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