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A fundamental tool of Differential Galois Theory is the assignment of analgebraic group to each finite-dimensional differential module overdifferential field in such a way that the category of differential modules itgenerates is equivalent, as a symmetric monoidal category, to the category ofrepresentations of the group. Its underlying set is then recognized as thegroup of differential automorphisms of the Picard-Vessiot field extension ofthe base field for this differential module. These results can be obtained bymeans of a Tannaka reconstruction process, applied to the abelian category offinite-dimensional differential modules. In this paper, we explore thepossibility of extending this theory when the differential field is replaced bya more general differential ring $A$. In this case, it is reasonable to dealwith differential modules which are finitely generated and projective over $A$.A major obstacle is that this category is not abelian, in contrast with theclassical case when $A$ is a field. To overcome this difficulty, we developsome 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 anequivalence of categories whenever a canonical ring homomorphism, which isintroduced in this work, with domain in the aforementioned finite dual and withvalues in the convolutional ring, is injective. Module-theoretical sufficientconditions to get its injectivity are investigated. This machinery is appliedto differential modules and their Picard-Vessiot theory.

Finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory