A $p$-Adic 6-Functor Formalism in Rigid-Analytic Geometry

Abstract

We develop a full 6-functor formalism for $p$-torsion étale sheaves in rigid-analytic geometry. More concretely, we use the recently developed condensed mathematics by Clausen--Scholze to associate to every small v-stack (e.g. rigid-analytic variety) $X$ with pseudouniformizer $π$ an $\infty$-category $\mathcal D^a_\square(\mathcal O^+_X/π)$ of "derived quasicoherent complete topological $\mathcal O^+_X/π$-modules" on $X$. We then construct the six functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ in this setting and show that they satisfy all the expected compatibilities, similar to the $\ell$-adic case. By introducing $φ$-module structures and proving a version of the $p$-torsion Riemann-Hilbert correspondence we relate $\mathcal O^+_X/π$-sheaves to $\mathbb F_p$-sheaves. As a special case of this formalism we prove Poincaré duality for $\mathbb F_p$-cohomology on rigid-analytic varieties. In the process of constructing $\mathcal D^a_\square(\mathcal O^+_X/π)$ we also develop a general descent formalism for condensed modules over condensed rings.