Geodesic
Representations of étale Lie groupoids and modules over Hopf algebroids
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 3, pp. 653-672.

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The classical Serre-Swan's theorem defines 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. 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 Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.
DOI : 10.1007/s10587-011-0037-7
Classification : 16D40, 16D90, 19L47, 22A22, 58H05
Keywords: étale Lie groupoids; Hopf algebroids; representations; modules; equivalence; Morita category 
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Kališnik, Jure. Representations of étale Lie groupoids and modules over Hopf algebroids. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 3, pp. 653-672. doi : 10.1007/s10587-011-0037-7. http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0037-7/

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