Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

TL;DR

This paper explores the structure of noncommutative spectra related to matrix algebras, introduces a rank filtration, and describes the rational stabilization, connecting to $K$-theory and noncommutative CW-spectra.

Contribution

It introduces a rank filtration on the category of noncommutative suspension spectra of matrix algebras and describes its properties and implications for rational and localized spectra.

Findings

The rank filtration lifts the classical rank filtration of connective $K$-theory.

The rank filtration stabilizes rationally after the first stage.

An explicit model of the rationalized noncommutative spectra category is provided.

Abstract

In a companion paper [ABS1] we introduced the stable $\infty$-category of noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$ denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. In this paper we investigate the structure of $\mathcal{M}$, and derive some consequences regarding the structure of $\mathtt{NSp}$. To begin with, we introduce a rank filtration of $\mathcal{M}$. We show that the mapping spectra of $\mathcal{M}$ map naturally to the connective $K$-theory spectrum $ku$, and that the rank filtration of $\mathcal{M}$ is a lift of the classical rank filtration of $ku$. We describe the subquotients of the rank filtration in terms of complexes of direct-sum decompositions which also arose in the study of $K$-theory and of Weiss's orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of $\mathtt{NSp}$ as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the $p$-localization and the chromatic localization of $\mathcal{M}$.