Hostname: page-component-5c6d5d7d68-ckgrl Total loading time: 0 Render date: 2024-08-17T10:04:34.505Z Has data issue: false hasContentIssue false
  • English
  • Français

FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

Published online by Cambridge University Press:  02 August 2012

THOMAS HÜTTEMANN  and
DAVID QUINN
THOMAS HÜTTEMANN
Affiliation:
School of Mathematics and Physics, Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland, UK e-mail: t.huettemann@qub.ac.uk
DAVID QUINN
Affiliation:
School of Mathematics and Physics, Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland, UK e-mail: DavidQuinnMath@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 ⊗LR((x)) and C ⊗LR((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.

Keywords

Primary 55U15Secondary 18G35
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 145 - 160
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Brown, K., Homological criteria for finiteness, Comment. Math. Helv. 50 (1975), 129–135.Google Scholar
2.Hughes, B. and Ranicki, A., Ends of complexes, Cambridge tracts in mathematics, vol. 123 (Cambridge University Press, Cambridge, UK, 1996).Google Scholar
3.Mac Lane, S., Homology, Classics in mathematics (Springer-Verlag, Berlin, Germany, 1995), Reprint of the 1975 edition.Google Scholar
4.Ranicki, A., The algebraic theory of finiteness obstruction, Math. Scand. 57 (1) (1985), 105–126.CrossRefGoogle Scholar
5.Ranicki, A., Finite domination and Novikov rings, Topology 34 (3) (1995), 619–632.Google Scholar
6.Rosenberg, J., Algebraic K-theory and its applications, Graduate texts in mathematics, vol. 147 (Springer-Verlag, New York, 1994).Google Scholar
7.Schütz, D., Finite domination, Novikov homology and nonsingular closed 1-forms, Math. Z. 252 (3) (2006), 623–654.CrossRefGoogle Scholar