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

Min, Yu

doi:10.2140/ant.2021.15.1043

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

Yu Min

Vol. 15 (2021), No. 4, 1043–1076

DOI: 10.2140/ant.2021.15.1043

Abstract

We prove that for any proper smooth formal scheme $𝔛$ over $𝒪_{K}$ , where $𝒪_{K}$ 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 $i e < p - 1$ . 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 $i e < p - 1$ , which generalizes the case of schemes under the condition $(i + 1) e < p - 1$ proven by Fontaine and Messing (1987) and Caruso (2008).

Keywords

integral $p$-adic Hodge theory, prismatic cohomology

Mathematical Subject Classification

Primary: 14F30

Milestones

Received: 11 April 2020

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

Vol. 15, No. 4, 2021