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

Author

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

Abstract

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.

Keywords

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

Received 10 March 2021

Received revised 11 March 2022

Accepted 16 March 2021

Published 18 May 2022