EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
                              By Ioanid Rosu
            with an appendix by Allen Knutson and Ioanid Rosu



       Abstract.For T an abelian compact Lie group, we give a description of T-*
 *equivariant
       K-theory with complex coefficients in terms of equivariant cohomology. I*
 *n the appen-
       dix we give applications of this by extending results of Chang-Skjelbred*
 * and Goresky-
       Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.


                              1.Introduction

  Let T be an abelian compact Lie group, not necessarily connected.  Let X be a
compact T -equivariant manifold, or more generally a finite T -CW complex. We d*
 *enote
by H*T(X) the T -equivariant (Borel) cohomology of X, as described in Atiyah and
Bott [1], and by K*T(X) the T -equivariant K-theory of X, as described in Segal*
 * [16]. All
the cohomology theories in this paper have complex coefficients, unless otherwi*
 *se noted.
For example, K*T(X) = K*T(X, Z)  Z C. Also, when X is a point, we write K*Tinst*
 *ead
of K*T(X), and similarly for H*T.
  The goal of this paper is to describe K*T(X) in terms of H*T(X). When T is th*
 *e trivial
group, this is easy: it is a classical result that the Chern character ch : K*(*
 *X) ! H*(X)
is an isomorphism (in this case one only needs to tensor with Q). In general ho*
 *wever it
is not true that the equivariant version of the Chern character

                           chT : K*T(X) ! H**T(X)

is an isomorphism. (For the definition of chT see Lemma 3.1 and the discussion *
 *before
it.)
  The good news is that there is still a way in which one can describe K*T(X) i*
 *n terms
of H*T(X). Details will be given later, but for now let us outline the main ste*
 *ps of this
description. Let CT = Specm K*Tbe the complex algebraic group of the maximal id*
 *eals
of K*T. The construction of CT is functorial in T , and if H ,! T is a compact *
 *subgroup
of T , we can identify CH as a subgroup of CT via the map CH ! CT. If ff is a p*
 *oint of
CT, denote by H(ff) the smallest compact subgroup H of T such that CH contains *
 *ff.
Denote by Xff= XH(ff), the subspace of X fixed by all elements of H(ff). Denote*
 * by O
the sheaf of algebraic functions on CT, and by Oh the sheaf of holomorphic func*
 *tions.
Then we will define a sheaf, denoted by K*T(X), whose stalk at a point ff 2 CT *
 *is

                             K*T(X)ff= H *T(Xff) ,

where H *T(-) is the extension of H*T(-) by the ring of holomorphic germs at ze*
 *ro on
H*T. Moreover, the transition functions of K*T(X) will be also defined entirely*
 * using the
equivariant cohomology of X.  Now, if   denotes the global sections functor, we*
 * will
show that there exists an isomorphism

                        T : K*T(X)   O  Oh ~= K*T(X) .

This is the sense in which equivariant K-theory with complex coefficients can b*
 *e de-
scribed in terms of equivariant cohomology.

                                     1
2            EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

  We should say a few words about the isomorphism  T.  If we denote by chT the
equivariant Chern character (to be defined below), we will see that  T is esent*
 *ially a
sheaf version of chT.  However, chT has to be twisted (translated) in an approp*
 *riate
sense, to take into account the point ff over which the germ of chT is taken.
  One may call this a de Rham model for equivariant K-theory, but it would be a*
 * slight
misnomer, since we do not describe K*T(X) at the level of cocycles.  As a matte*
 *r of
fact, we do more: we describe the classes themselves, as sections in a sheaf bu*
 *ild from
ordinary equivariant cohomology. But, if one were really intent on giving a de *
 *Rham
model, one could use the sheaf model to define a cocycle in K-theory as a colle*
 *ction of
germs of ordinary closed differential forms, and use a similar definition for c*
 *oboundaries.
We do not purse this avenue because it would only obscure the purely topologica*
 *l nature
of our description of K*T(X).
  There were previous attempts to give a de Rham type of model for K*T(X).  The
earliest version appeared in Baum, Brylinski and MacPherson [4]. The ideas were*
 * further
developed in Block and Getzler [5], and Duflo and Vergne [10]. In fact, both th*
 *e idea of
describing equivariant K-theory as a sheaf and twisting the equivariant Chern c*
 *haracter
are present in Duflo and Vergne. The problem in their paper is that they cannot*
 * prove
the Mayer-Vietoris property for their cohomology theory, because they work with*
 * C1
functions. The advantage of our approach is that we work with coherent analytic*
 * sheaves
over the affine (Stein) manifold CT, and in this case the global section functo*
 *r is exact.
  The present paper is inspired mainly by Grojnowski's preprint [12].  In this *
 *semi-
nal work, he uses ideas from the papers mentioned above to define equivariant e*
 *lliptic
cohomology with complex coefficients. His model starts with an elliptic curve E*
 *, and con-
structs for every torus T a complex variety ET and a coherent analytic sheaf El*
 *l*T(X) over
ET, whose stalk at each point is defined in terms of equivariant cohomology. Th*
 *e sheaf
Ell*T(X) is then defined by Grojnowski to be the ("delocalized") complex T -equ*
 *ivariant
cohomology of X. Interestingly enough, equivariant K-theory is never explicitly*
 * men-
tioned in Grojnowski's preprint, although he was most likely aware that an anal*
 *ogous
construction to that of Ell*T(X) should lead to equivariant K-theory, if the el*
 *liptic curve
E is replaced by the multiplicative group C = C \ {0}.  The main contribution o*
 *f the
present paper is to do exactly that: it starts with the multiplicative group C,*
 * out of
which it defines the base complex variety CT, and constructs a coherent analyti*
 *c sheaf
K*T(X) over CT. Then, the ring of global sections in K*T(X) turns out to be a f*
 *aithfully
flat extension of equivariant K-theory1.
  A simple exercise shows that if one starts instead with the additive group A *
 *= C,
the resulting sheaf is nothing else but ordinary equivariant cohomology. An imp*
 *ortant
conclusion is that, when working with complex coefficients, the difference betw*
 *een equi-
variant cohomology, K-theory and elliptic cohomology stems mainly from the fact*
 * that
these theories are associated to different complex groups of dimension one: the*
 * additive,
the multiplicative, and the elliptic groups, respectively.
  The results of this paper can be extended in several directions. First, we ca*
 *n describe
K*T(X) directly, instead of describing its faithfully flat extension K*T(X)  O *
 *Oh. But in
order to do that, one needs to define algebraic sheaves Fff(see Definition 2.9)*
 * instead of
holomorphic ones. And this can only be done using completions, because the loga*
 *rithm
___________
  1Besides its contribution to equivariant K-theory, one can regard the present*
 * paper as giving a

rigorous definition for Grojnowski's equivariant elliptic cohomology: for this,*
 * it is enough to change the
base manifold CT to Grojnowsi's ET. See also Rosu [15] for a definition of equi*
 *variant elliptic cohomology
when T = S1.
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY             3

map is not algebraic.  The construction therefore becomes more complicated, and*
 * we
decided to relegate it to another paper. Second, if G is a nonabelian connected*
 * compact
Lie group, and T its maximal torus, then K*G(-) = K*T(-)W , where W is the Weyl
group, so one can also describe K*G(-) using Borel equivariant cohomology. Thir*
 *d, we
can prove a similar result whenever the coefficient ring R of the cohomology th*
 *eories
involved is an algebra over Q adjoined the roots of unity. We need R to be a Q-*
 *algebra
because the logarithm map is only defined over Q, and we need to invert the roo*
 *ts of
unity because we want to split R[z]=<zn - 1> into a direct sum of n copies of R.
  While the details of the sheafifying process are somewhat technical, in princ*
 *iple the
construction allows one to infer some results in equivariant K-theory from the *
 *corre-
sponding ones in equivariant cohomology.  In the Appendix we give examples of t*
 *his,
extending results of Chang-Skjelbred [9] and Goresky-Kottwitz-MacPherson [11] f*
 *rom
equivariant cohomology to equivariant K-theory.


                  2. A sheaf-valued cohomology theory

  The purpose of this section is to define a sheaf valued T -equivariant cohomo*
 *logy
theory, which we denote by K*T(-).  In the next sections we are going to show t*
 *hat
global sections of this sheaf are essentially equivariant K-theory. We already *
 *knew that
K*T(X) can be regarded as a coherent sheaf over CT = Specm K*T(because it is a *
 *K*T-
module). The novelty is that K*T(X) can be completely described in terms of ord*
 *inary
equivariant cohomology (since we will show that K*T(X) is).  Let us start with *
 *a few
definitions.2.


