Crystalline representations and Wach modules in the imperfect residue field case

For an absolutely unramified extension L/Qₚ with imperfect residue field, we define and study Wach modules in the setting of (φ,Γ) -modules for L . Our main result establishes a direct equivalence between the category of lattices inside crystalline representations of the absolute Galois group of L and the category of integral Wach modules for L . Moreover, we provide a direct relation between a rational Wach module equipped with the Nygaard filtration and the filtered φ -module of its associated crystalline representation.