The Emerton–Gee stack of rank one (φ,Γ)-modules

We give a classification of rank one (φ,Γ) -modules with coefficients in a p -adically complete Zₚ -algebra. As a consequence, we obtain a new proof of the explicit description of the Emerton–Gee stack of (φ,Γ) -modules in the rank one case. In fact, our method also applies in the context of rank one étale φ -modules (i.e. in the absence of a Γ -action), generalizing another result of Emerton–Gee.