Relations between algebraic K-theories

Abstract

In chapter II we have seen that there exist different constructions of the algebraic K-theory. It will be proved below that the algebraic K-theories of Quillen defined by plus construction and by Q-construction, and the algebraic K-theory of Swan are all isomorphic. On the other hand, we know that Waldhausen’s algebraic K-theory generalizes Quillen’s algebraic K-theory of exact categorics. We conclude that there are in fact two different algebraic K-theories up to isomorphism: Quillen’s algebraic K-theory and Karoubi-Villamayor’s algebraic K-theory, but it will be shown below that they agree on a wide class of rings including regular rings. Recall also the existence of Milnor’s algebraic K-theory, Volodin’s algebraic K-theory (not treated by us) which is also isomorphic to Quillen’s K-theory, and the negative algebraic K-theory of Bass.