  A fundamental tool of Differential Galois Theory is the assignment of an
algebraic group to each finite-dimensional differential module over
differential field in such a way that the category of differential modules it
generates is equivalent, as a symmetric monoidal category, to the category of
representations of the group. Its underlying set is then recognized as the
group of differential automorphisms of the Picard-Vessiot field extension of
the base field for this differential module. These results can be obtained by
means of a Tannaka reconstruction process, applied to the abelian category of
finite-dimensional differential modules. In this paper, we explore the
possibility of extending this theory when the differential field is replaced by
a more general differential ring $A$. In this case, it is reasonable to deal
with differential modules which are finitely generated and projective over $A$.
A major obstacle is that this category is not abelian, in contrast with the
classical case when $A$ is a field. To overcome this difficulty, we develop
some fundamental results concerning the \emph{finite dual}, hereby introduced,
of a cocommutative Hopf algebroid, and a canonical monoidal functor sending
(differential) modules to comodules. This functor is proved to be an
equivalence of categories whenever a canonical ring homomorphism, which is
introduced in this work, with domain in the aforementioned finite dual and with
values in the convolutional ring, is injective. Module-theoretical sufficient
conditions to get its injectivity are investigated. This machinery is applied
to differential modules and their Picard-Vessiot theory.
