Multivariable $(\ddot{I}{†}_q,{\mathcal{O}}_K^{\times })$-modules associated to $p$-adic representations of $\mathrm{Gal}(\overline{K}/K)$

Abstract

Let $K$ be an unramified extension of ${\mathbb{Q}}_p$, and $E$ a finite extension of $K$ with ring of integers ${\mathcal{O}}_E$. We associate to every finite type continuous ${\mathcal{O}}_E$-representation $\ddot{I}$ of $\mathrm{Gal}(\overline{K}/K)$ an étale $(\ddot{I}{†}_q,{\mathcal{O}}_K^{\times })$-module $D_{A_{\mathrm{mv},E}}^{(0)}(\ddot{I})$ over $A_{\mathrm{mv},E}$, where $A_{\mathrm{mv},E}$ is the $p$-adic completion of a completed localization of the Iwasawa algebra ${\mathcal{O}}_E[[{\mathcal{O}}_K]]$. Furthermore, we prove that the functor $D_{A_{\mathrm{mv},E}}^{(0)}$ is fully faithful and exact. This functor is a $p$-adic analogue of $D_A^{(0)}$ in the recent work of Breuil, Herzig, Hu, Morra and Schraen.

48 pages, some typos are fixed