Crystalline representations and Wach modules in the imperfect residue field case

For an absolutely unramified field extension $L\mathrm{/}{\mathbb{Q}}_p$ with imperfect residue field, we define and study Wach modules in the setting of $(\phi ,\mathrm{\Gamma })$-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 $\phi$-module of its associated crystalline representation.