Abstract：

For an absolutely unramified extension K/Q_p with perfect residue field, by the works of Fontaine, Colmez, Wach, and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of G_K (the absolute Galois group of K). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of G_K are equivalent to the category of prismatic F-crystals on the absolute prismatic site of the ring of integers of K. The goal of this talk is to present a generalisation of these results to a "small" relative base ring and discuss a direct construction of the categorical equivalence between relative Wach modules and prismatic F-crystals over the absolute prismatic site of the base ring. If time permits, we will also mention relationships between relative Wach modules, q-connections and filtered phi-modules with connections.