**
P. Ebeling, F. Keune**

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

**abstract:**

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.