying that its sections o* *ver the set [ H] of subgroups of H is ssT*(E[ H]+ ^ E). Using idempotents from the Burnside* * ring of H this may be split up into stalks ssT*(E ^ E) one for each subgroup K H* * (it turns out that these are in independent of H, as is required for consistency). * *Now if H has order n, the infinite sphere S(1zn) is a model for E[ H], and hence there is a * *long exact sequence n T . .-.! ssT*(E) -! ssT*(S1z ^ E) -! ss*(E[ H]+ ^ E) -! . .:. n T azn Since ssT*(S1z ^ E) = lim ss*(S ^ E) we may conclude there is a short exact * *sequence ! a 0 -! E*VT(H)=e(zn)1 - ! ssT*(E[ H]+ ^ E) -! e(zn)-power torsion(E*VT(H)) -! 0 * * n where E*VT(H)is the ring graded by multiples of zn with azn-th component ssT0(S* *az ^ 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 avai* *lable data. If this is to determine E we must show in addition that Mt(E) is formal. In our ca* *se this will hold because V and T are in even degrees. It is convenient for calculation to d* *educe Ms(E). q Lemma 3.4. If Mt(E) = (tF*V - ! T ) has surjective structure map, then Mt(E) is* * formal and 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 1 0 1 0 1 N tF* V T 0 -! Ms(E) = @ # A -! @ # A -! @ # A -! 0 tF* V tF* V 0 in As. Proof: To see that Mt(E) is formal, it is only necessary to remark that T is th* *e quotient of an E-divisible group and therefore injective [6, 5.3.1]. Finally, we should record that spheres and suspensions in the algebraic and t* *opological contexts correspond. Lemma 3.5. [6, 5.8.3] Suppose W is a virtual representation with W T= 0 and let* * w = dim C(W ). The object modelling the sphere SW with V T= 0 in As is the algebr* *aic sphere Sw : 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 w* *ith a periodicity element u of degree 2. We may therefore shift even degree elements * *into degree zero. For example uc is the degree 0 counterpart of c. For the rest of the pape* *r we use c to denote the degree 0 version. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. 9 4. The affine case: T-equivariant cohomology theories from additive and multiplicative 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 co* *nstruction of a cohomology theory EG*T(.) associated to a one dimensional affine group scheme* * G. This will serve to illustrate the algebraic categories described in Section 3 and al* *so complete the motivation of our construction for elliptic curves. The additive group scheme Ga and the multiplicative group scheme Gm are affin* *e, and therefore the construction of associated cohomology theories is considerably si* *mpler than that for elliptic curves. Nonetheless the general features are the same, and i* *t is useful to have seen the phenomena first in a familiar setting. It turns out that the asso* *ciated 2-periodic T-equivariant theories are concentrated in even degrees and (EGa)0T(X) = H*(ET xT X) and (EGm )0T(X) = K0T(X); and models for these theories were given in [6]. We will repeat the answer here* * in our present language. We start by summarizing the properties we want of such a construction, and th* *en observe that the algebraic categories of Section 3 immediately gives a unique construct* *ion. o The subgroup T[n] of order n corresponds to the subgroup G[n] of element* *s of order dividing n o The family F of finite subgroups corresponds to the set G[tors] of eleme* *nts of torsion points. n o The suspension Saz ^ EG corresponds to the sheaf O(aG[n]) and more gener* *ally, suspension by zn correspondsnto tensoring with O(G[n]). o The inclusion S0 -! Sz which induces multiplication by the Euler class * *(in the presence of a Thom isomorphism) corresponds to O -! O(G[n]). o We extend the notation to allow n azn S1z := lim S ! a to correspond to the sheaf O(1G[n]) := lim O(aG[n]) ! a and "EF := lim Sazn ! a;n to correspond to O(1G[tors]) := lim O(aG[n]): ! a;n (This description of "EF requires us to be working rationally; more gene* *rally one only has "EF = lim SV .) ! V T=0 We need to say more about Euler classes. Consider the subgroup T[n]nof order* * n. The natural geometricnconstruction is the Euler class induced by S0 -! Sz . Pullin* *g back a Thom class for Sz gives the function e(zn) in R, which vanishes at all subgrou* *ps of T[n]. 