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

By L. Mann. 2022

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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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/89d4406bbd530d16ac5514fc1862766e7bbfa589