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Representations of étale Lie groupoids and modules over Hopf algebroids. (English) Zbl 1249.22003

An equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space is described in classical Serre-Swan’s theorem. In the presented paper these results were extended for obtaining a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of the finite type and of the constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and it is shown that the given correspondence represents a natural equivalence between them.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
16T05 Hopf algebras and their applications
16D40 Free, projective, and flat modules and ideals in associative algebras
16D90 Module categories in associative algebras
19L47 Equivariant \(K\)-theory
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