### Abstract

Abstract Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x ^{-1}]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C - L R((x)) and C-L R((x^{-1})) are acyclic, as has been proved by Ranicki (Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619-632). Here R((x))=R[[x]][x ^{-1}] and R((x ^{-1})) = R[[x ^{-1}]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

Original language | English |
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Pages (from-to) | 145-160 |

Number of pages | 16 |

Journal | Glasgow Mathematical Journal |

Volume | 55 |

Issue number | 1 |

Early online date | 2 Aug 2012 |

DOIs | |

Publication status | Published - Jan 2013 |

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### Keywords

- 2000 Mathematics Subject Classification Primary 55U15
- Secondary 18G35

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Glasgow Mathematical Journal*,

*55*(1), 145-160. https://doi.org/10.1017/S0017089512000419

**Finite domination and novikov rings. Iterative approach.** / Hüttemann, Thomas; Quinn, David.

Research output: Contribution to journal › Article

*Glasgow Mathematical Journal*, vol. 55, no. 1, pp. 145-160. https://doi.org/10.1017/S0017089512000419

}

TY - JOUR

T1 - Finite domination and novikov rings. Iterative approach

AU - Hüttemann, Thomas

AU - Quinn, David

PY - 2013/1

Y1 - 2013/1

N2 - Abstract Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x -1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C - L R((x)) and C-L R((x-1)) are acyclic, as has been proved by Ranicki (Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619-632). Here R((x))=R[[x]][x -1] and R((x -1)) = R[[x -1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

AB - Abstract Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x -1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C - L R((x)) and C-L R((x-1)) are acyclic, as has been proved by Ranicki (Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619-632). Here R((x))=R[[x]][x -1] and R((x -1)) = R[[x -1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

KW - 2000 Mathematics Subject Classification Primary 55U15

KW - Secondary 18G35

UR - http://www.scopus.com/inward/record.url?scp=84870954115&partnerID=8YFLogxK

U2 - 10.1017/S0017089512000419

DO - 10.1017/S0017089512000419

M3 - Article

VL - 55

SP - 145

EP - 160

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 1

ER -