Crystalline representations and Wach modules in the relative case II

TL;DR

This paper extends the theory of Wach modules to the relative case, establishing categorical equivalences, purity results, and criteria for crystallinity in relative crystalline representations, with interpretations as modules with $q$-connections.

Contribution

It generalizes Wach modules to the relative setting, proving categorical equivalence with lattices in relative crystalline representations and connecting them to $q$-connections.

Findings

Categorical equivalence between relative Wach modules and lattices in relative crystalline representations.

A purity statement for relative crystalline representations.

A criterion for checking the crystallinity of relative $p$-adic representations.

Abstract

We study relative Wach modules generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking crystallinity of relative $p$-adic representations. Furthermore, we interpret relative Wach modules as modules with $q$-connections, and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonical $q$-deformation (after inverting $p$) of the filtered $(\varphi,\partial)$-module associated to the representation.