Representations of étale Lie groupoids and modules over Hopf algebroids

Jure Kališnik

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 653-672
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Kališnik, Jure. "Representations of étale Lie groupoids and modules over Hopf algebroids." Czechoslovak Mathematical Journal 61.3 (2011): 653-672. <http://eudml.org/doc/196325>.

@article{Kališnik2011,
abstract = {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.},
author = {Kališnik, Jure},
journal = {Czechoslovak Mathematical Journal},
keywords = {étale Lie groupoids; Hopf algebroids; representations; modules; equivalence; Morita category; étale Lie groupoid; Hopf algebroids; representation; module; equivalence; Morita category},
language = {eng},
number = {3},
pages = {653-672},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Representations of étale Lie groupoids and modules over Hopf algebroids},
url = {http://eudml.org/doc/196325},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Kališnik, Jure
TI - Representations of étale Lie groupoids and modules over Hopf algebroids
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 653
EP - 672
AB - 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.
LA - eng
KW - étale Lie groupoids; Hopf algebroids; representations; modules; equivalence; Morita category; étale Lie groupoid; Hopf algebroids; representation; module; equivalence; Morita category
UR - http://eudml.org/doc/196325
ER -

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