On a Discrepancy among Picard-vessiot Theories in Positive Characteristics

There is a serious discrepancy among literature on the Picard-Vessiot theory in positive characteristics (for iterative differential fields). We consider descriptions of Galois correspondence in four approaches to this subject: Okugawa’s result [7], Takeuchi’s Hopf algebraic approach [11] (and [3]), the result of Matzat and van der Put [6], and the model theoretic approach by Pillay [8]. In the three approaches except Takeuchi’s one, Galois correspondence is described between closed subgroups of algebraic matrix groups and its fixed fields. But such a description has a problem that the Galois correspondence may not be bijective there. We explain this problem in the first section by giving an explicit example. We should use affine group schemes instead of algebraic matrix groups to obtain a suitable Galois correspondence, as in Takeuchi’s approach. But intermediate fields are not necessarily fixed fields there. In the second section, we give some sufficient conditions for intermediate artinian simple module algebras to be fixed algebras in the context of the unified Picard-Vessiot theory developed by the author and Masuoka [3].