                              2.1. Definitions


  First we want a simpler description of CT = Specm K*T. Denote by ^Tthe Pontrj*
 *agin

dual of T , i.e. ^T= Hom (T, S1).  For example, if T = (S1)p x G, where G is a *
 *finite

abelian group, then ^T~=ZpxG (the isomorphism is not natural, however). Denote *
 *by C
the multiplicative algebraic group C \ {0}. Although it might generate some con*
 *fusion,
we will use additive notation for C throughout the paper. The following straigh*
 *forward
lemma gives two alternate descriptions of CT.

Proposition 2.1. There is a natural isomorphism of algebraic varieties

                              CT ~=Hom Z(T^, C) .

If T is connected (i.e. a torus), and  T is its integer lattice, then there is *
 *a natural
isomorphism
                                CT ~= T  Z C .

Proof.It is easy to see that K*T= C[T^], the group algebra of ^T. We define a m*
 *ap

                        : Hom Z(T^, C) ! Specm C[T^] = CT

by noting that ff 2 Hom Z(T^, C) extends to a non-zero C-algebra map ff0 : C[T^*
 *] ! C.
We then take  (ff) = ker(ff0), which is a maximal ideal of C[T ]. Since the dom*
 *ain and
codomain of   both take products of groups to products of varieties, it suffice*
 *s to check
that   is an isomorphism when T = S1 or T = Zn, which we leave to the reader.
  For the second statement, notice that when T is a torus, there is a natural i*
 *somorphism
^T-~!  *T.
___________
  2For a similar definition in the case of equivariant elliptic cohomology, see*
 * Rosu [15]. The discussion

there is only for T = S1, but it generalizes easily with the same formalism as *
 *here.
4            EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

  Suppose T ~=(S1)p x G, where G is a finite abelian group. Then one can apply *
 *the
previous proposition to obtain that CT ~=Cpx G. This formula shows that CT is i*
 *n fact
an algebraic group (and it is the complexification of T ). Just as we did for C*
 *, in the rest
of the paper we are going to use additive notation for CT.

Definition 2.2. Let tC be the complexification of the Lie algebra of T . Then t*
 *he expo-
nential map exp : C ! C extends to a complex algebraic group map

                                exp: tC ! CT  .

To be more precise, this map is the composite map tC =  T ZC !  T ZC = CT0 ! CT,
where T 0is the connected component of T containing the identity. (For the iden*
 *tification
 T  Z C = CT0, use the previous proposition.)  Note that when T is connected, t*
 *he
exponential map is surjective.

Definition 2.3. We call a neighborhood U of zero in CT "small" if the above def*
 *ined
exponential map, exp : tC ! CT, has a local inverse on U. We call a neigborhood*
 * V of
zero in tC "small" if it is of the form exp-1(U), for U a small neighborhood of*
 * zero in
CT.

  Let A be a collection of compact subgroups of T . Define a relation on CT as *
 *follows:
ff  A fi if, for any H 2 A, fi 2 CH implies ff 2 CH . The relation  A is reflex*
 *ive and
transitive, but not antisymmetric, so it is not an order relation. When it is c*
 *lear what A
is, we will omit it and write simply ff   fi. The next definition singles out a*
 * special class
of open covers of CT, called adapted covers. These will be used below in the de*
 *finition
of the sheaf K*T(X).

Definition 2.4. Let A be a collection of compact subgroups of T , and let U = (*
 *Uff)ff2CT
be an open cover indexed by the points of CT.  Then U is called ä dapted to A" *
 *if it
satisfies the following conditions:

    1. ff 2 Uff, and Uff- ff is small.
    2. If Uff\ Ufi6= ;, then either ff   fi or fi   ff.
    3. If ff   fi, and for some H 2 A ff 2 CH but fi =2CH , then Ufi\ CH = ;.
    4. If Uff\ Ufi6= ;, and both ff and fi belong to CH for some H 2 A, then ff*
 * and fi
       belong to the same connected component of CH .

Proposition 2.5. If A is a finite collection of compact subgroups of T , then t*
 *here exists
a cover U of CT adapted to A. Any refinement of U is still adapted.

Proof.Define H = {CH | H 2 A}, and H0 = the set of all connected components of *
 *the
elements in H. Put a metric on CT which yields its usual topology. Denote this *
 *metric
by "dist".
  Let ff 2 CT.  If ff 2 C for all C 2 H0 (this is possible only when T is conne*
 *cted),
then choose Uffsuch that ff 2 Uff, and Uff- ff is small. If, on the contrary, t*
 *here exists
a connected component C 2 H0 such that ff =2C, then take Uffa ball of center ff*
 * and
radius d, with
                          d < 1_2 min   dist(ff, D) ,
                               D2H0,ff=2D
and such that Uff- ff is small.
  We show that U = (Uff)ff2CTis adapted: Condition 1 is trivially satifsfied. T*
 *o prove
Condition 2, let ff and fi be such that Uff\ Ufi6= ;. Suppose we have neither f*
 *f   fi, nor
fi   ff. Then by the definition of   there exist two compact subgroups K and L *
 *of T
such that ff 2 CK \ CL and fi 2 CL \ CK . But from the definition of Uffit foll*
 *ows that Uff
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY             5

is a ball of center ff and radius d < 1_2dist(ff, CL)   1_2dist(ff, fi). Simila*
 *rly, Ufiis a ball

of center fi and radius less than 1_2dist(ff, fi), so Uffand Uficannot possibly*
 * intersect,
contradiction.
  Condition 3 is obviously satisfied, by construction.
  Finally, to show Condition 4, let ff, fi 2 CH be such that Uff\ Ufi6= ;. Supp*
 *ose ff and
fi belong to different connected components of CH . Then by the same type of re*
 *asoning
as above, it follows that the radii of Uffand Ufiare smaller than 1_2dist(ff, f*
 *i), so Uffand
Uficannot possibly intersect, which again leads to a contradiction.

  Let ff 2 CT. The construction of CT is functorial, so if H is any compact sub*
 *group
of T , we get an inclusion map CH ! CT. If ff 2 Im(CH ! CT), we say that ff 2 C*
 *H .
For ff 2 CH , denote by H(ff) the smallest compact subgroup H of T such that ff*
 * 2 CH .
The fact that there exists a smallest H such that ff 2 CH  is implied by the fo*
 *rmula
CK \CL = CK\L , which follows from an easy diagram chase. This also implies tha*
 *t H(ff)
is the intersection of all compact subgroups H such that ff 2 CH . Another imme*
 *diate
consequence is the following formula, which will be useful later:

(1)                        H(ff)   K () ff 2 CK  .

  Let X be a space with a T -action.  If K is a compact subgroup of T , denote *
 *by
XK   X the subspace of points fixed by K. Also, if ff 2 CT, define

                                Xff= XH(ff).

Now we want to define the notion of a cover adapted to a finite T -CW complex. *
 *So let
X be a finite T -CW complex. We know that there exists a finite collection A = *
 *(Hi)i
of compactSsubgroups of T such that X has an equivariant cell decomposition of *
 *the
form X =  iDnix (T=Hi). Here we denoted by Dn the open disk in dimension n, and
by D0 a point. The group T acts trivially on Dni, and by left multiplication on*
 * T=Hi.
Notice that if K is a compact subgroupSof T , then the subcomplex of X fixed by*
 * K has
a decomposition of the form XK =  i:K HiDnix (T=Hi). Taking K = H(ff), we get
                                 [
(2)                       Xff=       Dnix (T=Hi) .
                               i:ff2CHi

We say that the cover U = (Uff)ff2CTof CT is ä dapted to X" if U is adapted to *
 *the
collection A of the isotropy groups Hi appearing in a T -equivariant cell decom*
 *position
of X. Since this collection is finite, Proposition 2.5 implies that there alway*
 *s exists a
cover adapted to X.
  Next we discuss a few useful results in equivariant cohomology. We start with*
 * a well-
known proposition which says that the ring of coefficients of (complex) T -equi*
 *variant
cohomology is the polynomial algebra on tC, the complex Lie algebra of T .

Proposition 2.6. If T is an abelian compact Lie group, there is a natural isomo*
 *rphism

                                S(t*C) -~! H*T ,

where S(-) denotes the symmetric algebra, and t*Cis the dual of tC.


Proof.T^= Hom (T, S1) is the group of irreducible characters of T , so for ~ 2 *
 *^Tconsider
the complex vector bundle
                               V~ = ET xT C ,

over the classifying space BT , where the map T ! C is given by ~. Then the fir*
 *st Chern
class of V~ gives a natural isomorphism c1 : ^T! H2(BT, Z).
6            EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

  If T is a torus, we saw in Proposition 2.1 that ^Tcan be identified with the *
 *dual of the
integer lattice  *T, so by tensoring the map c1 with C, we get the natural isom*
 *orphism
c1 : t*C=  *T  C ! H2(BT, C).  Taking symmetric products, we get the desired
isomorphism.
  If T is a general (non-connected) compact abelian Lie group, the isomorphism *
 *still
holds, since both domain and codomain depend only on the connected component of*
 * T
containing the identity, which is a torus.

