A <em>p</em>-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 $pi$ an $infty$-category $mathcal D_square^a(mathcal O^+_X/pi)$ of ``derived quasicoherent complete topological $mathcal O^+_X/pi$-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 $varphi$-module structures and proving a version of the $p$-torsion Riemann-Hilbert correspondence we relate $mathcal O^+_X/pi$-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 $mathcal D_square^a(mathcal O^+_X/pi)$ we also develop a general descent formalism for condensed modules over condensed rings.