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On the K _ ~ functors

The functors K0 and K1 have well-known descriptions in terms of projectives and automorphisms, respectively. The purpose of this paper is to give analogous descriptions of the K−i-functors of Bass [1]. In fact given a ring R, we give two descriptions of K−i(R), i ≥ 0; one as a Whitehead construction on a certain category of R-modules (elements represented by automorphisms), and one as a Grothendieck construction on a related category of Rmodules (elements represented by objects). The categories in question are associated with the category of Z-graded R-modules and bounded homomorphisms in the following sense: