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# 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

#### 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

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