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

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