On Modules Over Valuations

Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

To any smooth manifold X an algebra of smooth valuations V ∞ (X) was associated in [Alesker, Israel J. Math. 156, 311–339 (2006); Adv. Math. 207(1), 420–454 (2006); Theory of Valuations on Manifolds, IV. New Properties of the Multiplicative Structure (2007); Alesker, Fu, Trans. Am. Math. Soc. 360(4), 1951–1981 (2008)]. In this note we initiate a study of V ∞ (X)-modules. More specifically we study finitely generated projective modules in analogy to the study of vector bundles on a manifold. In particular it is shown that for a compact manifold X there exists a canonical isomorphism between the K-ring constructed out of finitely generated projective V ∞ (X)-modules and the classical topological K 0-ring constructed out of vector bundles over X.

Keywords

Vector Bundle Topological Space Direct Summand Euler Characteristic Full Subcategory 
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Notes

Acknowledgements

I thank M. Borovoi for useful discussions on non-abelian cohomology, and F. Schuster for numerous remarks on the first version of the paper. Partially supported by ISF grant 701/08.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Sackler Faculty of Exact Sciences, Department of MathematicsTel Aviv UniversityTel AvivIsrael

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