\(K_1\) by generators and relations
type: Item
instance of: scholarly article
title: \(K_1\) by generators and relations
zbMATH Open document ID: 0923.19001
DOI: 10.1016/S0022-4049(97)00056-X
published in: Journal of Pure and Applied Algebra
publication date: 1999-10-14T00:00:00Z
review text: The author obtains a presentation for the algebraic \(K\)-theory group \(K_1({\mathcal A})\), where \({\mathcal A}\) is an exact category. The generators are double short exact sequences; that is, they are pairs of short exact sequences with the same objects but possibly with different morphisms. An automorphism \(\alpha \colon A \to A\) yields a double short exact sequence \[ 0 \to 0 \to A \to A \to 0, \quad 0 \to 0 \to A \to A \to 0, \] where the endomorphisms of \(A\) are the identity and \(\alpha\), and a double short exact sequence can be regarded as a generalised automorphism. The members of \(K_1({\mathcal A})\) can be regarded as equivalence classes of double short exact sequences, because every member of \(K_1({\mathcal A})\) is the image of a generator.
Mathematics Subject Classification ID: 19B99
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 1308230
zbMATH Keywords: exact category
zbMATH Keywords: double short exact sequence
zbMATH Keywords: \(K_1\)
author: Alexander Nenashev
reviewed by: Richard John Steiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308230