On duality between etale groupoids and Hopf algebroids

摘要

For any etale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra C-c(infinity) (G) of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra C-c(infinity) (M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C-c(infinity) (M) we construct the associated spectral etale Lie groupoid G(sp)(A) over M such C that G(sp)(C-c(infinity)(G)) is naturally isomorphic to G. Both these constructions are functorial, and C-c(infinity) is fully faithful left adjoint to G(sp). C We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid C-c(infinity)(G) of an etale Lie groupoid G. (c) 2006 Elsevier B.V. All rights reserved.