  We now define an algebra homomorphism

                               h : H*T! OhtC,0

by taking a polynomial in H*T= S(t*C) and sending it to its germ at zero. The m*
 *ap is

injective, so we can consider H*Tas a subring of OhtC,0. Let V be a small neigh*
 *borhood of

zero in tC. Then, since the ring H*T  OhtC,0consists of the germs of global hol*
 *omorphic

functions on tC, the map h factors through the inclusion OhtC(V ) ,! OhtC,0, so*
 * we can
define a map, also denoted by h,

                              h : H*T! OhtC(V ) .

Let U be a small neighborhood of zero in CT, and let V = exp-1(U), where exp: t*
 *C ! CT
is the exponential map.  Via the exponential, we have the following identificat*
 *ions:
OhCT,0~-!OhtC,0and OhCT(U) -~! OhCT(V ).

Definition 2.7. Let U be a small neighborhood of zero in CT. Via the identifica*
 *tions
above, we define the following two maps, and denote them also by h (the second *
 *one is
the correstriction of the first):

                   h : H*T! OhCT,0 and   h : H*T! OhCT(U) .

  Now we define a few cohomology theories that we are going to use throughout t*
 *he
paper.  Let X be a finite T -CW complex.  We define the holomorphic T -equivari*
 *ant
cohomology of X to be
                         H *T(X) = H*T(X)  H*TOhCT,0,

where the map h : H*T! OhCT,0is given in Definition 2.7.  It is indeed a cohomo*
 *logy

theory, because by Proposition 2.8 the map H*T! OhCT,0is flat.  The theory is n*
 *ot

Z-graded anymore; however, it can be thought of as Z=2-graded, by its even and *
 *odd
part.
  Let H**T(X) be the completion of H*T(X) with respect to the augmentation ideal
I = ker(H*T! C).  Since H*T(X) is a finitely generated module over the Noetheri*
 *an
ring H*T, a simple result on completions (see for example Matsumura [14], Theor*
 *em 55)
implies that
                          H**T(X) ~=H*T(X)  H*TH**T.

The ring H*Tis a polynomial ring, so we have the following well-known results f*
 *rom
algebra (they are sometimes called GAGA results, since they first appeared in S*
 *erre's
GAGA [17]).

Proposition 2.8. H *T= OhCT,0and H**Tare flat over H*T. If U is a small neighbo*
 *rhood

of zero, then OhCT(U) is flat over H*T.
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY             7

Proof.All the results we mention in this proof are from Matsumura [14]. Identif*
 *y the
complex Lie algebra tC with X = Cn.  We denote by O the algebraic structure she*
 *af
of Cn, and by Oh the analytic structure sheaf. Let O^0be the completion of the *
 *local
ring O0 with respect to its maximal ideal.  It is sufficient to show that the n*
 *atural
maps O(X) ! Oh0and O(X) ! O^0are flat, and that for any U open in X, the map
O(X) ! Oh(U) is flat. We start by noticing that O^0is the completion of O(X) wi*
 *th
respect to its maximal ideal at zero, so by Corollary 1 of Theorem 55, we know *
 *that O^0
is flat over O(X). The completion of Oh0with respect to its maximal ideal is al*
 *so O^0,
so Oh0! O^0is flat. It is in fact faithfully flat, because it is local: see The*
 *orem 3 (4.D).
Now one can check directly by the definition of flatness that having O(X) ! O^0*
 *flat
and Oh0! O^0faithfully flat implies that O(X) ! Oh0is flat. Notice also that th*
 *e same
proof can be used to show that O0 ! Oh0is flat.
  Next let U   X be an open set. By the local characterization of flatness, The*
 *orem
(3.J), in order to show that O(X) ! Oh(U) is flat, we have to show that for any*
 * x 2 U
the natural map Ox ! Ohxis flat. But we have already shown this when x = 0, and*
 * the
proof for general x is the same.
  Now in order to prove the proposition, just transfer the results we have prov*
 *ed via
the exponential map, exp: Cn = tC ! CT. This is where we need U small.

  In particular, it follows that H*T(X) and H *T(X) can be regarded as subrings*
 * of
H**T(X).


                         2.2. Construction of K*T


  Let X be a finite T -CW complex. Fix U a cover adapted to X, which exists bec*
 *ause of
Proposition 2.5. We are going to define a sheaf F = FU over CT whose stalk at f*
 *f 2 CT is
isomorphic to H *T(Xff). Recall that in order to give a sheaf F over a topologi*
 *cal space, it
is enough to give an open cover (Uff)ffof that space, and a sheaf Fffon each Uf*
 *ftogether
with isomorphisms of sheaves OEfffi: Fff|Uff\Ufi-! Ffi|Uff\Ufi, such that OEfff*
 *fis the identity

function, and the cocycle condition OEfiflOEfffi= OEffflis satisfied on Uff\ Uf*
 *i\ Ufl. If U is
a subset of CT, denote by U + ff = {x + ff | x 2 U} the translation of U by ff.

Definition 2.9. Define a presheaf Fffon Uffby taking, for any open U   Uff,

                      Fff(U) = H*T(Xff)  H*TOhCT(U - ff) ,

where the restriction maps are induced from those of OhCT.  The ring map h : H**
 *T!

OhCT(U - ff) is given in Definition 2.7.


Proposition 2.10. Fffis a coherent sheaf of OhCT-modules.

Proof.First we show that Fffis a sheaf of H*T-modules. If (Ui)i is an open cove*
 *r of a
topological space Y , denote by Uij= Ui\ Uj, etc. Then a presheaf G is a sheaf *
 *if and
only if for any m > 0 and any finite cover (Ui)i=1...mthe following sequence is*
 * exact
                        Y        r1 Y         r2
         0 -! G(Y ) -r0!  G(Ui) -!    G(Uij) -!  . .-.! G(U1...m) -! 0 ,
                         i          i<j
                        Q
where r0(s)i= s |Ui, r1   isi i,j= si |Uij- sj |Uij, etc. Denote this sequence *
 *by M(G)o.

Since OhCTis a sheaf of H*T-modules on Y = Uff, the sequence M(OhCT)o is exact.*
 * Now, by

Proposition 2.8, each H*T-module in the sequence M(OhCT)o is flat. Therefore, M*
 *(OhCT)o

can be thought as a flat resolution of the last term to the right, OhCT(U1...m)*
 *. Using the
8            EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

Flat Resolution Lemma (see Weibel [19], Lemma 3.2.8), one can easily show that *
 *by
tensoring M(OhCT)o with any H*T-module, we also get an exact sequence. In parti*
 *cular,

by tensoring with H*T(Xff) we obtain M(Fff)o, which therefore is exact. But tha*
 *t means
that Fffis a sheaf (of H*T-modules).

  To show that Fffis a sheaf of OhCT-modules, if U   Uff, we need an action of *
 *f 2

OhCT(U) on Fff(U). The translation map tff: U - ff ! U, which takes x to x + ff*
 * gives

a map t*ff: OhCT(U) ! OhCT(U - ff), which sends f(u) to f(u + ff). Then we take*
 * the

result of the action of f 2 OhCT(U) on ~   g 2 Fff(U) = H*T(Xff)  H*TOhCT(U - f*
 *f) to be

~   (t*fff . g). It is an easy exercise to show that Fffis coherent: one uses t*
 *hat H*T(Xff)
is a finitely generated H*T-module.

  Next, for the second part of the definition of F, we glue the different sheav*
 *es Fff.
Let ff, fi 2 CT such that Uff\ Ufi6= ;.  Since the cover U is adapted to A, Con*
 *dition
2 from Definition 2.4 implies that either ff   fi or fi   ff.  Without loss of *
 *generality
we can assume that ff   fi. Then it is clear that H(ff)   H(fi), and hence Xfi *
 * Xff.
Denote by i : Xfi! Xffthe inclusion map. Now we want to investigate conditions *
 *in
which the restriction map i* : H*T(Xff) ! H*T(Xfi) becomes an isomorphism. The *
 *next
proposition is esentially the Localization Theorem of Borel and Hsiang (see Ati*
 *yah and
Bott [1]).

Proposition 2.11. Let X be a finite T -CW complex, and U = (Uff)ff2CTa cover ad*
 *apted
to X. Let ff   fi be two points in CT. Let U   Uff\ Ufi, and assume Uff\ Ufinon*
 *empty.
Then the restriction map

          i*   1 : H*T(Xff)  H*TOhCT(U - ff) ! H*T(Xfi)  H*TOhCT(U - ff)

is an isomorphism.

