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). It is about descriptions of Galois correspondence. We should use affine group schemes instead of algebraic matrix groups to obtain a suitable Galois correspondence. If one adheres to algebraic matrix groups, then his correspondence may not be bijective. 
Katsutoshi Amano