EQUIVARIANT K-THEORY AND KK-THEORY
3
condition, p is a prime ideal with R(G) a principal ideal
domain, then there is a natural short exact sequence which
determines
KG(A$B)
in terms of K*(A) and
KG(B)
.
The following special case deserves special attention.
THEOREM 6.1. (Hodgkin spectral sequence) . Let G be a compact Lie
group satisfying the Hodgkin condition, let H be a closed
subgroup, and let B be a G-algebra. Then there is a spectral
u
sequence of R(G)-modules which strongly converges to K#(B) with
E* S
Tor*(G)(R(H),K*(B)).
In particular, there is a strongly convergent spectral sequence
E* = Tor*(G,(Z,KG(B)) == K^B).
This reduces to Kasparov's generalization [Ka3, 17] of the
Pimsner-Voiculescu sequence [PV] by taking
G = Tr,
B = ZrixA.
The Pimsner-Voiculescu sequence is recovered by setting r = 1.
Our other basic theorem is a generalization of the Universal
Coefficient Theorem of [RSI, RS2] which determined KK^A^) in
terms of K^A) and K^fB). More precisely, we proved that there
was a natural short exact sequence of the form
0 •Ext*(K#(A),K#(B)) •KKj^B) Y Hom2(K*(A)fK#(B) ) —» 0
which splits unnaturally. The map Y is the natural Kasparov
pairing. It generalizes to the equivariant setting to yield a
natural map
Y: KKG(A,B) HomR(G)(KG(A),KG(B)).
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