Crystalline part of the Galois cohomology of crystalline representations

TL;DR

This paper constructs syntomic complexes using Wach modules to compute the crystalline part of Galois cohomology for certain p-adic representations, advancing the understanding of their cohomological properties.

Contribution

It introduces new syntomic complexes based on Wach modules that accurately compute the crystalline Galois cohomology for unramified extensions of erent from previous methods.

Findings

The syntomic complex computes the crystalline Galois cohomology.

Wach modules descend to a smaller period ring, enabling new constructions.

The new complexes match the crystalline part of Galois cohomology.

Abstract

For $p \geqslant 3$ and an unramified extension $F/\mathbb{Q}_p$ with perfect residue field, we define a syntomic complex with coefficients in a Wach module over a certain period ring for $F$. We show that our complex computes the crystalline part of the Galois cohomology (in the sense of Bloch and Kato) of the associated crystalline representation of the absolute Galois group of $F$. Furthermore, we establish that Wach modules of Berger naturally descend over to a smaller period ring studied by Fontaine and Wach. This enables us to define another syntomic complex with coefficients, and we show that its cohomology also computes the crystalline part of the Galois cohomology of the associated representation.