The Emerton--Gee stack of rank one $(\ddot{I}†,\hat{I}“)$-modules

Abstract

We give a classification of rank one $(\ddot{I}†,\hat{I}“)$-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 $(\ddot{I}†,\hat{I}“)$-modules in the rank one case. In fact, our method also applies in the context of rank one \' etale $\ddot{I}†$-modules (i.e. in the absence of a $\hat{I}“$-action), generalizing another result of Emerton--Gee.

Final version