K-Theory of Free Rings
作者:
S. M. Gersten,
期刊:
Communications in Algebra
(Taylor Available online 1974)
卷期:
Volume 1,
issue 1
页码: 39-64
ISSN:0092-7872
年代: 1974
DOI:10.1080/00927877408548608
出版商: Taylor & Francis Group
数据来源: Taylor
摘要:
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 Knrefers 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) → Knk-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).
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