  In this paper we establish a duality between etale Lie groupoids and a class
of non-necessarily commutative algebras with a Hopf algebroid structure. For
any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of
smooth functions with compact support on G has a natural coalgebra structure
over C_c(M) which makes it into a Hopf algebroid. Conversely, for any Hopf
algebroid A over C_c(M) we construct the associated spectral etale Lie groupoid
G_sp(A) over M such that G_sp(C_c(G)) is naturally isomorphic to G. Both these
constructions are functorial, and C_c is fully faithful left adjoint to G_sp.
We give explicit conditions under which a Hopf algebroid is isomorphic to the
Hopf algebroid C_c(G) of an etale Lie groupoid G. We also demonstrate that an
analogous duality exists between sheaves on M and a class of coalgebras over
C_c(M).
