Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over OK and k[[u]], respectively. In this paper, we show the ramification subgroups of G and H in the sense of Abbes–Saito are naturally isomorphic to each other when they are associated to the same Kisin module.