8 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

We proceed next to the study of K^(-;Z ), equivariant

K-theory mod n. We show that the ring of R(G)-linear homology

operations is the exterior algebra over R(G) on the Bockstein

element. Finally, we classify admissible multiplications; these

correspond to elements of R(G)$Z and (if n is odd) exactly one

of them is (graded) commutative.

We note that if G is a finite group of odd order then M.

Bokstedt (unpublished thesis [Bo]) has obtained a short exact UCT

sequence for KQ(X), where X is a finite G-CW-complex. These

results would seem to be orthogonal to our own, as R(G) is always

of infinite homological dimension if G is of finite order.

According to I. Madsen and M. Rothenberg [Mad], Bokstedt also has

results in the compact Lie case which are along the same lines as

our own. We have not seen these results.

We wish to thank the Mathematical Sciences Research

Institute for supporting us in 1984-5, during which time this

paper was completed. We are grateful to A. Wassermann for telling

us about Theorem 3.7 (ii).