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 ${\mathbb{Z}}_p$ 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 $W(k)[\text{\hspace{0.17em}⁣}[t_1,\cdots ,t_d]\text{\hspace{0.17em}⁣}]$ and where the base ring is a $p$-adically complete ring that is étale over the Tate Algebra $W(k)⟨t_1^{\pm 1},\cdots ,t_d^{\pm 1}⟩$.