Duality for $p$-adic geometric pro-étale cohomology
Pierre Colmez, Sally Gilles, Wiesława Nizioł
TL;DR
The paper proves a conjectured Poincaré duality for $p$-adic geometric pro-étale cohomology in the category of Topological Vector Spaces, by geometrizing pro-étale cohomology as solid quasi-coherent sheaves on the Fargues-Fontaine curve and relating it to syntomic, Hyodo-Kato, and filtered de Rham cohomologies. The strategy untangles Frobenius via the FF curve, proves dualities for Hyodo-Kato and de Rham theories, and then descends them to the TVS setting using derived pushforwards and fully-faithfulness results. A key outcome is a canonical duality ${\mathbb R}_{pro\acute{e}t}(X_C,{\bf Q}_p) \simeq {\mathrm R}{\mathcal H om}_{TVS}({\mathbb R}_{pro\acute{e}t,c}(X_C,{\bf Q}_p(d))[2d],{\bf Q}_p)$, together with a Künneth formula and a robust framework to extend dualities to pro-étale and syntomic contexts. The results illuminate how $p$-adic analytic cohomology interacts with topological and condensed structures, providing a foundation for arithmetic dualities and potential applications to $p$-adic Langlands and geometric questions in non-Archimedean geometry.
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.