P. Ebeling, F. Keune

On the Equivalence of Quillen's and Swan's K-Theories

The K-theory of rings can be defined in terms of nonabelian derived functors [see the second author, "Nonabelian derived functors and algebraic K-theory", in: Springer-Verlag, Lecture Notes in Mathematics 341 (1973) 166--176; see also the books of H. Inassaridze, "Algebraic K-theory" (Mathematics and its Applications 311, Kluwer Publ. Group, 1995) and "Non-abelian homological algebra and its applications" (Mathematics and its Applications 421, Kluwer Publ. Group, 1997) for a similar approach]. In fact both Swan's theory and Quillen's theory can be described this way. The equivalence of both K-theories was proved by S. M. Gersten ["K-theory of free rings", Commun. Algebra 1 (1974) 39--64]. In this paper we give a proof using these descriptions that involve nonabelian derived functors.