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.
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Notes
- 1.
All manifolds are assumed to be countable at infinity, i.e. presentable as a union of countably many compact subsets. In particular they are paracompact.
References
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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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Alesker, S. (2012). On Modules Over Valuations. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_2
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DOI: https://doi.org/10.1007/978-3-642-29849-3_2
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