$(\ddot{I}†,\hat{I}“)$-modules over relatively discrete algebras

Abstract

In this paper, we study $(\ddot{I}†,\hat{I}“)$-modules over rings which are "combinations of discrete algebras and affinoid ${\mathbb{Q}}_p$-algebras", and prove basic results such as the existence of a fully faithful functor from the category of Galois representations, the deperfection of $(\ddot{I}†,\hat{I}“)$-modules over perfect period rings, and the dualizability of the cohomology of $(\ddot{I}†,\hat{I}“)$-modules, and the classification of $(\ddot{I}†,\hat{I}“)$-modules of rank $1$. This work is motivated by the categorical $p$-adic Langlands correspondence for locally analytic representations, as proposed by Emerton-Gee-Hellmann, and the $G{L}_1$ case, as formulated and proved by Rodrigues Jacinto-Rodríguez Camargo.

83 pages. Classification results added