A spectrum level rank filtration in algebraic K-theory

Abstract

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)).