On modules over valuations

TL;DR

This paper explores modules over smooth valuations on manifolds, establishing a connection between finitely generated projective modules over valuations and classical vector bundles, and showing an isomorphism between their K-rings on compact manifolds.

Contribution

It introduces the study of modules over smooth valuations and proves a canonical isomorphism between the valuation-based K-ring and the classical topological K^0-ring on compact manifolds.

Findings

Existence of a canonical isomorphism between valuation-based and classical K-rings on compact manifolds.

Development of a theory of finitely generated projective modules over smooth valuations.

Analogy between modules over valuations and vector bundles on manifolds.

Abstract

Recently an algebra of smooth valuations was attached to any smooth manifold. Roughly put, a smooth valuation is finitely additive measure on compact submanifolds with corners which satisfies some extra properties. In this note we initiate a study of modules over smooth valuations. 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 on a compact manifold there exists a canonical isomorphism between the $K$-ring constructed out of finitely generated projective modules over valuations and the classical topological $K^0$-ring constructed out of vector bundles.