Finite domination and NOVIKOV homology over strongly Z2-graded rings

Thomas Hüttemann; Luke Steers

doi:10.1017/prm.2023.64

Finite domination and NOVIKOV homology over strongly Z²-graded rings

Thomas Hüttemann^*
, Luke Steers

^*Corresponding author for this work

School of Mathematics and Physics

Research output: Contribution to journal › Article › peer-review

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Abstract

Let R be a strongly Z² -graded ring, and let C be a bounded chain complex of finitely generated free R-modules. The complex C is R_(0,0)-finitely dominated, or of type FP over R_(0,0), if it is chain homotopy equivalent to a bounded complex of finitely generated projective R_(0,0)-modules. We show that this happens if and only if C becomes acyclic after taking tensor product with a certain eight rings of formal power series, the graded analogues of classical NOVIKOV rings. This extends results of RANICKI, QUINN and the first author on LAURENT polynomial rings in one and two indeterminates.

Original language	English
Pages (from-to)	1786-1818
Number of pages	33
Journal	Proceedings of the Royal Society of Edinburgh. Section A. Mathematics
Volume	154
Issue number	6
Early online date	28 Jul 2023
DOIs	https://doi.org/10.1017/prm.2023.64
Publication status	Published - Dec 2024

Keywords

algebraic mapping torus
finite domination of chain complexes
Mather trick

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1017/prm.2023.64Licence: CC BY

Finite domination and NOVIKOV homology over strongly Z2-graded rings
Copyright 2023 The Authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited.
Final published version, 1.19 MBLicence: CC BY

Algebraic finite domination
Steers, L. (Author), Huettemann, T. (Supervisor) & Shkarin, S. (Supervisor), Sept 2017
Student thesis: Doctoral Thesis › Doctor of Philosophy

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2 Article

Finite domination and Novikov homology over strongly Z-graded rings
Huettemann, T. & Steers, L., 22 Sept 2017, In: Israel Journal of Mathematics. 221, 2, p. 661–685
Research output: Contribution to journal › Article › peer-review

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Finite domination and Novikov rings: Laurent polynomial rings in two variables
Huttemann, T. & Quinn, D., May 2014, In: Journal of Algebra and its Applications. 14, 4, 44 p., 1550055.
Research output: Contribution to journal › Article › peer-review

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Cite this

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title = "Finite domination and NOVIKOV homology over strongly Z2-graded rings",

abstract = "Let R be a strongly Z2 -graded ring, and let C be a bounded chain complex of finitely generated free R-modules. The complex C is R(0,0)-finitely dominated, or of type FP over R(0,0), if it is chain homotopy equivalent to a bounded complex of finitely generated projective R(0,0)-modules. We show that this happens if and only if C becomes acyclic after taking tensor product with a certain eight rings of formal power series, the graded analogues of classical NOVIKOV rings. This extends results of RANICKI, QUINN and the first author on LAURENT polynomial rings in one and two indeterminates. ",

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N2 - Let R be a strongly Z2 -graded ring, and let C be a bounded chain complex of finitely generated free R-modules. The complex C is R(0,0)-finitely dominated, or of type FP over R(0,0), if it is chain homotopy equivalent to a bounded complex of finitely generated projective R(0,0)-modules. We show that this happens if and only if C becomes acyclic after taking tensor product with a certain eight rings of formal power series, the graded analogues of classical NOVIKOV rings. This extends results of RANICKI, QUINN and the first author on LAURENT polynomial rings in one and two indeterminates.

AB - Let R be a strongly Z2 -graded ring, and let C be a bounded chain complex of finitely generated free R-modules. The complex C is R(0,0)-finitely dominated, or of type FP over R(0,0), if it is chain homotopy equivalent to a bounded complex of finitely generated projective R(0,0)-modules. We show that this happens if and only if C becomes acyclic after taking tensor product with a certain eight rings of formal power series, the graded analogues of classical NOVIKOV rings. This extends results of RANICKI, QUINN and the first author on LAURENT polynomial rings in one and two indeterminates.

KW - algebraic mapping torus

KW - finite domination of chain complexes

KW - Mather trick

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DO - 10.1017/prm.2023.64

M3 - Article

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JO - Proceedings of the Royal Society of Edinburgh. Section A. Mathematics

JF - Proceedings of the Royal Society of Edinburgh. Section A. Mathematics

IS - 6

ER -

Finite domination and NOVIKOV homology over strongly Z2-graded rings

Abstract

Keywords

ASJC Scopus subject areas

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Student theses

Algebraic finite domination

Research output

Finite domination and Novikov homology over strongly Z-graded rings

Finite domination and Novikov rings: Laurent polynomial rings in two variables

Cite this

Finite domination and NOVIKOV homology over strongly Z²-graded rings