Integral p-adic Hodge theory of formal schemes in low ramification

We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie<p-1$. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze, we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies $ie<p-1$, which generalizes the case of schemes under the condition $(i+1)e<p-1$ proven by Fontaine-Messing and Caruso.