  Given a finitely generated and projective Lie-Rinehart algebra, we show that
there is a continuous homomorphism of complete commutative Hopf algebroids
between the completion of the finite dual of its universal enveloping Hopf
algebroid and the associated convolution algebra. The topological Hopf
algebroid structure of this convolution algebra is here clarified, by providing
an explicit description of its topological antipode as well as of its other
structure maps. Conditions under which that homomorphism becomes an
homeomorphism are also discussed. These results, in particular, apply to the
smooth global sections of any Lie algebroid over a smooth (connected) manifold
and they lead a new formal groupoid scheme to enter into the picture. In the
Appendix we develop the necessary machinery behind complete Hopf algebroid
constructions, which involves also the topological tensor product of filtered
bimodules over filtered rings.
