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 𝔛 over 𝒪K, where 𝒪K is the ring of integers in a complete discretely valued nonarchimedean extension K of 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 (2018, 2019), 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 and Messing (1987) and Caruso (2008).

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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