Crystalline representations and Wach modules in the imperfect residue field case

For an absolutely unramified extension L/Q₏ with imperfect residue field, we define and study Wach modules in the setting of (, ) -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 -module of its associated crystalline representation.