EQUIVARIANT K-THEORY AND KK-THEORY

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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)).