Mod $p$ Poincar\'e duality for $p$-adic period domains

TL;DR

This paper introduces a new class of p-adic rigid analytic varieties satisfying Poincaré duality with mod p coefficients, including p-adic period domains, and computes their étale cohomology, extending previous results.

Contribution

It defines primitive comparison varieties satisfying Poincaré duality, encompassing almost proper and p-adic period domains, and computes their étale cohomology with mod p coefficients.

Findings

Established Poincaré duality for new class of varieties

Computed étale cohomology of p-adic period domains

Extended previous duality and cohomology results to broader classes

Abstract

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincar\'e duality for \'etale cohomology with mod $p$-coefficients : the varieties satisfying "primitive comparison with compact support". We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincar\'e duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the \'etale cohomology with $\mathbb{F}_p$-coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld's symmetric spaces. The arguments used in this paper rely crucially on Mann's six functors formalism for solid $\mathcal{O}^{+,a}/\pi$ coefficients.