  As left adjoint to the dual algebra functor, Sweedler's finite dual
construction is an important tool in the theory of Hopf algebras over a field.
We show in this note that the left adjoint to the dual algebra functor, which
exists over arbitrary rings, shares a number of properties with the finite
dual. Nonetheless the requirement that it should map Hopf algebras to Hopf
algebras needs the extra assumption that this left adjoint should map an
algebra into its linear dual. We identify a condition guaranteeing that
Sweedler's construction works when generalized to noetherian commutative rings.
We establish the following two apparently previously unnoticed dual
adjunctions: For every commutative ring $R$ the left adjoint of the dual
algebra functor on the category of $R$-bialgebras has a right adjoint. This
dual adjunction can be restricted to a dual adjunction on the category of Hopf
$R$-algebras, provided that $R$ is noetherian and absolutely flat.
