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arXiv · 2501.01609

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

Jonathan Campbell·Alexander Kupers·Inna Zakharevich
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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.

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Jonathan Campbell, Alexander Kupers, Inna Zakharevich. 2025-01-03. A stable rank filtration on direct sum $K$-theory. https://arxiv.org/abs/2501.01609

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