K-THEORY OF ENDOMORPHISM-RINGS AND OF RINGS OF INVARIANTS

This paper gives the following description of K-0 of the endomorphism ring of a finitely generated projective module. THEOREM. Let T be a ri ng and P a finitely generated, projective T-module. Let I be the trace ideal of P. Then K-0(End P-T) is isomorphic to a subgroup of K-0(T, I ). If, further, the natural map K-1(T) --> K-1(T/I) is subjective then K-0(End P-T) is isomorphic to the subgroup of K-0(T) generated by the direct summands of P-n, for n is an element of N. As a corollary we c an determine K-0 of the ring of invariants for many free linear action s. In particular, the following result is proved. THEOREM. Let V be a fixed-point-free linear representation of a finite group G over a fiel d k of characteristic zero and let S(V) be the symmetric algebra of V. Let K be any finite-dimensional k-vector space. Then K-0(S(V)(G) x(k) S(K)) = [[S(V)(G) x(k) S(K)]]. Similar results are given for suitable noncommutative versions of S(V). (C) 1997 Academic Press.