A stable rank filtration on direct sum $K$-theory

In the literature, there are two standard rank filtrations on $K$-theory: an unstable'' one which is traditionally defined through the homology of $GL_n$, and a stable’’ one which was defined by Rognes using the simplicial structure on Waldhausen’s $S_{\bullet }$-construction. In this paper we give an alternate stable rank filtration, which uses the simplicial structure present in a $\mathrm{\Gamma }$-space construction of $K$-theory; we investigate this in the case of convenient addition categories,'' and show that in good situtations where a notion of rank’’ is present, the filtration quotients will be homotopy coinvariants of certain highly-connected suspension spectra. This approach generalizes Rognes’s results on the common basis complex, and produces an alternate spectral sequences converging to the homology of algebraic $K$-theory.