Homology, Homotopy and Applications
Volume 22 (2020)
$K_1$-groups via binary complexes of fixed length
Pages: 203 – 213
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