Duality for $p$-adic geometric pro-\'etale cohomology

TL;DR

This paper establishes a Poincaré duality for $p$-adic geometric pro-étale cohomology of smooth rigid analytic varieties, connecting it to dualities on the Fargues-Fontaine curve and classical Serre duality.

Contribution

It proves a conjectured duality for $p$-adic cohomology, linking it to dualities on the Fargues-Fontaine curve and classical Serre duality, using comparison theorems and sheaf theory.

Findings

Proves $p$-adic geometric pro-étale cohomology satisfies Poincaré duality.

Reduces duality to Serre duality on Stein varieties.

Connects cohomology duality to sheaves on the Fargues-Fontaine curve.

Abstract

We prove that $p$-adic geometric pro-\'etale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincar\'e duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizio{\l}, from a Poincar\'e 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 Poincar\'e 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.