Analytic de Rham stacks of Fargues-Fontaine curves
Johannes Anschütz, Guido Bosco, Arthur-César Le Bras, Juan Esteban Rodríguez Camargo, Peter Scholze
TL;DR
The paper develops a comprehensive framework for analytic de Rham stacks in p-adic geometry, extending beyond perfectoid spaces by introducing bounded Q_p–algebras, Gelfand rings, and a perfectoidization construction. It defines the analytic de Rham stack as the right adjoint to perfectoidization and proves strong descent properties, enabling a robust 6-functor formalism and coherent cohomology theories for analytic stacks. The approach yields a geometric construction of Hyodo–Kato objects for relative Fargues–Fontaine curves and provides a path to interpreting Hyodo–Kato cohomology via stack-cohomology in a sequel, with a new p-adic monodromy theorem proven geometrically without differential equations. The framework unifies derived Berkovich spaces, dagger/overconvergent structures, and HK-stacks, and lays the groundwork for cohomology computations and comparisons in p-adic Hodge theory, including Hyodo–Kato cohomology and monodromy phenomena.
Abstract
We define and initiate the study of analytic de Rham stacks of relative Fargues-Fontaine curves. To this end, we develop a theory of analytic de Rham stacks with sufficiently strong descent and approximation properties. Specializing to the de Rham stack of the Fargues-Fontaine curve attached to $\mathbb{C}_p$, we apply the general theory to obtain a new geometric proof of the $p$-adic monodromy theorem, avoiding any reliance on earlier results on $p$-adic differential equations. Building on the foundations established here, we plan in a sequel to investigate the cohomology of de Rham stacks of relative Fargues-Fontaine curves in geometric situations and, in particular, provide a stack-theoretic definition of Hyodo-Kato cohomology.