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

Jonathan Campbell, Alexander Kupers, Inna Zakharevich

TL;DR

The paper develops a stable rank filtration on direct sum $K$-theory by leveraging a $\Gamma$-space construction and a poset-indexed valuation framework on $K$-theory for convenient addition categories. It generalizes Rognes's rank filtration to settings with a genuine poset filtration, showing that filtration quotients can be realized as homotopy coinvariants of certain suspension spectra and enabling a new decomposition into suspension-spectrum terms. Central results express filtered pieces $F_pK(\mathcal{A})/F_{<p}K(\mathcal{A})$ as $K(S_A)_{h\mathrm{Aut}(A)}$, where $S_A$ is a $\Gamma$-space, and yield a theorem guaranteeing that $K(S_A)$ is a suspension spectrum after suitable valuation. These insights reproduce and extend Rognes's common-basis analysis, supply new spectral sequences converging to the rationalized $K$-groups of commutative rings and inner product spaces over ordered fields, and extend to Waldhausen categories to recover an alternate model of the common basis complex.

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 $Γ$-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.

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

TL;DR

The paper develops a stable rank filtration on direct sum -theory by leveraging a -space construction and a poset-indexed valuation framework on -theory for convenient addition categories. It generalizes Rognes's rank filtration to settings with a genuine poset filtration, showing that filtration quotients can be realized as homotopy coinvariants of certain suspension spectra and enabling a new decomposition into suspension-spectrum terms. Central results express filtered pieces as , where is a -space, and yield a theorem guaranteeing that is a suspension spectrum after suitable valuation. These insights reproduce and extend Rognes's common-basis analysis, supply new spectral sequences converging to the rationalized -groups of commutative rings and inner product spaces over ordered fields, and extend to Waldhausen categories to recover an alternate model of the common basis complex.

Abstract

In the literature, there are two standard rank filtrations on -theory: an ``unstable'' one which is traditionally defined through the homology of , and a ``stable'' one which was defined by Rognes using the simplicial structure on Waldhausen's -construction. In this paper we give an alternate stable rank filtration, which uses the simplicial structure present in a -space construction of -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 -theory.
Paper Structure (2 sections, 8 theorems, 9 equations)

This paper contains 2 sections, 8 theorems, 9 equations.

Table of Contents

  1. Valuations and filtrations on $\mathbf{Set}$-valued functors
  2. Basic definitions

Key Result

Theorem A

Let $\mathcal{A}$ be a convenient addition category equipped with a functor $\nu:\mathcal{A} \rightarrow \mathcal{P}$ for a poset $\mathcal{P}$; suppose further that $\nu$ reflects isomorphisms. There is a filtration $F_p$ (indexed over $p\in \mathcal{P}$) on the spectrum $K(\mathcal{A})$. Let $\mat where $A\in \mathcal{A}$ is such that $\nu(A) = p$, and where $S_A$ is a certain $\Gamma$-space.

Theorems & Definitions (15)

  • Theorem A: See Theorem \ref{['thm:graded-part-SA']}
  • Theorem B: See Corollary \ref{['cor:filtered-spectrum-susp']}
  • Theorem C: See Corollary \ref{['cor:KSA']}
  • Theorem D: See Section \ref{['sec:rogneswald']}
  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Definition 1.5
  • ...and 5 more