In the proof of $p$-adic crystalline comparison theorem, the approach of Fontaine and Messing utilized syntomic cohomology. The key innovation in their paper was comparison between syntomic cohomology and p-adic étale cohomology via (Fontaine-Messing) period map. This program was successfully completed by Tsuji who gave a complete proof of the comparison theorem. Few years ago, Colmez and Nizioł gave a new interpretation of the (local) Fontaine-Messing period map in terms of complex of $(\varphi, \Gamma)$-modules and used it to prove the semistable comparison theorem for p-adic formal schemes. The goal of this talk is to generalize the local result of Colmez and Nizioł to an interesting class of coefficients, arising from relative Wach modules studied by the speaker.
