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

Abstract

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