Stable K-theory as a derived functor

Abstract

Let R be a ring with unit and let Rf denote the category of finitely generated free R-modules. Let 39 denote the abelian category of functors T: RF x Rf -+ Ab. For a given T E 93 we shall denote by T, the colim T(R”, R”). We define the stable K-theory groups of a given ring R with coefficients in T E 97 by the formula K”(R, T) = H,(F, T,) where F is a homotopy fiber of the plus-construction map BGl(R) + BGl(R)+, see [16] and [2] for discussions on stable K-theory. The action of xl(F) on T, is given by the homomorphism Z,(F) + Gl(R) and the conjugation action of Gl(R) on T,. The category 93 has enough projective objects and it is an easy exercise with Yoneda’s lemma to show that the functors