Syntomic complex and p-adic nearby cycles

Abhinandan,

doi:10.2140/ant.2026.20.17

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Syntomic complex and $p$-adic nearby cycles

Abhinandan

Vol. 20 (2026), No. 1, 17–108

DOI: 10.2140/ant.2026.20.17

Abstract

In local relative $p$ -adic Hodge theory, we show that the Galois cohomology of a finite-height crystalline representation (up to a twist) is essentially computed via the (Fontaine–Messing) syntomic complex with coefficients in the associated $F$ -isocrystal. In global applications, for smooth ( $p$ -adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine–Laffaille module and the $p$ -adic nearby cycles of the associated étale local system on the (rigid) generic fibre.

Keywords

$p$-adic Hodge theory, crystalline cohomology, syntomic complex, $(\varphi,\Gamma)$-modules

Mathematical Subject Classification

Primary: 11S25, 14F20, 14F30

Secondary: 14F40

Milestones

Received: 28 January 2023

Revised: 9 October 2024

Accepted: 23 December 2024

Published: 28 November 2025

Authors

Abhinandan
Institut de Mathématiques de Jussieu-Paris Rive Gauche Sorbonne Université Paris France

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Vol. 20, No. 1, 2026