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

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 ${\mathbf{B}}_{\mathrm{st}}^{+}$-twisted Hyodo-Kato and filtered ${\mathbf{B}}_{\mathrm{dr}}^{+}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result.

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