Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

$K_1$-groups via binary complexes of fixed length

Pages: 203 – 213

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a12

Authors

Daniel Kasprowski (Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Bonn, Germany)

Bernhard Köck (School of Mathematical Sciences, University of Southampton, Highfield, Southampton, United Kingdom)

Christoph Winges (Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Bonn, Germany)

Abstract

We modify Grayson’s model of $K_1$ of an exact category to give a presentation whose generators are binary acyclic complexes of length at most $k$ for any given $k \geqslant 2$. As a corollary, we obtain another, very short proof of the identification of Nenashev’s and Grayson’s presentations.

Keywords

exact category, binary acyclic complex, Nenashev relation

2010 Mathematics Subject Classification

Primary 19D06. Secondary 18E10, 19B99.

Full Text (PDF format)

Copyright © 2019, Daniel Kasprowski, Bernhard Köck and Christoph Winges. Permission to copy for private use granted.

Winges acknowledges support by the Max Planck Society and Wolfgang Lück’s ERC Advanced Grant “KL2MG-interactions” (no. 662400). Kasprowski and Winges were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1, Project-ID 390685813.

Received 15 May 2019

Received revised 4 July 2019

Accepted 8 July 2019

Published 20 November 2019