4 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

THEOREM 9.2. (Universal Coefficient Spectral Sequence) Let G be a

compact Lie group which satisfies the Hodgkin condition. For A €

BG and B a G-algebra, there is a spectral sequence of

R(G)-modules which strongly converges to KK^AjB) with

E^'* * Extg(G)(K^(A),K^(B)).

The spectral sequence has the canonical grading, so that

Ext*?/«v(K (A),K.(B)) has homological degree p and total degree

K\u) S t

p+s+t (mod 2). The edge homomorphism

KK^(A,B) • E°'* = HomR(G)(K*(A),K*(B))

is the map Y. The spectral sequence is natural with respect to

pairs (A,B) in the category. If G has rank r then E^

=

0 for

pr+l and E ,„ = E .

^ r+2 oo

COROLLARY. Let G be a compact Lie group which satisfies the

~ G

Hodgkin condition. Suppose that A € BQ and that either K^fA) is

R(G)-projective or that K#(B) is R(G)-injective. Then there is a

natural isomorphism

Y(A,B): KK^(A,B) • HomR(G}(K^(A),K^(B)).

We apply the Universal Coefficient spectral sequence (9.2)

in several ways. First we consider KK -equivalence. In our

previous work we showed that if A and B are C -algebras in C with

K^fA) = K^fB), then A is KK-equivalent to B. The equivariant

version of this is false in general: in Example 10.6 we construct

commutative ¥-algebras A and B such that K^fA) s K#(B) as

R(T)-modules but K^fA) * K^B), so that A and B can't be

K-equivalent, much less KK -equivalent.

On the positive side, we are able to offer some results.

Here is a theorem whose hypotheses are frequently satisfied in

practice.

THEOREM 10.3. Suppose that G is a Hodgkin group, A and B are

~ G

G-algebras in B

Q

with K^(A) £

KG(B)

#

£ M, and suppose that M has

homological or injective dimension ^ 1. Then A and B are

KK -equivalent (and the equivalence covers the given