Crystalline part of the Galois cohomology of crystalline representations

Abstract: For $p \geq 3$ 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.