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arXiv · 2411.12163

Duality for $p$-adic geometric pro-étale cohomology

Pierre Colmez·Sally Gilles·Wiesława Nizioł
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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 $\mathbf{B}^+_{\rm dr}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result.

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Pierre Colmez, Sally Gilles, Wiesława Nizioł. 2025-10-06. Duality for $p$-adic geometric pro-étale cohomology. https://arxiv.org/abs/2411.12163

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