Proof.The case when Xff= Xfiis trivial, so we assume the contrary.  Then there
exists a compact subgroup L of T such that ff 2 CL and fi =2CL.  Then Condition*
 * 3
from Definition 2.4 implies that Ufi\ CL = ;.  But CL = ff + CL, so we deduce t*
 *hat
(U - ff) \ CL = ;. This means that for every x 2 U - ff we have x =2CL.
  Now we will prove that whenever x =2CL, i* becomes an isomorphism if tensored*
 * with
OhCT,x. This is enough for us, because we can regard the statement to be proved*
 * as a
statement about sheaves, and we are proving the isomorphism stalkwise.
  Now notice that Equation (2) implies that Xffis built from Xfiby adding cells*
 * of
the form Dn x (T=L), with ff 2 CL and fi =2CL.  So it is enough to show that, w*
 *hen
tensored with OhCT,xover H*T, the ring H*TDn x (T=L), Sn-1 x (T=L)  becomes zer*
 *o.

Or, equivalently, it is enough to have H*T(T=L) = H*Lbecome zero after tensorin*
 *g with

OhCT,x. We therefore proceed to show that H*L H*TOhCT,xis the zero algebra.
  Via the correspondence log: tC ! CT, it is enough to show that if lC is the c*
 *omplex-
ification of the Lie algebra of L, and x =2lC, then H*L H*TOhtC,x= 0. Denote by*
 * IL the
kernel of the surjective map of algebras H*T! H*L. It is clear that IL kills H**
 *Las an
H*T-module. The vector space lC is the vanishing set of IL, so since x =2lC, th*
 *ere exists

a polynomial p in IL such that p(x) 6= 0. This means that p is invertible in Oh*
 *tC,x, hence

1   p is invertible in H*L H*TOhtC,x. By the balancing property of the tensor p*
 *roduct,
we know that 1   p = 0, because p goes to zero via the map H*T! H*L. So we foun*
 *d an

element in H*L H*TOhtC,xwhich is invertible and zero at the same time. This can*
 * only

happen if H*L H*TOhtC,xis the zero algebra.
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY             9

  Let us now consider the translation map tfi-ff: U - fi ! U - ff.  This induce*
 *s a

map t*fi-ff: OhCT(U - ff) ! OhCT(U - fi). Eventually we want to produce a trans*
 *lation

map H*T(Xfi)  H*TOhCT(U - ff) ! H*T(Xfi)  H*TOhCT(U - fi). But the problem is t*
 *hat
t*fi-ffis not a map of H*T-modules, so we cannot simply tensor it with the iden*
 *tity map

of H*T(Xfi).  However, we hope to find some other group K so that t*fi-ffis a m*
 *ap of
H*K-modules.

Lemma 2.12. Let X be a finite T -CW complex, and U = (Uff)ff2CTa cover adapted *
 *to
X. Let ff   fi be two points in CT and U   Uff\ Ufi, and assume Uff\ Ufinonempt*
 *y.
Let H = H(fi) and K = T=H. Then the translation map

                       t*fi-ff: OhCT(U - ff) ! OhCT(U - fi)

is a map of H*K-algebras. Here the H*T-algebras OhCT(U -ff) and OhCT(U -fi) are*
 * regarded
as H*K-algebras via the natural algebra map H*K ! H*Tinduced by the quotient map
T ! K.

Proof.Since fi 2 CH , it follows that ff 2 CH .  Let H0 be the connected compon*
 *ent
of H containing the identity.  Because both ff and fi belong to CH , Condition *
 *4 from
Definition 2.4 implies that ff and fi belong to the same connected component of*
 * CH , or
equivalently that fi - ff 2 CH0. Let hC be the complexification of the Lie alge*
 *bra of H.
The exponential map exp: hC ! CH0 is a surjective group homomorphism, because H0
is a torus. Consider an element in the coset exp-1(fi -ff)   hC. Denote it also*
 * by fi -ff.
Let   be the global sections functor. Using the exponential map, it is enough t*
 *o show
that
                             t*fi-ff:  OhtC!  OhtC

is a map of H*K-modules. This is equivalent to showing that t*fi-ff(p) = p for *
 *all p 2 H*K.

But this is easy, since we can identify H*Kwith the ring of polynomial function*
 *s f(u)
in H*Tsuch that f(u + a) = f(u) for any a in the complexification of the Lie al*
 *gebra of
K.

  Here is a well-known result in equivariant cohomology theory (see for example*
 * Atiyah
and Bott [1] for ordinary cohomology and Segal [16] for K-theory).

Proposition 2.13. Let h*T(-) be either ordinary equivariant cohomology, or equi*
 *variant
K-theory. Let Z be a finite T -CW complex which is fixed by a compact subgroup *
 *H of
T . Let K = T=H. Then there exists a natural isomorphism

                          h*K(Z)  h*Kh*T-~! h*T(Z) .

  The last step in the construction of the gluing maps OEfffiis the following p*
 *roposition:

Proposition 2.14. Under the hypotheses of Lemma 2.12, the translation map t*fi-*
 *ff
extends to a natural isomorphism

           ø*fi-ff: H*T(Xfi)  H*TOhCT(U - ff) ! H*T(Xfi)  H*TOhCT(U - fi) .


Proof.Xfiis fixed by H = H(fi), so by Proposition 2.13 we know that H*T(Xfi)  H*
 **T

OhCT(U - ff) ~=H*K(Xfi)  H*KOhCT(U - ff), with K = T=H. Therefore it is enough *
 *to
define a map

            OE : H*K(Xfi)  H*KOhCT(U - ff) ! H*K(Xfi)  H*KOhCT(U - fi) .
10           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

But t*fi-ffwas shown in Lemma 2.12 to be a map of H*K-algebras, so we can just *
 *take
OE = 1   t*fi-ff.
  More formally, we define ø*fi-ffas the composition of the following isomorphi*
 *sms:
                                                  1 t*fi-ff
H*T(Xfi) H*TOhCT(U -ff) ! H*K(Xfi) H*KOhCT(U -ff)  -!   H*K(Xfi) H*KOhCT(U -fi)*
 * !

H*T(Xfi)  H*TOhCT(U - fi).

Definition 2.15. Under the same hypotheses as Lemma 2.12, define OEfffi: Fff(U)*
 * !
                                                     * 1                       *
 * fi*fi-ff
Ffi(U) as the composite map H*T(Xff) H*TOhCT(U -ff) i-! H*T(Xfi) H*TOhCT(U -ff)*
 * -!

H*T(Xfi)  H*TOhCT(U - fi), where the second map was defined above.

Proposition 2.16. OEfffiis an isomorphism.

Proof.The first map is an isomorphism by Proposition 2.11, and the second map i*
 *s an
isomorphism by Proposition 2.14.

Proposition 2.17. The maps OEfffiverify the cocycle condition, i.e. OEfiflO OEf*
 *ffi= OEfffl, on
every triple intersection Uff\ Ufi\ Ufl6= ;.

Proof.Without loss of generality, we can assume ff   fi   fl. We need to show t*
 *hat the
following diagram is commutative:

             H*T(Xff)  H*TOhCT(U - ff)


                        i* 1


             H*T(Xfi)  H*TOhCT(U - ff)
                                      * 1
                        fi*fi-ff     i


             H*T(Xfi)  H*TOhCT(U - fi)  H*T(Xfl)  H*TOhCT(U - ff)


                        i* 1         fi*
                                       fi-ff
             H*T(Xfl)  H*TOhCT(U - fi)


                        fi*fl-fi


             H*T(Xfl)  H*TOhCT(U - fl)


In this diagram the only map which has not been defined is ø*fi-ff: H*T(Xfl) H**
 *TOhCT(U -

ff) ! H*T(Xfl)  H*TOhCT(U - fi). But since fi   fl it follows that H(fi) fixes *
 *Xfl, so we

can replace Xfiby Xflin Corollary 2.14, and obtain a map which we also denote by
ø*fi-ff.

  Since H = H(fi) fixes Xfiand Xfl, we can use Proposition 2.13 to replace (to *
 *pick an
example) H*T(Xfi)  H*TOhCT(U - fi) by H*K(Xfi)  H*KOhCT(U - fi), where K = T=H.*
 * We
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY            11

have to show that the following diagram is commutative:

            H*K(Xfi)  H*KOhCT(U - ff)
                                      * 1
                        1 t*fi-ff    i


            H*K(Xfi)  H*KOhCT(U - fi)   H*K(Xfl)  H*KOhCT(U - ff)


                        i* 1        1 t*
                                       fi-ff
            H*K(Xfl)  H*KOhCT(U - fi)


Now observe that both composites are equal to i*   t*fi-ff.


Definition 2.18. Let X be a finite T -CW complex, and U = (Uff)ff2CTan adapted
cover of CT. We then define a sheaf F = K*T(X) on CT by gluing the sheaves Fffd*
 *efined
in 2.9 via the gluing maps OEfffidefined in 2.15.

