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

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.