THIS PAPER introduces a new filtration of the algebraic K-theory spectrum KR of a ring R, and investigates the subquotients of this filtration. KR is constructed from the category $ P (R) of finitely generated projective R-modules, and its homotopy groups are the algebraic K-groups of R as defined by Quillen [14]. There is also a free K-theory K/R, constructed from the weakly cofinal subcategory 9 (R) of B (R) consisting of finitely generated free R-modules. Inclusion induces a covering map KfR+ KR, which in turn induces an isomorphism on 71i for i> 0 [7, 173. In particular the higher free K-groups~ iKfR for i> 0 agree with Quillen’s K-groups.

We construct a sequence of ‘unstable’algebraic K-theory spectra (FkKR}, filtering KfR. We will assume that R has the invariant dimension property [12] so that it makes sense to talk about the rank of a finitely generated free R-module. Then for a fixed rank k 2 0, FkKR is constructed as a subspectrum of K/R, built from free R-modules of rank less than or equal to k. As k increases, we obtain an increasing rank Jiltration {FkKR} k of spectra converging to KfR. It turns out that each subquotient spectrum F, KR/F, _, KR is a homotopy orbit spectrum D (Rk)/hGLk R for some spectrum-with-GLkR-action D (Rk). Furthermore we prove that D (Rk) is stably equivalent to the suspension spectrum on a finite dimensional GLkR-COIIIpkX D (Rk), which we call the stable building of Rk. Here is a description of the stable building, related to Volodin and Wagoner’s constructions of K-theory [20, 213: Dejinition 14.5’. Let Xv’DY (Rk) be a simplicial set with q-simplices the (q+ 1)-tuples {M,,..., M4} of free, proper, nontrivial submodules Mi c Rk, satisfying the following condition: There exists an R-basis s?# for Rk for which each submodule Mi has a subset of _@ as an R-basis. The stable building D (R’) is the suspension C (X-’D’(Rk)).