Crystalline representations and Wach modules in the relative case

We study the notion of Wach modules in the relative setting, generalizing the arithmetic case. Over an unramified base, for a p-adic representation admitting such structure, we examine the relationship between its relative Wach module and filtered (φ,∂)-module. Moreover, we show that such a representation is crystalline (in the sense of Faltings-Brinon), and one can recover its filtered (φ,∂)-module from the relative Wach module. Conversely, for low Hodge-Tate weights [0,p-2], we construct relative Wach modules from free relative Fontaine-Laffaille modules (in the sense of Faltings).