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FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

Published online by Cambridge University Press:  02 August 2012

THOMAS HÜTTEMANN
Affiliation:
School of Mathematics and Physics, Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland, UK e-mail: [email protected]
DAVID QUINN
Affiliation:
School of Mathematics and Physics, Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland, UK e-mail: [email protected]
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Abstract

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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 CLR((x)) and CLR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(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.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

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