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]).
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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