10 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU Evidently, if we take cd to be the function vanishing to the first order on the* * group of order d and taking the value 1 elsewhere, we have Y e(zn) = cd; d|n so that we may view cd as a universal cyclotomic function.n We have already motivatednthe idea that S0 -! Sz should correspond to O -! O* *(G[n]). The Thom class for Sz corresponds to a generating section of O(G[n]) and hence* * e(zn) should correspond to a function O(zn) defining G[n] in G. Now choose a coordinate y =: O(z) at e 2 G. We may then take O(zn) := [n](y) := n*(y): so that O(zn) is a function vanishing to first order on G[n]. Next, we may a decompose the divisor G[n]: X G[n] = G d|n where G is the divisor of points of exact order d. Now we define a function * *OE := OEG vanishing to the first order on G recursively by the condition Y O(zn) = OE : d|n the formula defines OE directly for n = 1, and for larger values of n, it is* * defined by dividing O(zn) by the previously defined OE. P Definition 4.1.PGiven a virtual complex representation V = nanzn, we define a* * divisor by D(V ) = nanG[n]. We say that a 2-periodic T-equivariant cohomology theory * *E*T(.) is of type G if eEiT(SV ) ~=Hi(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 ins* *ist V is an actual representation, and reformulate the other case as the isomorphism EeTi(SV ) ~=Hi(G; O(D(V )): Now we require these isomorphisms to be natural for inclusions0j : V - ! V 0of * *represen- tations. First note that such a map induces a map SV -! SV of T-spaces and h* *ence maps 0 i V j* : eEiT(SV ) -! eET(S ) and 0 j* : eETi(SV ) -! eEiT(SV ): On the other hand we have inclusion of divisors D(V ) -! D(V 0), inducing maps O(-D(V 0)) -! O(-D(V )) and O(D(V )) -! O(D(V 0)): The induced maps in sheaf cohomology are required to j* and j*. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *11 Theorem 4.2. Given a commutative 1-dimensional affine group scheme G over a rin* *g con- taining Q, and a coordinate y at e 2 G there is a 2-periodic cohomology theory * *EG*T(.) of type G. Furthermore, EG*Tis in even degrees and G = spec(EG0T). The construct* *ion is natural for isomorphisms. Remark 4.3. The construction is also natural for quotient maps p : G -! G=G[n] * *in the sense that there is a map p* : E(G=G[n]) -! inflTT=T[n]EG of T-spectra, where E* *G is viewed as a T=T[n]-spectrum and inflated to a T-spectrum. More precisely, if y is a coordinate on G then its norm a2G[n]Tay is a coordi* *nate on G=G[n] (where Ta denotes translation by a). Using these coordinates, we obtain * *equivariant spectra EG=G[n] and EG. As a first step to maps between them, note that we have* * maps p*V: V (G=G[n]) -! V G and p*T: T (G=G[n]) -! T G corresponding to pullback of * *functions. However p*Vand p*Tdo not give a map of T-spectra E(G=G[n]) -! EG; for example t* *he non-equivariant part of E(G=G[n]) corresponds to functions on G=G[n] with suppo* *rt at the identity, and these pull back to functions on G supported on G[n], which corres* *pond to the part of EG with isotropy contained in T[n]. The answer is to view the circle of* * equivariance of EG as T=T[n], and then to use the inflation functor studied in Chapters 10 a* *nd 24 of [6] to obtain a T-spectrum. Proof: The construction was motivated in Section 2. We take V G = O(1tors); T G = O(1tors)=O; and use the map qG : tF* O(1tors) -! O(1tors)=O given by ________ s=e(W ) f 7-! s . f=O(W ): We must explain how T G is a module over R, and why fi is a map of R-modules.