Relative Poincar\'e duality in nonarchimedean geometry

TL;DR

This paper proves a conjecture on how derived pushforwards and Verdier duality interact in nonarchimedean geometry, leading to new duality results and simplified proofs of p-adic Poincaré duality.

Contribution

It establishes the compatibility of derived pushforwards with Verdier duality in rigid-analytic spaces, advancing the understanding of duality in nonarchimedean geometry.

Findings

Proved conjecture of Bhatt-Hansen on pushforward and duality

Derived pushforwards commute with Verdier duality for Zariski-constructible complexes

Simplified proofs of p-adic Poincaré duality and preservation of Fp-local systems

Abstract

We prove a conjecture of Bhatt-Hansen that derived pushforwards along proper morphisms of rigid-analytic spaces commute with Verdier duality on Zariski-constructible complexes. In particular, this yields duality statements for the intersection cohomology of proper rigid-analytic spaces. In our argument, we construct cycle classes in analytic geometry as well as trace maps for morphisms that are either smooth or proper or finite flat, with appropriate coefficients. As an application of our methods, we obtain new, significantly simplified proofs of $p$-adic Poincar\'e duality and the preservation of $\mathbf{F}_p$-local systems under smooth proper higher direct images.