#### Vol. 15, No. 4, 2021

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

### Yu Min

Vol. 15 (2021), No. 4, 1043–1076
##### Abstract

We prove that for any proper smooth formal scheme $\mathfrak{𝔛}$ over ${\mathsc{𝒪}}_{K}$, where ${\mathsc{𝒪}}_{K}$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of ${ℚ}_{p}$ 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. 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 $ie, which generalizes the case of schemes under the condition $\left(i+1\right)e proven by Fontaine and Messing (1987) and Caruso (2008).

##### Keywords
integral $p$-adic Hodge theory, prismatic cohomology
Primary: 14F30
##### Milestones
Revised: 16 September 2020
Accepted: 17 October 2020
Published: 29 May 2021
##### Authors
 Yu Min Institut de Mathématiques de Jussieu-Paris Rive Gauche Campus Pierre et Marie Curie (Jussieu) Paris France