  Notice that we can remove the dependence of K*T(X) on the cover U as follows:
Let U and V be two covers adapted to X.  Then any common refinement W is going
to be adapted as well, and the corresponding maps of sheaves FU ! FW    FV are
isomorphisms on stalks, hence isomorphisms of sheaves.  Therefore we can omit t*
 *he
subscript U.

Theorem 2.19. K*T(-) is a T -equivariant cohomology theory on finite complexes,*
 * with

values in the category of coherent holomorphic sheaves of Z=2-graded OhCT-algeb*
 *ras.

Proof.We start by showing that the construction of K*T(X) is natural. Let X and*
 * Y
be two finite T -CW complexes, and f : X ! Y a T -equivariant map between them.
We want to define a map of sheaves f* : K*T(Y ) ! K*T(X) with the properties th*
 *at
1*X= 1K*T(X)and (fg)* = g*f*. Consider two T -cell decompositions of X and Y , *
 *and

let A be the collection of all compact subgroups L of T such that Dn x (T=L) ap*
 *pears
in the cell decomposition of either X or Y . Let U be a cover adapted to A. Sin*
 *ce f is
T -equivariant, for each ff we get by restriction a map fff: Xff! Y ff. This in*
 *duces for
all U   Uffa map f*ff  1 : H*T(Y ff)  H*TOhCT(U - ff) ! H*T(Xff)  H*TOhCT(U - f*
 *f), which

commutes with restrictions.  Therefore, we obtain a sheaf map f*ff: Fff(Y ) ! F*
 *ff(X).
To get our global map f*, we only have to check that the maps f*ffglue well, i.*
 *e. that
they commute with the gluing maps OEfffi.  This follows easily from the natural*
 *ity of
ordinary equivariant cohomology, and from the naturality in X of the isomorphism
H*T(Xfi) ~= H*K(Xfi)  H*KH*T.  (See Proposition 2.13 with Z = Xfi, H = H(fi) and

K = T=H.)
  Let (X, A) be a pair of finite T -CW complexes, i.e. A is a closed subspace o*
 *f X
and the inclusion map A ! X is T -equivariant.  We then define K*T(X, A) as the
kernel of the map j* : K*T(X=A) ! K*T(point), where j : point = A=A ! X=A is the
inclusion map. If f : (X, A) ! (Y, B) is a map of pairs of finite T -CW complex*
 *es, then
f* : K*T(Y, B) ! K*T(X, A) is defined as the unique map induced on the correspo*
 *nding
kernels from f* : K*T(Y ) ! K*T(X).
  The last definition we need is of the coboundary map.  If (X, A) is a pair of*
 * finite
T -CW complexes, we want to define ffi : K*T(A) ! K*+1T(X, A).  This is obtaine*
 *d by
gluing the maps

        ffiff  1 : H*T(Aff)  H*TOhCT(U - ff) ! H*+1T(Xff, Aff)  H*TOhCT(U - ff)*
 * ,
12           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

where ffiff: H*T(Aff) ! H*+1T(Xff, Aff) is the usual coboundary map. The maps f*
 *fiff  1
glue well, because ffiffis natural.
  To check the usual axioms of a cohomology theory (naturality, exact sequence *
 *of a pair,
and excision) for K*T(-), recall that it was obtained by gluing the sheaves Fff*
 *along the
maps OEfffi. Since the sheaves Fffwere defined using H*T(Xff), the properties o*
 *f ordinary

T -equivariant cohomology pass on to K*T(-), as long as tensoring with OhCT(U -*
 * ff) over
H*Tpreserves exactness. But this is implied by Proposition 2.8.

  Now, CT = Specm K*Tis a nonsingular affine complex variety, so it is Stein.  *
 *By a
generalization of Theorem B of Cartan, the sheaf cohomology vanishes on CT in p*
 *ositive
dimensions for any coherent sheaf. (See Gunning and Rossi [13].) This implies t*
 *hat  ,
the global sections functor, is exact, so we get the following result:

Corollary 2.20.  K*T(-) is an T -equivariant cohomology theory on finite comple*
 *xes
with values in the category of Z=2-graded algebras.

  We want to investigate a few more useful properties of K*T(-).

Proposition 2.21. The value of the theory on a point is given by K*T~=OhCT.

Proof.Notice that if X is a point, translation by ff induces an isomorphism of *
 *the
corresponding sheaf Fffon Uffto OhCT |Uff. Via this isomorphism, the gluing map*
 * OEfffiis

the usual restriction of holomorphic functions. Therefore K*T~=OhCT.

Proposition 2.22. Let   be the global sections functor. Then the ring  K*Tis fa*
 *ithfully
flat over K*T.

Proof.Recall that we defined CT as the complex algebraic variety Specm K*T.  As*
 * in
the proof of Proposition 2.8, denote by O the algebraic structure sheaf of CT, *
 *and by
Oh the analytic structure sheaf. Let X = CT. We need to show that the natural m*
 *ap
O(X) ! Oh(X) is faithfully flat. We first show it is flat: By the local charact*
 *erization
of flatness (e.g. Theorem (3.J) in Matsumura [14]), it is enough to show that f*
 *or any
x 2 U, the natural map Ox ! Ohxis flat. But this is true, and the proof goes ex*
 *actly as
in Proposition 2.8.
  To prove that O(X) ! Oh(X) is in fact faitfully flat, we also need to show th*
 *at the
induced map of spectra of maximal ideals, Specm Oh(X) ! Specm O(X) is surjective
(see Theorem 3 (4.D) in [14]). This is clearly true, since maximal ideals of O(*
 *X) and
Oh(X) correspond to the points x 2 X.

  Since K*T(X) was obtained from gluing the sheaves Fff, its stalks are the sam*
 *e as
those of Fff.

Proposition 2.23. If X is a finite T -CW complex and ff 2 CT , then the stalk o*
 *f the
K*T(X) at ff is

                        K*T(X)ff= H*T(Xff)  H*TOhCT,0.

Proposition 2.24. Let A and B are compact abelian Lie groups, X is a finite A-CW
complex, and Y is a finite B-CW complex. Then

                     K*AxB(X x Y ) ~= K*A(X)  C  K*B(Y ) .

Proof.This follows from CAxB ~= CA xCB and from H*AxB(X xY ) ~=H*A(X) C H*B(Y ),
where the last isomorphism comes from the Künneth formula.
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY            13

  We now return to the usual equivariant K-theory, and define its holomorphic e*
 *xtension

                          K*T(X) = K*T(X)  K*T K*T.

Since K*T(-) is a T -equivariant cohomology theory, it follows from Lemma 2.22 *
 *that
K *T(-) is a T -equivariant cohomology theory, with values in Z=2-graded algebr*
 *as.


                   3. The equivariant Chern character

  Let T be a compact abelian Lie group and X a finite T -CW complex. In this se*
 *ction
we construct a natural isomorphism

                          CHT : K *T(X) !  K*T(X) .

The main ingredient in the definition of CHT is the construction of a natural r*
 *ing map
CHT : K0T(X, Z) !  K*T(X).  Let E ! X be a complex T -vector bundle.  We want
CHT(E) to be a section in K*T(X), so we need to know its germ at a point ff 2 C*
 *T.
The germ CHT(E)ffshould be an element of H *T(Xff), so we would be quite tempted
to define it as the equivariant Chern character chT of the restriction of E to *
 *Xff(we
will define the equivariant Chern character in the next subsection). However, t*
 *here are
a few problems.  The first one is that in order to get chT to take values in H *
 **T(-),
we need some sort of holomorphicity result on chT. This is done in Lemma 3.1. T*
 *he
second problem is that when we vary ff 2 CT, the equivariant Chern character do*
 *es
not glue well. To solve this problem, we modify the definition of CHT by twisti*
 *ng chT
accordingly: see Definition 3.4.


                             3.1. Preliminaries


  Let E ! X be a complex T -vector bundle.  Then the equivariant Chern character
chT(E) can be defined as ch(ET), where ch is the usual Chern character, and ET *
 *! XT
is the Borel construction.  This a priori is an element of H**T(X), because the*
 * Chern
character is defined using the power series ex (see the Appendix of Rosu [15] f*
 *or a
discussion of equivariant characteristic classes). So we get a map

                           chT : K*T(X) ! H**T(X) .

Also, denote the n-th Chern class of ET by cn(E)T. This belongs to the noncompl*
 *eted
ring H*T(X).

Lemma 3.1. Let E ! X be a complex T -vector bundle over a finite T -CW com-
plex. Then its T -equivariant Chern character, chT(E), which a priori is an ele*
 *ment of
H**T(X), actually lies in H *T(X).

