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

Abstract

We prove that for any proper smooth formal scheme $\mathfrak{𝔛}$ 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 . Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze (2018, 2019), we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies , which generalizes the case of schemes under the condition proven by Fontaine and Messing (1987) and Caruso (2008).