### Abstract

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C - L Rx, y[(xy) -1] and C - L R[x, x-1]y[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

Original language | English |
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Article number | 1550055 |

Journal | Journal of Algebra and its Applications |

Volume | 14 |

Issue number | 4 |

Early online date | 9 Jan 2015 |

DOIs | |

Publication status | Published - 29 May 2015 |

### Fingerprint

### Keywords

- algebraic mapping torus
- Finite domination
- multi-complex
- Novikov homology
- totalisation
- truncated product

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Journal of Algebra and its Applications*,

*14*(4), [1550055]. https://doi.org/10.1142/S0219498815500553

**Finite domination and Novikov rings : Laurent polynomial rings in two variables.** / Hüttemann, Thomas; Quinn, David.

Research output: Contribution to journal › Article

*Journal of Algebra and its Applications*, vol. 14, no. 4, 1550055. https://doi.org/10.1142/S0219498815500553

}

TY - JOUR

T1 - Finite domination and Novikov rings

T2 - Laurent polynomial rings in two variables

AU - Hüttemann, Thomas

AU - Quinn, David

PY - 2015/5/29

Y1 - 2015/5/29

N2 - Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C - L Rx, y[(xy) -1] and C - L R[x, x-1]y[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

AB - Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C - L Rx, y[(xy) -1] and C - L R[x, x-1]y[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

KW - algebraic mapping torus

KW - Finite domination

KW - multi-complex

KW - Novikov homology

KW - totalisation

KW - truncated product

UR - http://www.scopus.com/inward/record.url?scp=84930074918&partnerID=8YFLogxK

U2 - 10.1142/S0219498815500553

DO - 10.1142/S0219498815500553

M3 - Article

AN - SCOPUS:84930074918

VL - 14

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

IS - 4

M1 - 1550055

ER -