^-theory of a Space with Coefficients in a (discrete) Ring

In [2], [3], S. Gersten has introduced higher i£-groups of a ring which satisfy properties analogous to those of a generalized homology theory in a suitably defined homotopy category of rings [ l ] . In this announcement we use Gersten's i£-groups to define for a ring R a generalized cohomology theory KR( ), analogous to the AtiyahHirzebruch i£-theory, on the category of finite simplicial sets so that Kn(pt) =KlR, where K%R are Gersten's stable jK-groups of the ring R. If R is suitably restricted, in particular if it is commutative and regular, the theory KR( ) will have products and Adams operations. One may also define, using the continuous theory in [6], a K-theory K&{ ) with coefficients in a Banach ring A. This theory coincides with the Atiyah-Hirzebruch theory for A = 2?, C, or H. We give here an outline of proofs. A full account will appear elsewhere.