Delooping the \(K\)-theory of exact categories
type: Item
instance of: scholarly article
title: Delooping the \(K\)-theory of exact categories
zbMATH Open document ID: 1059.18007
author: Marco Schlichting
published in: Topology
publication date: 2004-09-23T00:00:00Z
review text: The author introduces the notion of a left (and right) s-filtering exact subcategory \({\mathcal A}\) of an exact category \({\mathcal U}\). This condition generalises to exact categories the notion of a Karoubi filtration [\textit{M. Cárdenas} and \textit{E. K. Pedersen}, K-theory 12, 165--191 (1997; Zbl 0903.18005)]. These conditions suffice for the formation of a quotient exact category \({\mathcal U}/{\mathcal A}\) and a fibration of K-Theory spectra \(K{\mathcal A} \longrightarrow K{\mathcal U} \longrightarrow K{\mathcal U}/{\mathcal A} \). As an application, the author constructs a left s-filtering embedding \({\mathcal E} \subset {\mathcal FE}\) for any idempotent complete exact category \({\mathcal E}\) with \(K{\mathcal FE}\) contractible, which implies that \(K{\mathcal E} \simeq \Omega K{\mathcal FE}/{\mathcal E}\). This delooping construction permits the definition of negative K-theory for any exact category, generalising the negative K-theories of \textit{H. Bass, M. Karoubi} and \textit{R. W. Thomason}.
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 2102616
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: lower algebraic K-theory
zbMATH Keywords: infinite loopspaces
zbMATH Keywords: exact categories
reviewed by: Victor P. Snaith
MaRDI profile type: Publication
full work available at URL: j.top.2004.01.005
OpenAlex ID: W1996216338
DOI: 10.1016/J.TOP.2004.01.005
description: scientific article; zbMATH DE number 2102616