Proof.First assume that E is a line bundle.  Then from the definition of the Ch*
 *ern
character we get chT(E) = ec1(E)T. We have already seen that c1(E)T 2 H*T(X). S*
 *ince
X is finite, H*T(X) is a finite module over H*T. Let a1, . .,.am be a set of ge*
 *nerators for

H*T(X). Choose a set of elements fkij2 H*Tso that

                               Xm
                      ai. aj =    fkijak  8 i, j, k 2 1, . .,.m .
                               k=1

Denote by c = c1(E)T. The element c is in H*T(X), so there exist elements gi2 H*
 **Tsuch
that
                            c = g1a1 + . .+.gm am  .
                           P   P                      P   P                   p
We can also calculate c2 =   k(  i,jgigjfkij)ak, c3 =   p(  i,j,k,lgigjglfkijfk*
 *l)ap, etc.
14           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

  Choose some coordinates on tC such that H*T= C[u1, . .,.up]. We say that a po*
 *lyno-
mial OE 2 C[u1, . .,.up] is dominated by another polynomial _ if all coefficien*
 *ts of _ are
positive and all the coefficients of OE are less in absolute value than the coe*
 *fficients of _.
Also, if ~,   2 H*T(X) then we can write ~ = ~1a1+. .+.~m am and   =  1a1+. .+.*
 * m am ,
with ~i,  i2 H*T. We then say that ~ is dominated by   if ~i is dominated by  i*
 * for all
i. Similar definitions of domination can be made for the power series ring H**T*
 *and for
H**T(X).

  Let ~ 2 H*T= C[u1, . .,.up] be a polynomial such that fkijand gi are dominated

by ~ for all i, j, k.  Then one can show by induction that cn is dominated by (*
 *2n -
2)~2n-1(a1 + . .+.amP), which in turn is dominated by 2n~2n-1(a1P+ . .+.am ). S*
 *o ec is
dominated by 1+   n 1__2__(n-1)!~2n-1(a1+. .+.am ) = 1+   n 02_n!~2n+1(a1+. .+.*
 *am ) =

1 + 2~e2~(a1 + . .+.am ). But this last element belongs to H*T(X)  H*TOhtC,0= H*
 * *T(X),

so ec also belongs to H *T(X).
  Second, assume E is a rank n complex T -vector bundle.  Let SE be the splitti*
 *ng
space of E (or the flag variety of E_see Bott and Tu [6]). By construction SE c*
 *omes
with an equivariant map ß : SE ! X. Then the splitting principle says that the *
 *pull-
back bundle ß*(E) decomposes as a sum of T -equivariant line bundles over SE . *
 * Say
ß*(E) ~=L1  . . .Ln. Calculation using the Leray-Serre spectral sequence shows *
 *that

             H*T(SE ) = H*T(X)[x1, . .,.xn] =<oei(x1, . .,.xn) - ci(E)T> ,

where oei(x1, . .,.xn) is the i'th symmetric polynomial in the xj's.  The class*
 *es xj =
c1(Lj)T are called the Chern roots of E. Moreover, we can identify H*T(X) as th*
 *e subring
of H*T(SE ) generated by the polynomials in H*T(X)[x1, . .,.xn] which are symme*
 *tric in

the xj's. By tensoring with OhtC,0or H**Tthe same statement is true about H *T(*
 *X) and
H**T(X).
  Now consider chT(E) = ex1. .e.xn. Since Lj is a line bundle and xj = c1(Lj)T,*
 * the
first part of the proof implies that exj2 H *T(SE ) for all j. Therefore chT(E)*
 * 2 H *T(SE ),
and since it is symmetric in the xj's it follows that chT(E) 2 H *T(X), which i*
 *s what we
wanted.

  We have just proved that chT(E) is the germ of a holomorphic class, i.e. an e*
 *lement
of H *T(X). By looking more carefully at the preceding proof, one can see in fa*
 *ct that
we proved a stronger result:

Corollary 3.2. With the same notations as in Lemma 3.1, chT(E) is a global holo*
 *mor-
phic class, i.e. an element of H*T(X)  H*T OhCT.

  Now, if we extend chT(E) on a small neighborhood U of 0 2 CT, we can regard it
as an element of H*T(X)  H*TOhCT(U). It is important to see what happens to chT*
 *(E)
when it is translated by the map ø*fi-fffrom Proposition 2.14.
  The basic case is when X is a point and E is given by a representation V~ of *
 *T , with
~ 2 ^T. Recall that CT = Hom (T^, C), and consider ff 2 CT. Then we translate c*
 *hT(V~)
via the map ø*ff= t*ff: OhCT(U) ! OhCT(U + ff).


Lemma 3.3. Let T be a compact Abelian group, and T 0the connected component con-
taining the identity. Let ff 2 CT0 and ~ 2 ^T. Then, with the notations above,

                           t*ffchT(V~) = ff(~)chT(V~) .
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY            15

Proof.The proof of this lemma is mostly formal, and just makes intensive use of*
 * the
identifications and definitions we have made so far. Start by using the exponen*
 *tial map

                   exp: tC =  T  Z C !  T  Z C = CT0 ! CT .

The element ff is in the image, so pick a 2 tC such that exp(a) = ff. Denote by*
 * l 2  *Tthe

element corresponding to ~ via the map ^T!  *T. Then one can apply l to a 2  T *
 * Z C
and get a complex number that we denote by l(a). Now it is easy to check the fo*
 *rmula
ff(~) = exp(l(a)). We also know that chT(V~) = exp c1(V~) . So, via the exponen*
 *tial
map, what we have to prove becomes

                           t*ac1(V~) = l(a) + c1(V~) ,

with the equality being now regarded in H*T. Let us look more closely at c1(V~)*
 *. We saw
in the proof of Proposition 2.6 that there is an identification H*T= S(t*C), an*
 *d that the
class c1(V~) 2 H*Tcan be identified to l if this is regarded in S(t*C) via  *T *
 * t*C  S(t*C).
Denote by l(-) the polynomial function in S(t*C) corresponding to l. Then we ha*
 *ve to
prove that
                             t*al(-) = l(a) + l(-) .

But this is obvious, it is just saying that l(-) is a linear function.


                        3.2. Construction of CHT


  We define a multiplicative natural map

                         CHT : K0T(X, Z) !  K*T(X) .

Let E ! X be a complex T -vector bundle.  Let ff 2 CT, and denote by H = H(ff).

Then ff 2 CH . By Proposition 2.1, CH ~=Hom  Z(H^, C), so we can think of ff as*
 * a group

map ff : ^H! C. The space Xffhas a trivial action of H, so the restriction E|Xf*
 *fof E
to Xffhas a fiberwise decomposition by irreducible characters of H:

                              E|Xff~= ~2H^E(~) ,

where E(~) is the T -vector bundle where h 2 H acts by complex multiplication w*
 *ith
~(h).
  It would be tempting to define the germ of CHT(E) at ff to be chT(E|Xff), but
these germs would not glue well to give a global section of K*T(X). Instead, we*
 * do the
following:

Definition 3.4. Let ff 2 CT and H = H(ff). Then the germ of CHT(E) at ff is def*
 *ined
to be                              X
                         CHT(E)ff=    ff(~)chTE(~) .

                                   ~2H^

Proposition 3.5. The germs CHT(E)ffglue to a global section CHT(E) 2  K*T(X).

Proof.We notice that, by Lemma 3.1, CHT(E)ffdoes indeed belong to H *T(Xff), wh*
 *ich
by Proposition 2.23 is the stalk of K*T(X) at ff. Fix (Uff)ff2CTa cover of CT a*
 *dapted to
X.
  Let ff, fi 2 CT with Uff\ Ufi6= ; and ff   fi.  This implies ff, fi 2 CH(fi)a*
 *nd also
H(ff)   H(fi).  Denote by L = H(ff) and H = H(fi).  Condition 4 of Definition 2*
 *.4
implies that fi - ff 2 CH0.  Now we have to prove that OEfffiCHT(E)ff= CHT(E)fi,
16           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY
               P                        P
i.e. that ø*fi-ff~2^Lff(~)chTE(~)|Xfi=    ~2H^fi(~)chTE(~).   Consider the surj*
 *ec-

tive map jP :  ^H !  ^Linduced by the inclusion L  ,!  H.   If ~  2  ^L,  we ha*
 *ve
E(~)|Xfi=   ~2j-1(~)E(~).  Therefore it is enough to show that for all ~ 2 H^ we

have ø*fi-ffff(~)chTE(~)  = fi(~)chTE(~), where ~ = j(~).  But this is equivale*
 *nt to

ø*fi-ffchTE(~) = (fi - ff)(~)chTE(~). Denote by fl = fi - ff 2 CH0. So it is en*
 *ough to

show that, for all fl 2 CH0 and ~ 2 ^H,

                          ø*flchTE(~) = fl(~)chTE(~) .

  Let K = T=H and Y = Xfi. Proposition 2.13 applied to equivariant K-theory giv*
 *es
a natural isomorphism
                         K*K(Y )  K*KK*T-~! K*T(Y ) .

