Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Non-commutative localisation and finite domination over strongly $\mathbb{Z}$-graded rings

Pages: 373 – 398

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a18


Thomas Hüttemann (Queen’s University Belfast, School of Mathematics and Physics, Mathematical Sciences Research Centre, Belfast, Northern Ireland, United Kingdom)


Let $R = \bigoplus^{\infty}_{ k =-\infty} R_k$ be a strongly $\mathbb{Z}$-graded ring, and let $C^{+}$ be a chain complex of modules over the positive subring $P = \bigoplus^{\infty}_{k=0} R_k$. The complex $C^{+} \oplus_P R_0$ is contractible (resp., $C^{+}$ is $R_0$-finitely dominated) if and only if $C^{+} \oplus_P L$ is contractible, where $L$ is a suitable non-commutative localisation of $P$. We exhibit universal properties of these localisations, and show by example that an $R_0$-finitely dominated complex need not be $P$-homotopy finite.


non-commutative localisation, finite domination, type FP, strongly graded ring, Novikov homology, algebraic mapping torus, Mather trick

2010 Mathematics Subject Classification

16E99, 16W50, 18G35, 55U15

Copyright © 2022, Thomas Hüttemann. Permission to copy for private use granted.

Received 10 March 2021

Received revised 11 March 2022

Accepted 16 March 2021

Published 18 May 2022