Duality for p-adic geometric pro-étale cohomology

2024/11/19 by Pierre Colmez, Colmez, Pierre, Sally Gilles +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.12163

openalex publication_date 2024/11/19 · openalex created_date 2024/11/21 · openalex updated_date 2026/08/01

Abstract

We prove that p-adic geometric pro-étale cohomology of smooth partially proper rigid analytic varieties over p-adic fields seen in the category of Topological Vector Spaces satisfies a Poincaré duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizioł, from a Poincaré duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincaré duality for \mathbf B⁺_{\rm st}-twisted Hyodo-Kato and filtered B⁺_{\rm dr}-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result.

Related

• Duality for Arithmetic p-adic Pro-étale Cohomology of Analytic Spaces
• Compactly supported p-adic pro-étale cohomology of analytic varieties
• On the p-adic pro-étale cohomology of Drinfeld symmetric spaces
• Topological Vector Spaces
• Arithmetic duality for p-adic pro-étale cohomology of analytic curves