Via the identification above, we can think of E(~) as a tensor product F   V (~*
 *0), with

F a K-bundle and ~02 ^Tsome element in the preimage of ~ via the map ^T! ^H. (At
least, we know that E(~) is generated by such elements.) Via the same identific*
 *ation,
translation by fl 2 CH0 becomes

                               ø*fl7! ø*i(fl)  ø*fl,

where ß(ø) is the image of fl via the natural map ß : CT  ! CK , and the second
fl is regarded in CT via the usual inclusion CH0 ! CT.  But notice that ß(ø) = *
 *0,
because of the exact sequence 0 ! CH  ! CT ! CK  ! 0.  Also, chTE(~) becomes
chK F   chTV (~0) 2 H *K(Y )  H *KH*T. So via the above correspondence we have

                       ø*flchTE(~) 7! chK F   ø*flchTV (~0) .

Since fl 2 CH0, it follows that fl 2 CT0, and it is sufficient for us to show t*
 *hat, for all

fl 2 CT0 and ~02 ^T,
                         ø*flchTV (~0) = fl(~)chTV (~0) .

But this is directly implied by Lemma 3.3, so we are done.

  We have just finished constructing a natural map CHT : K0T(X, Z) !  K*T(X).
By taking the suspension of X instead of X, this induces a map CHT : K*T(X, Z) !
 K*T(X). One can check easily that CHT is a ring map, since chT is. Because  K**
 *T(X)
is a C-algebra and CHT is a ring map, we can now extend CHT to a natural map of
C-algebras CHT : K*T(X) !  K*T(X). Finally, making X a point, we get a ring map
K*T,!  K*T, so if we extend CHT by this, we obtain the desired natural map

                          CHT : K *T(X) !  K*T(X) .

Theorem 3.6. CHT is an isomorphism of T -equivariant cohomology theories.

Proof.Because of the Mayer-Vietoris sequence, it is enough to verify the isomor*
 *phism
for "equivariant pointsö  f the form T=L, with L a compact subgroup of T . Choo*
 *se an
identification                                 r
                                              Y
                          T = (S1)p x (S1)q x    Zmi
                                              i=1
such that via this identification
                                     Yq      Yr

                         L = (S1)p x    Znix    Zli.
                                     j=1     i=1
                  Q q             Q r
Then T=L = (1)p x   j=1(S1=Zni) x   i=1(Zmi=Zli).
             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY            17

  We use now Proposition 2.24, which is also true if we replace  K by K -theory*
 * (be-
cause it is true for K-theory). Since the map CH commutes with the isomorphisms*
 * of
Proposition 2.24, it is enough to check that the following maps are isomorphism*
 *s:

   (a) CHS1 : K *S1!  K*S1;

   (b) CHS1 : K *S1(S1=Zn) !  K*S1(S1=Zn);
   (c) CHZm : K *Zm(Zm =Zl) !  K*Zm(Zm =Zl).

  To prove (a), notice that CS1 = C. Then we have K *S1= K*S1 K*S1 OhC=  OhC. By

Proposition 2.21,  K*S1=  OhC. Now notice that, by definition, the map CHS1 is *
 *the
identity.

  For (b), denote X = S1=Zn. Then we have K*S1(X) = K*S1(X) K*S1 K*S1= K*Zn K*S1

 OhC.  But we know that K*S1= C[z 1] and K*Zn= C[z 1]=<zn - 1>.  So we deduce

K *S1(X) =  OhC=<zn - 1>. This last ring can be identified with C[z 1]=<zn - 1>*
 *, since
the condition zn = 1 makes all power series finite.  In conclusion, K *S1(X) = *
 *K*Zn=

C[z 1]=<zn - 1>.
  Let us now describe the sheaf F = K*S1(X). Let ff 2 C. If ff =2Zn, Xff= ;, so*
 * the

stalk of F at ff is zero. If ff 2 Zn, Xff= X, and the stalk of F at ff is H*Zn *
 *H*S1OhC,0.

But H*Zn= C, concentrated in degree zero (H*Znis Z-torsion in higher degrees, s*
 *o the
components in higher degree disappear when we tensor with C). It follows that F*
 * is a
sheaf concentrated at the elements of Zn, where it has the stalk equal to C. Th*
 *en the
global sections of F are  K*S1(X) = C   . . .C, n copies, one for each element *
 *of Zn.

  The map CHS1 : K *S1(X) !  K*S1(X) comes from the ring map CHT : C[z 1]=<zn -
1> ! C . . .C. Since z generates the domain of CHT, it is enough to see where z*
 * is sent.
Let Zn = {1, ffl, ffl2, . .,.ffln-1}. Then z represents the standard irreducibl*
 *e representation
V  = V (ffl) = C of Zn, where ffl acts on C by complex multiplication with ffl,*
 * which is

regarded as an element of C. Notice that V corresponds to the element ~ = ffl 2*
 * cZn= Zn.
c1(V )S1 = 0, because c1(V )S1 lies in H2Zn= 0. Then chS1(V ) = ec1(V )S1= e0 =*
 * 1, and
the stalk of CHS1(V ) at ff 2 Zn is CHS1(V )ff= ff(ffl) = ff. Therefore CHS1 se*
 *nds z to
(1, ffl, ffl2, . .,.ffln-1) 2 C   . . .C. One can easily check that this map is*
 * an isomorphism.

  For (c), denote X = Zm =Zl.  As in (b), K *Zm(X) = K*Zl= C[z 1]=<zl - 1>, and

 K*Zm(X) = C   . . .C, l copies. The proof that CHZm is an isomorphism is the s*
 *ame
as above.


                         Appendix A.  Applications

  We now give applications of the construction in this paper. First we use the *
 *Chang-
Skjelbred theorem in equivariant cohomology to infer the corresponding result f*
 *or equi-
variant K-theory.  Then as a corollary we show how a result about the equivaria*
 *nt
cohomology of GKM manifolds can be extended to equivariant K-theory.  Along the
way we need a natural splitting that does not seem to have been noticed before *
 *in this
area.

Definition A.1. Let X be a compact T -manifold, for T a compact abelian Lie gro*
 *up.
We say that X is equivariantly formal if the equvariant cohomology spectral seq*
 *uence
collapses at the E2 term.

  Many interesting T -spaces are equivariantly formal; for example any subvarie*
 *ty of
complex projective space preserved by a linear action, or symplectic manifold w*
 *ith a
18           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

Hamiltonian action.  Our reference for equivariantly formal spaces is [11].  We*
 * need
three results about them: The first is that the map H*T(X) ! H*T(XT ) is an inj*
 *ection.
The second is that for any H compact subgroup of T , XH  is also equivariantly *
 *formal.
The third is due to Chang and Skjelbred [9] (see also [8] for a proof):

Theorem A.2. Let X be an equivariantly formal T -manifold, and let X1 be its eq*
 *ui-
variant 1-skeleton, i.e. X1 is the set of points in X with stabilizer of codime*
 *nsion at
most one.  We have inclusion maps i : XT ! X and j : XT ! X1.  Then the map
i* : H*T(X) ! H*T(XT ) is injective, and the maps i* and j* : H*T(X1) ! H*T(XT *
 *) have
the same image.

  The ring H*T(X1), in the notation of the above theorem, has not received much*
 * study.
It is typically much bigger than H*T(X), and though H*T(X) injects into it, it *
 *does not
inject into H*T(XT ). These phenomena can be seen in the case of T 2acting on X*
 * = CP2,
where X1 is a cycle of three CP1's and therefore has H1, not seen in either H*T*
 *(X) or
H*T(XT ).

Lemma  A.3. In  the  notation  of  Theorem  A.2,  there  is  a  natural  identi*
 *fication
H*T(X1) = H*T(X)   kerj*.

Proof.By Theorem A.2 the images of the two maps i* : H*T(X) ! H*T(XT ) and j* :
H*T(X1) ! H*T(XT ) are the same. But i* is injective, so we can identify H*T(X)*
 * with
the image of i*.  This implies that j* factors through a map H*T(X1) ! H*T(X), *
 *and
this yields a splitting H*T(X1) = H*T(X)   kerj*.

  This natural splitting sheafifies, allowing us to extend both results to K-th*
 *eory.

Theorem A.4. We use the same notations as in Theorem A.2. Then i* : K*T(X) !
K*T(XT ) is injective, and the maps i* and j* : K*T(X1) ! K*T(XT ) have the sam*
 *e image.

