On duality between étale groupoids and Hopf algebroids (Q878676)

To any étale Lie groupoid \(G\) over a smooth manifold \(M\), we can associate a Hopf algebroid by taking the groupoid convolution algebra \({\mathcal C}_C^\infty(G)\) of smooth functions with compact support on \(G\) with its natural coalgebra structure. In this paper the author provides explicit conditions for identifying when a Hopf algebroid arises from a Lie groupoid in this way. In addition, the author gives a complementary construction for a Hopf algebroid \(A\) over \({\mathcal C}_C^\infty(M)\): the `associated spectral étale Lie groupoid' \({\mathcal G}_{sp}(A)\) has the property that setting \(A = {\mathcal C}_C^\infty(G)\) yields the original Lie groupoid \(G\) in a functorial way, and \({\mathcal G}_{sp}\) is a right adjoint to the functor \({\mathcal C}_C^\infty\).