* * We make T G into a module over R by letting cd act as OEd. Since any function only* * has finitely many poles, all but finitely many cd act as the identity on any element of T G,* * and since poles are of finite order, T G is a E-torsion module. The definition of the map* * qG shows it is an R-map. Finally, we must show that the homotopy groups of the resulting object are as* * required in 4.1. By 3.4 we have Ms(EG) = (fiG : NG -! tF*V G), where NG = ker(tF*V G -! T G* *), and we need to calculate [SW ; EG]T*= [Sw ; Ms(EG)]*: Since qG is epimorphic, fiG is monomorphic, and T G is injective. Thus by 3.4 w* *e have the explicit injective resolution 0 1 0 1 0 1 NG tF* V G T G 0 -! Ms(EG) = @ # A -! @ # A -! @ # A -! 0: tF* V G tF* V G 0 Now, applying 3.3 we see Ext(Sw ; Ms(EG)) = 0 since any torsion element t 2 T G* * lifts to f 2 V G and hence to 1=e(W ) O(W )f. It is immediate from the definition that Hom (Sw ; Ms(EG)) = {c-w f | f=O(W ) regular}: 12 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU By construction the divisor associated to the function O(V ) is D(V ), so f=O(V* * ) is regular if and only if f 2 O(-D(V )) as required. Remark 4.4. In the above proof we made use of the fact that the Euler class O(W* * ) exists as a function in V G. The point of this comment will become apparent when we treat* * the elliptic case which behaves rather differently: there the Euler class is given by differ* *ent functions at different points, corresponding to the fact that the cohomology theory is no* *t complex orientable, 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 th* *e coproduct x 7-! 1 x + x 1. We choose x as a coordinate about the identity, zero. The gr* *oup Ga[n] of points of order dividing n is defined by the vanishing of O(zn) = nx, so the* * identity is the only element of finite order over Q-algebras. This case becomes rather degenera* *te in that it only 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(1tors) = tF* Q[x; x-1] -! Q[x; x-1]=Q[x] = O(1tors)=O _______ s=e(V ) f 7-! s . f=O(V ): Here O = Q[x] and O(zn) = nx. The ring O(1tors) = Q[x; x-1] of functions with p* *oles only at points of finite order is obtained by inverting the Euler class of z. Accord* *ingly, 2-periodic Borel 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 defined by the coproduct z 7-! z z. We choose y = 1 - z as a coordinate about the ide* *ntity element, 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 O* *(zn) = 1 - zn. Proposition 4.6. [6, 13.4.4] The model of equivariant K-theory in the torsion m* *odel is formal, concentrated in even degrees and in each even degree is the map tF* O(1tors) -! O(1tors)=O _______ s=e(V ) f 7-! s . f=O(V ): Here O = Q[z; z-1] and O(zn) = 1 - zn. The ring O(1tors) of functions with pole* *s only at points of finite order is obtained by inverting all Euler classes. Accordingly,* * equivariant K theory 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 c* *ompletes the circle by establishing the universality of the motivation described in Sect* *ion 2. However, since we must work over Q, there is little difference from the additive group a* *bove. Suppose given a commutative one dimensional formal group bGover a ring k containing Q, * *with a coordinate y. We may identify the ring of functions on bGwith k[[x]], and the g* *roup structure is the coproduct x 7-! F (x1; 1x). The group bG[n] of points of order dividing * *n is defined by the vanishing of O(zn) = [n](x) so the identity is the only element of finit* *e order over RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *13 Q-algebras. We may now make the direct analogue of the construction in 4.2. T* *his case becomes 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 c* *omplex orientable theory E*(.) in the torsion model is formal, concentrated in even de* *grees and in each even degree is the map tF* O(1tors) = tF* E0((x)) -! E0((x))=E0[[x]] = O(1tors)=O _______ s=e(V ) f 7-! s . f=O(V ): Here O = E0[[x]] and O(zn) = [n](x). The ring O(1tors) = E0[[x]][1=x] = E0((x)* *) of functions with poles only at points of finite order is obtained by inverting th* *e Euler class of z. Accordingly, 2-periodic E-Borel cohomology is the theory associated to the f* *ormal group of E in the sense of 4.2. 