On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid GG over a smooth manifold MM, the groupoid convolution algebra Cc∞(G) of smooth functions with compact support on GG has a natural coalgebra structure over the commutative algebra Cc∞(M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid AA over Cc∞(M) we construct the associated spectral étale Lie groupoid Gsp(A) over MM such that Gsp(Cc∞(G)) is naturally isomorphic to GG. Both these constructions are functorial, and Cc∞ is fully faithful left adjoint to Gsp. We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid Cc∞(G) of an étale Lie groupoid GG.