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

Abstract

AbstractIn a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable $$\infty $$ ∞ -category of noncommutative CW-spectra, which we denoted $$\mathtt {NSp}$$ NSp . Let $${\mathcal {M}}$$ M denote the full spectrally enriched subcategory of $$\mathtt {NSp}$$ NSp whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that $$\mathtt {NSp}$$ NSp is equivalent to the $$\infty $$ ∞ -category of spectral presheaves on $${\mathcal {M}}$$ M . In this paper we investigate the structure of $${\mathcal {M}}$$ M , and derive some consequences regarding the structure of $$\mathtt {NSp}$$ NSp . To begin with, we introduce a rank filtration of $${\mathcal {M}}$$ M . We show that the mapping spectra of $${\mathcal {M}}$$ M map naturally to the connective K-theory spectrum ku, and that the rank filtration of $${\mathcal {M}}$$ M is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces 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 $$\mathtt {NSp}$$ 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 $${\mathcal {M}}$$ M .