5.Elliptic curves. In this section we record the well known facts about elliptic curves that wil* *l play a part in 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 point e) over an algebraically closed field k of characteristic 0 and let O = OA be i* *ts sheaf of regular functions. Note that O = k, so the sheaf contains a great deal more inf* *ormation than its ring of global sections. A divisor on A is a finite Z-linear combinat* *ion of points, and associated to any rational function f on A we have the divisor div(f) = Por* *dP (f)(P ), where ordP(f) 2 Z is the order of vanishing of f at P . In the usual way, if D * *is a divisor on A, we write O(D) for the associated invertible sheaf. Its global sections are g* *iven by O(D) = {f | div(f) -D} [ {0}; so that for a point P , the global sections of O(-P ) are the functions vanishi* *ng 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; .) = (.) and H1(A; .), whic* *h are related by Serre duality. This takes a particularly simple form since the canonical div* *isor is zero on an elliptic curve: H0(A; O(D)) = H1(A; O(-D))_; where (.)_ = Hom k(.; k) denotes vector space duality. From the Riemann-Roch theorem we deduce that the canonical divisor is 0 and t* *he coho- mology of each line bundle: ae deg D ifdeg(D) 1 dim(H0(A; O(D)) = 0 ifdeg(D) -1 and ae | degD| ifdeg(D) -1 dim(H1(A; O(D)) = 0 ifdeg(D) 1: For the trivial divisor one has dim(H0(A; O)) = dim(H1(A; O)) = 1: 14 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU Now if D = PnP(P ) is a divisor of degree 0, we may form the sum P (D) = PnPP i* *n A, and D is linearly equivalent to (P (D)) - (e). If P (D) = e then the sheaf O(D)* * has the same cohomology 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 (O(*e)) = {O(ne)}n0 . Indeed, this is * *the basis of the proof in [15, III.3.1] that any elliptic curve is a subvariety of * *P2 defined by a Weierstrass equation. We choose a basis {1; x} of O(2e) and a extend it to a ba* *sis {1; x; y} of O(3e). Now observe that since O(6e) is 6-dimensional, there is a relation b* *etween the seven elements 1; x; x2; x3; y; xy and y2: this is the Weierstrass equation* *, and it may be verified that A is the closure in P2 of the plane curve it defines. The graded * *ring (O(*e)) has generator Z of degree 1 corresponding to the constant function 1 in O(e), X* * of degree 2 corresponding to x, and Y of degree 3 corresponding to y. These three variabl* *es satisfy the homogeneous form of the Weierstrass equation. The statement that A is the p* *rojective closure of the plane curve defined by the Weierstrass equation may be restated * *in terms of Proj: A = Proj((O(*e))): 6. Coordinate data Our main theorem constructs a cohomology theory of type A from an elliptic cu* *rve together with suitable coordinate data. In this section we describe the data, and the c* *hoices of functions that they permit. Definition 6.1. Coordinate data for an elliptic curve is a choice of two functi* *ons xe with a pole of order 2 at the identity and nowhere else, and ye with a pole of order* * 3 at the identity and nowhere else. We also require that xe and ye only vanish at torsio* *n points. This coordinate data determines a local uniformizer te = xe=ye of Oe, and hence also* * a uniformizer tP 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; * *Z0away from e, so ue cannot be a constant. (iii) One popular choice of coordinate data involves choosing a point P of or* *der 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 0) + (P 00) where A[2] = {e; P; P 0; P 00}. Thus div(te) = (e) + (P ) - (P 0) - (P 00): The divisor A of points of exact order n will play a central role. Note th* *at X A[n] = A; d|n and X A[tors] = A: d1 The coordinate data allow us to specify a function defining the points of exact* * order d. