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

TL;DR

This paper develops a comprehensive 6-functor formalism for p-torsion étale sheaves in rigid-analytic geometry using condensed mathematics, establishing duality results and relating sheaves to F_p-sheaves.

Contribution

It introduces a new 6-functor formalism for p-torsion sheaves in rigid-analytic geometry, utilizing condensed mathematics and establishing a p-torsion Riemann-Hilbert correspondence.

Findings

Constructed the six functors with expected compatibilities.

Proved Poincaré duality for F_p-cohomology on rigid-analytic varieties.

Developed a descent formalism for condensed modules.

Abstract

We develop a full 6-functor formalism for $p$-torsion \'etale 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 $\mathcal D^a_\square(\mathcal O^+_X/\pi)$ 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 Poincar\'e duality for $\mathbb F_p$-cohomology on rigid-analytic varieties. In the process of constructing $\mathcal D^a_\square(\mathcal O^+_X/\pi)$ we also develop a general descent formalism for condensed modules over condensed rings.