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


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)


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.


exact category, binary acyclic complex, Nenashev relation

2010 Mathematics Subject Classification

Primary 19D06. Secondary 18E10, 19B99.

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