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

TL;DR

This paper extends Differential Galois Theory to differential rings by developing the finite dual of cocommutative Hopf algebroids, enabling a generalized Picard-Vessiot theory for modules over more complex algebraic structures.

Contribution

It introduces the finite dual of cocommutative Hopf algebroids and establishes a monoidal equivalence of categories, broadening the scope of differential Galois Theory beyond fields.

Findings

Developed fundamental results on the finite dual of cocommutative Hopf algebroids.

Established conditions for the injectivity of a canonical ring homomorphism.

Applied the machinery to extend Picard-Vessiot theory to differential modules over rings.

Abstract

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.