K-Theory of Endomorphism Rings and of Rings of Invariants

Abstract

This paper gives the following description ofK0of the endomorphism ring of a finitely generated projective module.Theorem.Let T be a ring and P a finitely generated,projective T-module. Let I be the trace ideal of P. Then K0(EndPT)is isomorphic to a subgroup of K0(T,I).If,further,the natural map K1(T)→K1(T/I)is surjective then K0(EndPT)is isomorphic to the subgroup of K0(T)generated by the direct summands of Pn,for n∈N.As a corollary we can determineK0of the ring of invariants for many free linear actions. In particular, the following result is proved.Theorem.Let V be a fixed-point-free linear representation of a finite group G over a field k of characteristic zero and let S(V)be the symmetric algebra of V.Let K be any finite-dimensional k-vector space. Then K0(S(V)G⊗kS(K))=〈[S(V)G⊗kS(K)]〉.Similar results are given for suitable noncommutative versions ofS(V).