K-Theory of Free Rings

Abstract

The object of this article is to establish the following result (Corollary 3.9 below): If R is a regular right noetherian ring and R{X} is the free associative algebra on the set X, then Kn(R) = Kn(R{X}), where Kn refers to the Quillen K-theory. The result can be stated in the equivalent form that Hn(G1(R),Z) = Hn(G1(R{X}),Z). From this result it follows that if F is a free ring without unit, then Kn(F) = 0, whence free rings are acyclic models for Quillen K-theory (3.11 below). This result in turn enables us to complete Anderson's work [1] in identifying the Quillen K-theory [11] and the K-theory proposed by Gersten [7] and Swan [18] for all rings. We also establish that the natural transformation Kn(R) → Kn k-v(R) between the Quillen theory and the K-theory of Karoubi and Villamayor is an isomorphism if R is a supercoherent (Definition 1.2) and regular (Definition 1.3) ring. From this result we can gain some information about the K-theory of group rings of free products of groups (Theorem 5.1).