Groupoid Representations and Modules over the Convolution Algebras

The classical Serre-Swan’s theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of étale Lie groupoids and the given correspondence represents a natural equivalence between them.