On \(K_1\) of an exact category
type: Item
instance of: scholarly article
title: On \(K_1\) of an exact category
zbMATH Open document ID: 0901.19001
DOI: 10.1023/A:1007718422686
published in: \(K\)-Theory
publication date: 1998-11-26T00:00:00Z
review text: Let \({\mathcal M}\) be an exact category, \(A\overset \alpha \rightarrowtail X\) and \(B\overset \beta \rightarrowtail Y\) two admissible monomorphisms, and \(\Theta:A\oplus X/A\oplus Y\to B\oplus Y/B\oplus X\) an isomorphism in \({\mathcal M}\). In 1994 the author constructed an element \(G(\alpha, \beta, \Theta)\) of \(K_1({\mathcal M})\), where \(G(\alpha, \beta, \Theta)\) is the homotopy class in \(\pi_1(| G{\mathcal M}|)\simeq K_1({\mathcal M})\) represented by the loop [see \textit{C. Sherman}, J. Algebra 163, No. 2, 568-582 (1994; Zbl 0798.19001)]. In this paper the author proves that every element of \(K_1({\mathcal M})\) is of this form. In addition, several geometric representations of \(G(\alpha, \beta, \Theta)\) are constructed, and then he uses these results to determine the description of elements arising from Bass's group \(K_1^{\text{det}}({\mathcal M})\).
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 19B99
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19E99
zbMATH DE Number: 1156511
zbMATH Keywords: \(K_1\)-group
zbMATH Keywords: exact category
zbMATH Keywords: homotopy
author: Clayton C. Sherman
MaRDI profile type: Publication
full work available at URL: a:1007718422686
OpenAlex ID: W2027436021
description: scientific article; zbMATH DE number 1156511