Lattices in crystalline representations and Kisin modules associated with iterate extensions

Cais and Liu extended the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. Based on their theory, we classify lattices in crystalline representations by Kisin modules with additional structures under a Cais-Liu's setting. Furthermore, we give a geometric interpretation of Kisin modules of height one in terms of Dieudonn\'e crystals of $p$-divisible groups, and show a full faithfulness theorem for a restriction functor on torsion crystalline representations.