Fontaine-Laffaille Theory over Power Series Rings

Let k be a perfect field of characteristic $p > 2$. We extend the equivalence of categories between Fontaine-Laffaille modules and Zₚ lattices inside crystalline representations with Hodge-Tate weights at most $p-2$ of Fontaine and Laffaille to the situation where the base ring is the power series ring over the Witt vectors W(k)[\![ t₁, ⋯ , td]\!] and where the base ring is a p-adically complete ring that is \'etale over the Tate Algebra W(k) t₁± 1, ⋯ , td± 1.