$(\varphi,Γ)$-modules over relatively discrete algebras
Yutaro Mikami
TL;DR
The paper develops a robust framework for $(\varphi,\Gamma)$-modules with coefficients in relatively discrete algebras, extending classical $p$-adic Hodge–theoretic structures to condensed and non-Fredholm settings. It proves a fully faithful functor from $G_K$-representations to $(\varphi,G_K)$-modules over a deperfected (and relatively discrete) period ring, and establishes canonical equivalences among categories of modules over perfect and imperfect period rings, including a deperfection procedure. It further constructs and analyzes cohomology theories for these modules, proving dualizability of $(\varphi,\Gamma_K)$-cohomology and providing a concrete presentation analogous to Bel24; a moduli interpretation for rank-1 objects is proposed, with caveats in the non-Fredholm case. Collectively, the results advance a condensed-mathematics approach to the categorical $p$-adic Langlands program, generalizing key correspondences and dualities beyond affinoid coefficients and offering tools for future moduli-theoretic and cohomological investigations. These developments have potential impact on understanding $p$-adic representations and their geometric realizations in a broader non-archimedean analytic setting.
Abstract
In this paper, we will consider $(\varphi,Γ)$-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 $(\varphi,Γ)$-modules over perfect period rings, and the dualizability of the cohomology of $(\varphi,Γ)$-modules. 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-Rodríguez Camargo.