Proof.Let ff 2 CT.  Any compact T -manifold admits a decompositionSas a finite *
 *T -
CW complex (see for example Allday and Puppe [2]).  Let X =   iDnix (T=Hi) be
suchPa cell decomposition.  We saw that if K is a compact subgroup of T , XK  =
         ni                                               ff    ff
  i:K HiD  x (T=Hi). In particular, this implies that (X1)  = (X )1.
  Let Y = Xff, which is again equivariantly formal. By Lemma A.3 there is a nat*
 *ural
identification H*T(Y1) = H*T(Y )   kerj*.  By Proposition 2.8 the map H*T! H *T*
 *is
flat, so tensoring with H *Tover H*Tyields a splitting H *T(Y1) = H *T(Y )   ke*
 *rj*. Now,
we observed above that (Y1)ff= (Y ff)1.  So we finally get a splitting H *T(X1)*
 *ff =
H *T(Xff)   kerj*. This is compatible with the gluing maps of the sheaf K*T(X),*
 * so we
get K*T(X1) = K*T(X)   kerj*.
  The upshot of the above discussion is that i* : K*T(X) ! K*T(XT ) is injectiv*
 *e (since
it is injective on stalks), and K*T(X1) = K*T(X)   kerj*. The global section fu*
 *nctor   is
left exact, so i* :  K*T(X) !  K*T(XT ) is injective and  K*T(X1) =  K*T(X)   k*
 *erj*.
This implies that i* and j* have the same image in  K*T(XT ), namely  K*T(X). (*
 *Notice
we couldn't have done this without using the splitting, because   is not right *
 *exact, so
it doesn't commute with the image functor.)
  Now recall that we have a natural isomorphism CHT : K *T(X) !  K*T(X). Trans-
lating the above results via CHT, we obtain that the maps i* : K *T(X) ! K *T(X*
 *T ) and
j* : K *T(X1) ! K *T(XT ) have the same image. But K *T(X) = K*T(X)  K*TK *Tand*
 * by
Lemma 2.22 the map K*T! K *Tis faithfully flat. So we obtain that i* and j* hav*
 *e the
same images in K*T(XT ), which is what we wanted.

             EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY            19

  An alternative proof of Theorem A.4 can be given by noticing that the sheaf m*
 *aps
i* : K*T(X) ! K*T(XT ) and j* : K*T(X1) ! K*T(XT ) have the same image (because
they have the same image at the level of stalks). But the global section functo*
 *r is exact,
since we work with coherent sheaves over Stein manifolds (see the comment before
Corollary 2.20). It follows that the maps i* : K *T(X) ! K *T(XT ) and j* : K **
 *T(X1) !
K *T(XT ) have the same image, and the proof proceeds as before.
  In [11] a special case of this is studied, in which XT is discrete and X1 is *
 *a union
of S2's; these are called balloon manifolds or GKM manifolds. (An interesting e*
 *xample
of GKM manifolds are toric varieties.) In this case it is easy to calculate the*
 * image of
restriction from H*T(X1), by reducing it to the case of H*T(S2). The theorem ab*
 *ove lets
us extend this result to K-theory.

Corollary A.5. Let X be a GKM manifold, and i* : K*T(X) ! K*T(XT ) the restrict*
 *ion
map.  Then i* is an injection; and a class ff 2 K*T(XT ) is in the image if for*
 * each
2-sphere B   X1 with fixed points N and S, the difference ff|N - ff|S 2 KT is a*
 * multiple
of the K-theoretic Euler class of the tangent space TN B. (Technically, we can *
 *only take
the Euler class once we orient TN B, but either orientation leads to the same c*
 *ondition
on ff.)

  If T = (S1)n, we can identify K*Twith Laurent polynomials, and this condition*
 * says
that the difference ff|S - ff|N of Laurent polynomials must be a multiple of 1 *
 *- w, where
w is the weight of the action of T on TN B. (Again, we can only speak of the we*
 *ight w
once we orient this R2-bundle over the point N, but it doesn't matter because b*
 *eing a
multiple of 1 - w is the same as being a multiple of 1 - w-1. In most examples *
 *one has
around a T -invariant almost complex structure with which to orient all these t*
 *angent
spaces simultaneously.)
  After finishing this paper, similar results in the algebraic case have appear*
 *ed in [18],
particularly with regard to the extension of Chang-Skjelbred's results to K-the*
 *ory. More
specifically, their Corollary 5.10 is the algebraic analogue of our Theorem A.4*
 *. Notice
also that their results, remarkably, hold over Z, while ours only hold over C. *
 *We thank
Angelo Vistoli for explaining his results to us.
  We also found out that Atiyah [3] proved a Chang-Skjelbred lemma in the conte*
 *xt
of equivariant K-theory.  He not only did it more or less at the same time as C*
 *hang
and Skjelbred, but his results are stronger:  Let G be a torus and X a compact *
 *G-
manifold. Denote by Xi the equivariant i-skeleton of X. Then Atiyah shows that *
 *the
long exact sequence of the pair (X \ Xi, X \ Xi+1) splits into short exact sequ*
 *ences

0 -! K*-1G(X \ Xi+1) -ffi!K*G(Xi+1\ Xi) -! K*G(X \ Xi) -! 0.  This in turn is
equivalent to the having a long exact sequence

     0 -! K*G(X) -! K*G(X0) -! K*+1G(X1 \ X0) -! K*+2G(X2 \ X1) -! . . .

As noted by Bredon [7], Atiyah's argument carries over to equivariant cohomolog*
 *y with
compact support and rational coefficients. In this context, the exactness up to*
 * the term
K*+1G(X1 \ X0) is just the assertion of the Chang-Skjelbred lemma. In cohomolog*
 *ical
setting, the above sequence may be thought of as the E1 term of the spectral se*
 *quence
coming from the filtration of the Borel construction XG by the subspaces (Xi)G *
 *. More-
over, its exactness is in fact equivalent to the equivariant formality of X, i.*
 *e. to the
freeness of H*G(X). (The direction Atiyah proved is the harder one.) We thank M*
 *atthias
Franz for pointing out these results to us.
20           EQUIVARIANT K-THEORY AND EQUIVARIANT COHOMOLOGY

A.1. Acknowledgements. We thank Victor Guillemin for persuading us to write this
paper, and Haynes Miller for constant guidance and support throughout this proj*
 *ect. We
also thank Michele Vergne for going carefully through the paper and suggesting *
 *several
corrections and improvements. Finally, we thank Lars Hesselholt, Payman Kassaei*
 * and
Behrang Noohi for helpful discussions.


Massachusetts Institute of Technology, Cambridge, MA
E-mail address: ioanid@math.mit.edu
University of California at Berkeley, CA
E-mail address: allenk@math.berkeley.edu


                                References

 [1]M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 *
 *(1984), 1-28.
 [2]C. Allday, V. Puppe, Cohomological methods in transformation groups, Cambri*
 *dge University Press,
   1993.
 [3]M. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathemat*
 *ics 401 (1974).
 [4]P. Baum, J.L. Brylinski, R. MacPherson, Cohomologie 'equivariante d'elocali*
 *s'ee, C.R. Acad. Sci.
   Paris, Serie I 300 (1985), 605-608.
 [5]J. Block, E. Getzler, Equivariant cyclic homology and equivariant different*
 *ial forms, Ann. Sci. 'Ec.
   Norm. Sup. 27 (1994), 493-527.
 [6]R. Bott, L. Tu, Differential forms in algebraic topology, Springer, 1982.
 [7]G. Bredon, The free part of a torus action and related numerical equalities*
 *, Duke Math. J. 41
   (1974), 843-854.
 [8]M. Brion, M. Vergne, On the localization theorem in equivariant cohomology,*
 * Mathematics ArXiv
   (1997), http: //xxx.lanl.gov, dg-ga/9711003.
 [9]T. Chang, T. Skjelbred, The topological Schur lemma and related results, An*
 *n. of Math. 100 (1974),
   307-321.
[10]M. Duflo, M. Vergne, Cohomologie 'equivariante et descente, Ast'erisque 215*
 * (1993), 3-108.
[11]M. Goresky, R. Kottwitz, R. MacPherson, Equivariant cohomology, Koszul dual*
 *ity, and the local-
   ization theorem, Invent. Math. 131 (1998), 25-83.
[12]I. Grojnowski, Delocalized equivariant elliptic cohomology, Yale University*
 * preprint, 1994.
[13]R. Gunning, H. Rossi, Analytic functions of several complex variables, Pren*
 *tice-Hall, 1965.
[14]H. Matsumura, Commutative Algebra, Benjamin/Cummings, 1980.
[15]I. Rosu, Equivariant elliptic cohomology and rigidity, Amer. J. of Math. 12*
 *3 (2001), 647-677.
[16]G. Segal, Equivariant K-theory, Publ. Math. I.H.E.S. 34 (1968), 129-151.
[17]J.-P. Serre, G'eom'etrie alg'ebrique et g'eom'etrie analytique, Anal. Inst.*
 * Fourier 6 (1956), 1-42.
[18]G. Vezzosi, A. Vistoli, Higher algebraic K-theory for actions of diagonaliz*
 *able groups, Mathematics
   ArXiv (2001), http: //xxx.lanl.gov, math.AG/0107174.
[19]C. Weibel, An introduction to homological algebra, Cambridge University Pre*
 *ss, 1994.