2022/06/04 by Lucas Mann, Mann, Lucas · 7 citations
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2206.02022
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 ∞-category \mathcal Da_\square(\mathcal O+X/π) of "derived quasicoherent complete topological \mathcal O+X/π-modules" on X. We then construct the six functors ⊗, \underlineHom, f^*, f_*, f_! and f^! in this setting and show that they satisfy all the expected compatibilities, similar to the ℓ-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 Fp-sheaves. As a special case of this formalism we prove Poincaré duality for \mathbb Fp-cohomology on rigid-analytic varieties. In the process of constructing \mathcal Da_\square(\mathcal O+X/π) we also develop a general descent formalism for condensed modules over condensed rings.