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *15 Lemma 6.3. Given a choice of coordinate data on the elliptic curve d, for each * *d 2, there is a unique function td with the properties (1) td vanishes exactly to the first order on A, (2) td is regular except at the identity e 2 A where it has a pole of order * *|A|, (3) t|A|etd takes the value 1 at e Proof: Consider the divisor A - |A|(e). Note that the sum of the points o* *f A in A is the identity: if d 6= 2 this is because points occur in inverse pairs, and* * if d = 2 it is because the A[2] is isomorphic to C2x C2. It thus follows from the Riemann-Roch* * theorem that there is a function f with A - |A|(e) as its divisor. This function * *(which satisfies the 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 ss = {d1; : :;:ds} of orders 2,* * there is again a unique function OE with analogous properties. Indeed, the good multiplica* *tive property of the normalization means we may take Y OE = OE: i This applies in particular to the set A[n] \ {e}. For some purposes, it is convenient to have a basis for functions with specif* *ied poles. We already have the basis 1; x; y; x2; xy; : :i:f all the poles are at the identit* *y. Multiplication by a function f induces an isomorphism f. : O(D) -! O(D + (f)) so we can translate the basis we have. Q n(b) Lemma 6.5. For the divisor D = d1 n(d)A let t*(D) := b2 tb . Multiplicati* *on by t*(D) gives an isomorphism ~= 0 t*(D). : H0(A; O(deg(D)(e)) -! H (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 in the 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 associa* *ted to an elliptic curve A are analogous to the affine case. The vertex V A = O(1tors) 16 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU consists of rational functions whose poles are all at torsion points, however t* *he torsion module is not simply the quotient of this by regular functions, but rather T A = (O(1tors)=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 dele* *ting torsion points. Thus localization only permits poles at torsion points: for exa* *mple OP is the subsheaf of O(1tors) consisting of functions regular at P . For any effective divisor D we may use the short exact sequence 0 -! O -! O(aD) -! Q(aD) -! 0 of sheaves to define the quotient sheaf Q(aD) for 0 a 1. The cohomology of Q(* *1D) is the cohomology of A with support on D. In fact we may reduce constructions to the case when the divisor D is a singl* *e point P . Evidently, Q(1P ) is a skyscraper sheaf concentrated at P , so we may localize * *at P to obtain 0 -! OP -! O(1P )P -! Q(1P ) -! 0: Notice that O(1P )P = O(1tors). Since A is a smooth curve, the local ring OP is a discrete valuation ring, an* *d if we choose a local uniformizer tP any element of (Q(1P )) may be represented by an element* * of the form a-1t-1P+ a-3t-3P+ . .+.a-nt-nP for suitable scalars a-i. Thus the sequence becomes 0 -! OP -! OP[1=tP] -! OP=t1P- ! 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(1P ) O(rP ) ~=Q(1P ): Note that it is immediate from the Riemann-Roch formula that for 0 a 1 the cohomology 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(1D) - ! Q(1D) # # O -! O(1(D + D0)) - ! Q(1(D + D0)) of sheaves, and hence a map Q(1D) -! Q(1(D + D0)). Proposition 7.3. If P; P 0are distinct points of A then the natural map ~= 0 Q(1P ) Q(1P 0) -! Q(1(P + P )) is an isomorphism. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *17 Proof: We apply the Snake Lemma to the diagram O O -! O(1P ) O(1Q) - ! Q(1P ) Q(1P 0) # # # O -! O(1(P + P 0)) - ! Q(1(P + P 0)) in the abelian category of sheaves on A. The first vertical is obviously surjec* *tive with kernel O. The kernel of the second vertical is also O, since if f and f0 are local sec* *tions of O(1P ) and O(1P 0) (ie f only has poles at P and f0 only at P 0) then f + f0 = 0 impli* *es that f and f0 are regular. Finally we must show that O(1(P + P 0)) is the sheaf quo* *tient of O -! O(1P ) O(1P 0). However, this may be verified stalkwise, where it is clea* *r. Let us now collect what we need for the construction. To give a Thom isomorph* *ism for Q(1A) we need to choose local uniformizers tP at each point P of exact order* * d. For example we explained in Section 6 how coordinate data determines a function td * *vanishing on A to the first order at all points of A, and we could take tP = td for* * all points P of exact order d. Corollary 7.4. The natural map gives an isomorphism M ~= Q(1A) -! Q(1tors); d and a choice of coordinate tP at each