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 $\pi$ 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 $\otimes$, $\underline{Hom}$, $f^{\ast }$, $f_{\ast }$, $f_{!}$ and $f^{!}$ in this setting and show that they satisfy all the expected compatibilities, similar to the $\ell$-adic case. By introducing $\phi$-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.

318 pages. Comments welcome!

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