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
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