P 2 A gives a Thom isomorphism ~= Td : Q(1A) O(A) -! Q(1A): The sheaf Q(1A) has no higher cohomology and its global sections are Q(1A) = V A={f | f is regular onA}: Remark 7.5. This corresponds to the fact that there is a rational splitting _ EF+ ' E H where E = 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 f* *ar has consisted entirely of motivation and repackaging of known results by way of pre* *paration. Theorem 8.1. Given an elliptic curve A over a field k of characteristic zero, a* *nd coordinate data (xe; ye), there is an associated 2-periodic rational T-equivariant cohomol* *ogy theory of type A, so that for any representation W with W T= 0 we have gEAiT(SW ) = Hi(A; O(-D(W ))) and gEATi(SW ) = Hi(A; O(D(W ))) where the divisor D(W ) is defined by taking X X D(W ) = anA[n] when W = anzn: n n 18 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU This association is functorial for isomorphisms of elliptic curves with coordin* *ate data. Remark 8.2. (i) The elliptic curve can be recovered from the cohomology theory.* * Indeed, we may form the graded ring gEA0T(S-*z) := {gEA0T(S-az)}a0 from the products S-az^ S-bz- ! S-(a+b)z, and the elliptic curve can be recover* *ed from the cohomology theory via 0 -*z A = Proj(gEAT(S )); as commented in Section 5. (ii) The coordinate data on A can therefore be recovered from suitable elements* * of homology: T 2z T 3z xe 2 gEA0(S ) and ye 2 gEA0(S ): Remark 8.3. We have not required the field to be algebraically closed. To see t* *he advantage of this, note that even for the multiplicative group, the individual points of * *order n are only defined over k if k contains appropriate roots of unity. However Gm [n] (define* *d by 1-zn) and hence also Gm (defined by the cyclotomic polynomial OEn(z)) are defined ove* *r Q. Hence equivariant K theory itself is defined over Q. For an elliptic curve A we requi* *re that there is a basis for O(aG)) consisting of functions defined over k. Proof: We must describe a vector space V = V A, an R-module T A and an R-map qA : tF* V A -! T A: It is easy to describe V A and T A; indeed, we take V A = O(1tors) consisting of rational functions whose poles are all at torsion points, and tor* *sion module T A = (Q(1tors)): The splitting M Q(1tors) ~= Q(1A) d of 7.4 gives M T A = T A d where T A = V A={f | f is regular onA}: It is not hard to describe the R-module structure on T A. The direct sum spli* *tting of T A corresponds to the splitting Y R ~= Q[c]; d and T A is a Q[c]-module where c acts as multiplication by the function td d* *efining A. Since the order of any pole is finite, T A is a torsion Q[c]-module. Notice * *that the definition of the Thom isomorphism is arranged so that the composite OED : Q(1D) = Q(1D) O -! Q(1D) O(D) ~=Q(1D) is multiplication by td. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *19 Definition 8.4. If u : F -! Z is a function positive almost everywhere, we defi* *ne M qA : cu V - ! T A = T A d by specifying its dth component _____ qA(cu f)d = tu(d)df: Lemma 8.5. The definition does determine an R-map qA : tF* V - ! T A. Proof: Since any function is regular at all but finitely many points, the map q* *A maps into the sum. L Now, R-maps q : tF* V -! dTd 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* * components of qA (ie qd(cs f) = qA(csffi(d) f)d) have these properties, and that the func* *tion they determine agrees with qA(cu f) wherever it is defined. Now we can check that the resulting homology and cohomology of spheres agrees* * with the cohomology of the corresponding divisors on the elliptic curve. Consider the complex representation W with W T= 0 and the corresponding funct* *ion w(H) = dimC(W H). We see from 3.3 and 3.5 (as in the proof of 4.2 that gEAT0(SW ) = ker(qA : cw V A -! T A) and EgA T1(SW ) = cok(qA : cw V - ! T A) and similarly with W replaced by -W . Since the kernel and cokernel are vector * *spaces over k, it is no loss of generality to extend scalars to assume it is algebraically * *closed. This is convenient because it is simpler to treat separate points of order n one at a t* *ime. The following two lemmas complete the proof. Lemma 8.6. If W is a representation with W T= 0 then gEAT0(SW ) = H0(A; O(D(W ))); and if W 6= 0, gEAT0(S-W ) = 0: Proof: By definition _____ qA(cw f)d = tw(d)df: Since the function td vanishes to exactly the first order on A, the conditio* *n that f lies in the kernel is that ordP(f) -w(d) for each point P of exact order d. Since * *D(W ) = Pw(dP)(P ) we have ker(qA : cw W -! T A) = {f 2 V A | div(f) + D(W ) 0} as required. Replacing W by -W , the second statement is immediate. 