On modules over valuations

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. © 2012 Springer-Verlag Berlin Heidelberg.