The Emerton--Gee stack of rank one $(\varphi,\Gamma)$-modules

We give a classification of rank one $(\varphi,\Gamma)$-modules with coefficients in a $p$-adically complete $\mathbf{Z}_p$-algebra. As a consequence, we obtain a new proof of Proposition 7.2.17 in {arXiv:1908.07185}, which gives an explicit description of the Emerton--Gee stack of $(\varphi,\Gamma)$-modules in the rank one case. In fact, our method also applies in the context of rank one \' etale $\varphi$-modules (i.e. in the absence of a $\Gamma$-action), generalizing another result of Emerton--Gee.