20 J.P.C.GREENLEES, M.J.HOPKINS, AND I.ROSU Remark 8.7. The proof is local and therefore shows the kernel is actually the s* *ubsheaf O(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 EgA T1(S-W ) = H1(A; O(-D(W ))); and if W 6= 0, gEAT1(SW ) = 0: Proof: We have to calculate cok(qA : c-w V A -! T A). The following proof that* * this is H1(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 sheav* *es 0 -! O(-D(W )) -! V A -! Q(-D(W )) -! 0: The exact sequence in cohomology ends OE 0 1 V A -! H (A; Q(-D(W ))) -! H (A; O(-D(W ))) -! 0; so it remains to observe that cok(OE) 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 2 V A; and almost allxP 2 k} is the space of adeles (for torsion points P ) and W (D) = {(xP) 2 W | ordP(xP) + ordP(D) 0}: Thus cok(OE) = W=(W (-D(W )) + V A) = cok(qA-w ) as required. Remark 8.9. It is possibleLto give a more explicit proof as follows. First, one* * checks any element (g1; g2; : :):2 dT A is congruent to one with g2 = g3 = . .=.0. No* *w, identify a subspace of the correct codimension in the image. Using divisors one sees the c* *okernel must be at least this big. Finally, the cokernel is naturally dual to H0(A; O(D(W ))* *, and hence naturally 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 = E* *G or EA) then E is a commutative ring spectrum. For the rest of this section we suppose E = (N - ! tF* V ), and that there is* * a short exact sequence fi F q 0 -! N -! t* V - ! Q -! 0: It is natural to use the geometric terminology, and talk of V as a space of sec* *tions (of an imagined 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 fi map is flat. RATIONAL S1-EQUIVARIANT ELLIPTIC COHOMOLOGY. * *21 Proof: Tensor product on As is defined termwise. First, note that tF* V is exac* *t for tensor product with objects P with E-1P ~=tF* W for some W , so the tensor product is * *exact on the 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 * *refer to it as an algebra of sections). If the product of two regular sections is regular then the associated object * *E admits a commutative and associative product. Proof: By hypothesis the product on tF* V takes N R N to N, and therefore gives* * a map E 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 re* *gular sections is regular. (i) If V is an elliptic algebra of functions then the product of two regular se* *ctions is regular. Proof: Suppose s and t are sections. We must show that if q(s) = 0 and q(t) = * *0 then q(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 in a satisfactory way. Lemma 9.5. Spanier-Whitehead duality for spheres corresponds to Serre duality i* *n the sense that the Serre duality pairing H1(A; O(-D(W ))) H0(A; O(D(W ))) -! H1(A; O) k k [S0; S-W ^ EA]T [S0; SW ^ EA]T [S0; EA]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 loo* *p groups" Trans. American Math. Soc. (to appear) [2]G. Carlsson and R.Cohen "The cyclic groups and the free loop space.' Comm. * *Math. 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Shipley "Rational torus equivariant cohomology the* *ories 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) 5* *3-73 [12]R. Hartshorne "Algebraic geometry" Springer-Verlag, 1977 [13]I. Rosu "Equivariant elliptic cohomology and rigidity." Thesis (1998), Prep* *rint (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: j.greenlees@sheffield.ac.uk Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA. E-mail address: mjh@math.mit.edu Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA. E-mail address: ioanid@math.mit.edu