On spectral representation of coalgebras and Hopf algebroids

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).