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Stable Rank Filtration on Direct-Sum Algebraic K-Theory

A filtration of the direct-sum K-theory spectrum of finitely generated projective modules over a commutative ring by stable rank, constructed from P-valuations on convenient addition categories. Each filtration stage is built via a Segal-style Gamma-space machine from a poset-indexed variant of the S-construction, the associated graded pieces are identified with homotopy orbit spectra recording the homology of general linear groups, and the filtration recovers and refines the rank filtration studied for topological K-theory.

Cyberman Text

A STABLE RANK FILTRATION ON DIRECT SUM K-THEORY JONATHAN CAMPBELL, INNA ZAKHAREVICH, AND APPENDIX

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 …