Syntomic complex and $p$-adic nearby cycles

TL;DR

This paper connects the Galois cohomology of crystalline representations with syntomic complexes in local $p$-adic Hodge theory and compares syntomic complexes with $p$-adic nearby cycles in global geometric contexts.

Contribution

It establishes a new link between Galois cohomology and syntomic complexes, and compares syntomic complexes with $p$-adic nearby cycles for smooth schemes.

Findings

Galois cohomology of crystalline representations is computed via syntomic complexes.

Comparison between syntomic complexes and $p$-adic nearby cycles is established for smooth schemes.

Provides a bridge between local $p$-adic Hodge theory and global geometric applications.

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 \'etale local system on the (rigid) generic fibre.