EQUIVARIANT K-THEORY AND KK-THEORY
5
isomorphism).
In the same vein, we prove the following theorem.
THEOREM 10.8. Let G be a compact Lie group satisfying the Hodgkin
~ G
condition, and let A BQ with K^fA) having homological or
injective dimension 1 (as an R(G)-module). Then
G ~ 0 1
a) A is KK -equivalent to a G-algebra in Bn of the form C eC ,
where K.(CJ) = 0 unless i=j.
b) If B is any G-algebra with K1(B) = 0, then there are split
exact sequences of the form
0 K^(A)SR(G)K^(B)
KJ(A8B)
—-TorJ^fKj^AKKJjfB)) 0
and
0 _•ExtJ(Q)(Kj_1(A),Kj(B)) —KK^(A,B) —*—
-HomR(G)(KJ(A),KQ(B)) 0.
c) In particular, there are split exact sequences
0 K°(A)0R(G)Z K0(A) TorJ{G)(K°(
and
0 ExtR(G)(K^(A),R(G)) KJj(A) —• HomR{Q) (KJ(A),R(G)) 0.
The splittings of these sequences are not natural.
When G is the trivial group, R(G) = Z and the Universal
Coefficient spectral sequence collapses to the short exact
sequence of [RSI, Theorem 4.2]. Specializing still further gives
the Universal Coefficient Theorem of Brown [Bl] for the
Brown-Douglas-Fillmore functor K (A). If one takes A = C (X), X a
locally compact G-space, and B = C, then one obtains a spectral
sequence of the form
Previous Page Next Page