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

TL;DR

This paper introduces a new stable rank filtration on algebraic K-theory using a $ ext{Gamma}$-space construction, generalizing previous filtrations and providing new spectral sequences for homology computations.

Contribution

It presents an alternative stable rank filtration based on $ ext{Gamma}$-spaces, extending Rognes's work and enabling new spectral sequences for algebraic K-theory.

Findings

Filtration quotients are homotopy coinvariants of highly-connected suspension spectra.

Generalizes Rognes's results on the common basis complex.

Produces new spectral sequences converging to algebraic K-theory homology.

Abstract

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 $\Gamma$-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.