Syntomic complex and $p$-adic nearby cycles

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.