$(\varphi,\Gamma)$-modules over relatively discrete algebras

In this paper, we study $(\varphi,\Gamma)$-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 $(\varphi,\Gamma)$-modules over perfect period rings, and the dualizability of the cohomology of $(\varphi,\Gamma)$-modules, and the classification of $(\varphi,\Gamma)$-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 $GL_1$ case, as formulated and proved by Rodrigues Jacinto-Rodr\'iguez Camargo.