Galois theory of Artinian simple module algebras

Abstract: This main purpose of this article is the unification of the Galois theory of algebraic differential equations by Umemura and the Galois theory of algebraic difference equations by Morikawa-Umemura in a common framework using Artinian simple D-module algebras, where D is a bialgebra. We construct the Galois hull of an extension of Artinian simple D-module algebras and define its Galois group, which consists of infinitesimal coordinate transformations fulfilling certain partial differential equations and which we call Umemura functor. We eliminate the restriction to characteristic 0 from the above mentioned theories and remove the limitation to field extensions in the theory of Morikawa-Umemura, allowing also direct products of fields, which is essential in the theory of difference equations. In order to compare our theory with the Picard-Vessiot theory of Artinian simple D-module algebras due to Amano and Masuoka, we first slightly generalize the definition and some results about them in order to encompass as well non-inversive difference rings. Finally, we give equivalent characterizations for smooth Picard-Vessiot extensions, describe their Galois hull and show that their Umemura functor becomes isomorphic to the formal scheme associated to the classical Galois group scheme after a finite \'etale base extension.