Syntomic complex and p-adic nearbycycles

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.