Crystalline representations and Wach modules in the relative case

TL;DR

This paper extends the theory of Wach modules to a relative setting over unramified bases, establishing their connection with crystalline representations and filtered modules, and constructing Wach modules from Fontaine-Laffaille modules for certain weights.

Contribution

It generalizes Wach modules to the relative case, relates them to filtered $(, abla)$-modules, and constructs Wach modules from Fontaine-Laffaille modules for low Hodge-Tate weights.

Findings

Wach modules in the relative setting are related to crystalline representations.

Crystalline representations can be recovered from their relative Wach modules.

Construction of Wach modules from Fontaine-Laffaille modules for weights [0, p-2].

Abstract

We study the notion of Wach modules in 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 $(\varphi, \partial)$-module. Moreover, we show that such a representation is crystalline (in the sense of Brinon), and one can recover its filtered $(\varphi, \partial)$-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).