  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.
